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\newlabel{(1.48)}{{2.3.18}{24}{Bounds and Covnvergence}{equation.2.3.18}{}}
|
||||
\newlabel{(1.48)@cref}{{[equation][18][2,3]2.3.18}{[1][24][]24}}
|
||||
\newlabel{cor:1.25.1}{{2.3.3.1}{24}{}{corollary.2.3.3.1}{}}
|
||||
\newlabel{cor:1.25.1@cref}{{[corollary][1][]2.3.3.1}{[1][24][]24}}
|
||||
\newlabel{(2.48)}{{2.3.26}{26}{}{equation.2.3.26}{}}
|
||||
\newlabel{(2.48)@cref}{{[equation][26][2,3]2.3.26}{[1][26][]26}}
|
||||
\newlabel{2.3.27}{{2.3.27}{26}{}{equation.2.3.27}{}}
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||||
\newlabel{2.3.27@cref}{{[equation][27][2,3]2.3.27}{[1][26][]26}}
|
||||
\newlabel{2.3.29}{{2.3.28}{26}{Bounds and Covnvergence}{equation.2.3.28}{}}
|
||||
\newlabel{2.3.29@cref}{{[equation][28][2,3]2.3.28}{[1][26][]26}}
|
||||
\citation{hjw2020}
|
||||
\citation{hjw2020}
|
||||
\citation{hjw2020}
|
||||
\newlabel{2.3.29}{{2.28}{32}{Bounds and Covnvergence}{equation.2.3.28}{}}
|
||||
\newlabel{2.3.29@cref}{{[equation][28][2,3]2.28}{[1][31][]32}}
|
||||
\newlabel{(2.3.33)}{{2.33}{34}{Bounds and Covnvergence}{equation.2.3.33}{}}
|
||||
\newlabel{(2.3.33)@cref}{{[equation][33][2,3]2.33}{[1][33][]34}}
|
||||
\newlabel{2.3.34}{{2.34}{34}{Bounds and Covnvergence}{equation.2.3.34}{}}
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||||
\newlabel{2.3.34@cref}{{[equation][34][2,3]2.34}{[1][34][]34}}
|
||||
\newlabel{(2.3.35)}{{2.35}{34}{Bounds and Covnvergence}{equation.2.3.35}{}}
|
||||
\newlabel{(2.3.35)@cref}{{[equation][35][2,3]2.35}{[1][34][]34}}
|
||||
\newlabel{2.3.34}{{2.3.34}{28}{Bounds and Covnvergence}{equation.2.3.34}{}}
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||||
\newlabel{2.3.34@cref}{{[equation][34][2,3]2.3.34}{[1][28][]28}}
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@ -1,4 +1,4 @@
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\chapter{Brownian Motion Monte Carlo}\label{chp:2}
|
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\chapter{Brownian Motion Monte Carlo}
|
||||
|
||||
\section{Brownian Motion Preliminaries}
|
||||
We will present here some standard invariants of Brownian motions. The proofs are standard and can be found in for instance \cite{durrett2019probability} and \cite{karatzas1991brownian}.
|
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|
|
@ -88,11 +88,11 @@ We will present here some standard invariants of Brownian motions. The proofs ar
|
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|
||||
|
||||
|
||||
\begin{definition}[Of $\mathfrak{k}$, the modified Kahane\textendash Kintchine constant]\label{def:1.17}
|
||||
\begin{definition}[Of $\mathfrak{k}$]\label{def:1.17}
|
||||
Let $p \in [2,\infty)$. We denote by $\mathfrak{k}_p \in \R$ the real number given by $\mathfrak{k}:=\inf \{ c\in \R \}$ where it holds that for every probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and every random variable $\mathcal{X}: \Omega \rightarrow \R$ with $\E[|\mathcal{X}|] < \infty$ that $\lp \E \lb \lv \mathcal{X} - \E \lb \mathcal{X} \rb \rp^p \rb \rp ^{\frac{1}{p}} \leqslant c \lp \E \lb \lv \mathcal{X} \rv^p \rb \rp ^{\frac{1}{p}}.$
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Primary Setting For This Chapter]\label{primarysetting} Let $d,m \in \mathbb{N}$, $T, \mathfrak{L},p \in [0,\infty)$, $\mathfrak{p} \in [2,\infty)$ $\mathfrak{m} = \mathfrak{k}_{\mathfrak{p}}\sqrt{\mathfrak{p}-1}$, $\Theta = \mathbb{Z}$, $g \in C(\mathbb{R}^d,\mathbb{R})$, assume for all $t \in [0,T],x\in \mathbb{R}^d$ that:
|
||||
\begin{definition}[Primary Setting]\label{primarysetting} Let $d,m \in \mathbb{N}$, $T, \mathfrak{L},p \in [0,\infty)$, $\mathfrak{p} \in [2,\infty)$ $\mathfrak{m} = \mathfrak{k}_{\mathfrak{p}}\sqrt{\mathfrak{p}-1}$, $\Theta = \mathbb{Z}$, $g \in C(\mathbb{R}^d,\mathbb{R})$, assume for all $t \in [0,T],x\in \mathbb{R}^d$ that:
|
||||
\begin{align}\label{(2.1.2)}
|
||||
\max\{|g(x)|\} \leqslant \mathfrak{L} \lp 1+\|x\|_E^p \rp
|
||||
\end{align}
|
||||
|
|
@ -103,7 +103,7 @@ and let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space. Let $\mathcal
|
|||
\end{align}
|
||||
and let let $U^\theta:[0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R}$, $\theta \in \Theta$ satisfy, $\theta \in \Theta$, $t \in [0,T]$, $x\in \mathbb{R}^d$, that:
|
||||
\begin{align}\label{(2.1.4)}
|
||||
U^\theta_m(t,x) = \frac{1}{m}\left[\sum^{m}_{k=1}g\left(x+\mathcal{W}^{\theta}_{T-t}\right)\right]
|
||||
U^\theta_m(t,x) = \frac{1}{m}\left[\sum^{m}_{k=1}g\left(x+\mathcal{W}^{(\theta,0,-k)}_{T-t}\right)\right]
|
||||
\end{align}
|
||||
\end{definition}
|
||||
\begin{lemma} \label{lemma1.1}
|
||||
|
|
@ -118,17 +118,17 @@ Assume Setting \ref{primarysetting} then:
|
|||
\end{enumerate}
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof} For (i) Consider that $\mathcal{W}^{\theta}_{T-t}$ are continuous random fields and that $g\in C(\mathbb{R}^d,\mathbb{R})$, we have that $U^\theta(t,x)$ is the composition of continuous functions with $m > 0$ by hypothesis, ensuring no singularities. Thus $U^\theta: [0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$ is a continuous random field.
|
||||
\begin{proof} For (i) Consider that $\mathcal{W}^{(\theta,0,-k)}_{T-t}$ are continuous random fields and that $g\in C(\mathbb{R}^d,\mathbb{R})$, we have that $U^\theta(t,x)$ is the composition of continuous functions with $m > 0$ by hypothesis, ensuring no singularities. Thus $U^\theta: [0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$.
|
||||
|
||||
\medskip
|
||||
|
||||
For (ii) observe that for all $\theta \in \Theta$ it holds that $\mathcal{W}^\theta$ is $\mathcal{B} \lp \lb 0, T \rb \otimes \sigma \lp W^\theta \rp \rp /\mathcal{B}\lp \mathbb{R}^d \rp$-measurable, this, and induction on prove item (ii).
|
||||
|
||||
\medskip
|
||||
Moreover observe that item (ii) and the fact that for all $\theta \in \Theta$ it holds that $\lp\mathcal{W}^{\lp \theta, \vartheta\rp}_{\vartheta \in \Theta}\rp$, $\mathcal{W}^\theta$ are independent establish item (iii).
|
||||
Moreover observe that item (ii) and the fact that for all $\theta \in \Theta$ it holds that $\lp\mathcal{W}^{\lp \theta, \vartheta\rp}_{\vartheta \in \Theta}\rp$, $\mathcal{W}^\theta$ are independently establish item (iii).
|
||||
|
||||
\medskip
|
||||
Furthermore, note that (ii) and the fact that for all $i,k,\mathfrak{i},\mathfrak{k} \in \mathbb{Z}$, $\theta \in \Theta$, with $(i,k) \neq (\mathfrak{i},\mathfrak{k})$ it holds that $\lp\mathcal{W}^{\lp\theta, i,k,\vartheta\rp}\rp_{\vartheta \in \Theta}$ and $\lp\mathcal{W}^{\lp\theta,\mathfrak{i},\mathfrak{k},\vartheta\rp}\rp_{\vartheta \in \Theta}$ are independent, establish item (iv).
|
||||
Furthermore, note that (ii) and the fact that for all $i,k,\mathfrak{i},\mathfrak{k} \in \mathbb{Z}$, $\theta \in \Theta$, with $(i,k) \neq (\mathfrak{i},\mathfrak{k})$ it holds that $\lp\mathcal{W}^{\lp\theta, i,k,\vartheta\rp}\rp_{\vartheta \in \Theta}$ and $\lp\mathcal{W}^{\lp\theta,\mathfrak{i},\mathfrak{k},\vartheta\rp}\rp_{\vartheta \in \Theta}$ are independent establish item (iv).
|
||||
|
||||
\medskip
|
||||
Hutzenhaler \cite[Corollary~2.5 ]{hutzenthaler_overcoming_2020} establish item (v). This completes the proof of Lemma 1.1.
|
||||
|
|
@ -154,7 +154,7 @@ We next claim that for all $s\in [0,T]$, $t\in[s,T]$, $\theta \in \Theta$ it hol
|
|||
|
||||
To prove this claim observe the triangle inequality and (\ref{(2.1.4)}), demonstrate that for all $s\in[0,T]$, $t\in[s,T]$, $\theta \in \Theta$, it holds that:
|
||||
\begin{align}\label{(1.18)}
|
||||
\mathbb{E}\lb \lv U^\theta \lp t,x+\mathcal{W}^\theta_{t-s}\rp \rv \rb \leqslant \frac{1}{m}\left[ \sum^{m}_{i=1}\mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{t-s}+\mathcal{W}^{\theta}_{T-t} \rp \rv \rb \rb
|
||||
\mathbb{E}\lb \lv U^\theta \lp t,x+\mathcal{W}^\theta_{t-s}\rp \rv \rb \leqslant \frac{1}{m}\left[ \sum^{m}_{i=1}\mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{t-s}+\mathcal{W}^{(\theta,0,-i)}_{T-t} \rp \rv \rb \rb
|
||||
\end{align}
|
||||
|
||||
Now observe that (\ref{(2.1.6)}) and the fact that $(W^\theta)_{\theta \in \Theta}$ are independent imply that for all $s \in [0,T]$, $t\in [s,T]$, $\theta \in \Theta$, $i\in \mathbb{Z}$ it holds that:
|
||||
|
|
@ -179,11 +179,11 @@ Combining (\ref{(1.16)}), (\ref{(1.20)}), and (\ref{(1.21)}) completes the proof
|
|||
\begin{enumerate}[label = (\roman*)]
|
||||
\item it holds that $t \in [0,T],x\in \mathbb{R}^d$ that:
|
||||
\begin{align}
|
||||
\mathbb{E}\lb \lv U^0 \lp t,x \rp \rv \rb + \mathbb{E}\lb \lv g \lp x+\mathcal{W}^{0}_{T-t} \rp \rv \rb < \infty
|
||||
\mathbb{E}\lb \lv U^0 \lp t,x \rp \rv \rb + \mathbb{E}\lb \lv g \lp x+\mathcal{W}^{(0,0,-1)}_{T-t} \rp \rv \rb < \infty
|
||||
\end{align}
|
||||
\item it holds that $t\in [0,T],x\in \mathbb{R}^d$ that:
|
||||
\begin{align}
|
||||
\mathbb{E}\lb U^0\lp t,x \rp \rb = \mathbb{E} \lb g \lp x+\mathcal{W}^{0}_{T-t}\rp\rb
|
||||
\mathbb{E}\lb U^0\lp t,x \rp \rb = \mathbb{E} \lb g \lp x+\mathcal{W}^{(0,0,-1)}_{T-t}\rp\rb
|
||||
\end{align}
|
||||
\end{enumerate}
|
||||
\end{corollary}
|
||||
|
|
@ -191,18 +191,18 @@ Combining (\ref{(1.16)}), (\ref{(1.20)}), and (\ref{(1.21)}) completes the proof
|
|||
\begin{proof}
|
||||
(i) is a restatement of Lemma \ref{lem:1.20} in that for all $t\in [0,T]$:
|
||||
\begin{align}
|
||||
&\mathbb{E}\left[ \left| U^0\left( t,x \right) \right| \right] + \mathbb{E} \left[ \left|g \left(x+\mathcal{W}^{0}_{T-t}\right)\right|\right] \nonumber\\
|
||||
&\mathbb{E}\left[ \left| U^0\left( t,x \right) \right| \right] + \mathbb{E} \left[ \left|g \left(x+\mathcal{W}^{(0,0,-1)}_{T-t}\right)\right|\right] \nonumber\\
|
||||
&<\mathbb{E} \left[ \left|U^\theta \lp t,x+\mathcal{W}^\theta_{t-s} \rp \right| \right] +\mathbb{E}\left[ \left|g \left(x+\mathcal{W}^\theta_{t-s}\right) \right| \right]+ \int^T_s \mathbb{E}\lb \lv U^\theta \lp r,x+\mathcal{W}^\theta_{r-s} \rp \rv \rb dr \nonumber\\
|
||||
&< \infty
|
||||
\end{align}
|
||||
|
||||
Furthermore (ii) is a restatement of Lemma \ref{lem:1.20} with $\theta = 0$, $m=1$, and $k=1$. This completes the proof of Corollary \ref{cor:1.20.1}.
|
||||
Furthermore (ii) is a restatement of (\ref{(1.14)}) with $\theta = 0$, $m=1$, and $k=1$. This completes the proof of Corollary \ref{cor:1.20.1}.
|
||||
\end{proof}
|
||||
|
||||
\section{Monte Carlo Approximations}
|
||||
|
||||
|
||||
\begin{lemma}\label{lem:1.21}Let $p \in (2,\infty)$, $n\in \mathbb{N}$, let $(\Omega, \mathcal{F}, \mathbb{P})$, be a probability space and let $\mathcal{X}_i: \Omega \rightarrow \mathbb{R}$, $i \in \{1,2,...,n\}$ be i.i.d. random variables with $\mathbb{E}[|\mathcal{X}_1|]<\infty$. Then it holds that:
|
||||
\begin{lemma}\label{lem:1.21}Let $p \in (2,\infty)$,$n\in \mathbb{N}$, let $(\Omega, \mathcal{F}, \mathbb{P})$, be a probability space and let $\mathcal{X}_i: \Omega \rightarrow \mathbb{R}$, $i \in \{1,2,...,n\}$ be i.i.d. random variables with $\mathbb{E}[|\mathcal{X}_1|]<\infty$. Then it holds that:
|
||||
\begin{align}
|
||||
\lp\E \lb \lv \E \lb \mathcal{X}_1 \rb-\frac{1}{n} \lp \sum^n_{i=1} \mathcal{X}_i \rp \rv^p \rb \rp^{\frac{1}{p}} \leqslant \lb \frac{p-1}{n}\rb ^{\frac{1}{2}}\left(\E\lb \lv \mathcal{X}_1-\E \lb \mathcal{X}_1 \rb \rv^p \rp \rb^{\frac{1}{p}}
|
||||
\end{align}
|
||||
|
|
@ -237,7 +237,7 @@ This completes the proof of the lemma.
|
|||
\end{corollary}
|
||||
|
||||
\begin{proof}
|
||||
Observe that e.g. \cite[Proposition~2.3]{grohsetal} and Lemma \ref{lem:1.21} establish (\ref{(1.26)}).
|
||||
Observe that e.g. \cite[Lemma~2.3]{grohsetal} and Lemma \ref{lem:1.21} establish (\ref{(1.26)}).
|
||||
\end{proof}
|
||||
|
||||
\begin{corollary}\label{cor:1.22.2}
|
||||
|
|
@ -252,7 +252,7 @@ This completes the proof of the lemma.
|
|||
\end{proof}
|
||||
\section{Bounds and Covnvergence}
|
||||
|
||||
\begin{lemma}\label{lem:1.21} Assume Setting \ref{primarysetting}. Then it holds for all $t\in [0,T]$, $x\in \mathbb{R}^d$
|
||||
\begin{lemma}\label{lem:1.21} Assume Setting \ref{def:1.18}. Then it holds for all $t\in [0,T]$, $x\in \mathbb{R}^d$
|
||||
\begin{align}
|
||||
&\left(\E\left[\left|U^0(t,x+\mathcal{W}^0_t)-\E \left[U^0 \left(t,x+\mathcal{W}^0_t \right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \nonumber\\
|
||||
&\leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E\left[ \lv g \lp x+\mathcal{W}^0_T \rp \rv^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}}\right]
|
||||
|
|
@ -264,19 +264,19 @@ This completes the proof of the lemma.
|
|||
G_k(t,x) = g\left(x+\mathcal{W}^{(0,0,-k)}_{T-t}\right)
|
||||
\end{align}
|
||||
\medskip
|
||||
Observe that the hypothesis that $(\mathcal{W}^\theta)_{\theta \in \Theta}$ are independent Brownian motions and the hypothesis that $g \in C(\mathbb{R}^d,\mathbb{R})$ assure that for all $t \in [0,T]$,$x\in \mathbb{R}^d$ it holds that $(G_k(t,x))_{k\in \mathbb{Z}}$ are i.i.d. random variables. This and Corollary \ref{cor:1.22.2} (applied for every $t\in [0,T]$, $x\in \mathbb{R}^d$ with $p \curvearrowleft \mathfrak{p}$, $n \curvearrowleft m$, $(X_k)_{k\in \{1,2,\..,m\}} \curvearrowleft (G_k(t,x))_{k\in \{1,2,...,m\}}$), with the notation of Corollary \ref{cor:1.22.2} ensure that for all $t\in [0,T]$, $x \in \mathbb{R}^d$, it holds that:
|
||||
Observe that the hypothesis that $(\mathcal{W}^\theta)_{\theta \in \Theta}$ are independent Brownian motions and the hypothesis that $g \in C(\mathbb{R}^d,\mathbb{R})$ assure that for all $t \in [0,T]$,$x\in \mathbb{R}^d$ it holds that $(G_k(t,x))_{k\in \mathbb{Z}}$ are i.i.d. random variables. This and Corollary \ref{cor:1.22.2} (applied for every $t\in [0,T]$, $x\in \mathbb{R}^d$ with $p \curvearrowleft \mathfrak{p}$, $n \curvearrowleft m$, $(X_k)_{k\in \{1,2,...,m\}} \curvearrowleft (G_k(t,x))_{k\in \{1,2,...,m\}}$), with the notation of Corollary \ref{cor:1.22.2} ensure that for all $t\in [0,T]$, $x \in \mathbb{R}^d$, it holds that:
|
||||
\begin{align}
|
||||
\left( \E \left[ \left| \frac{1}{m} \left[ \sum^{m}_{k=1} G_k(t,x) \right] - \E \left[ G_1(t,x) \right] \right| ^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}} \leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}}\left(\E \left[|G_1(t,x)|^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}}
|
||||
\end{align}
|
||||
\medskip
|
||||
Combining this, with (\ref{(1.12)}), (\ref{(2.1.4)}), and Item (ii) of Corollary \ref{cor:1.20.1} yields that:
|
||||
Combining this, with (1.16), (1.17), and item (ii) of Corollary \ref{cor:1.20.1} yields that:
|
||||
\begin{align}
|
||||
&\left(\E\left[\left|U^0(t,x) - \E \left[U^0(t,x)\right]\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} \nonumber\\
|
||||
&= \left(\E \left[\left|\frac{1}{m}\left[\sum^{m}_{k=1}G_k(t,x)\right]- \E \left[G_1(t,x)\right]\right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} \\
|
||||
&\leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}}\left(\E \left[\left| G_1(t,x)\right| ^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \\
|
||||
&= \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E \left[\left|g\left(x+\mathcal{W}^1_{T-t}\right)\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}}\right]
|
||||
\end{align}
|
||||
This and the fact that $\mathcal{W}^0$ has independent increments ensure that for all $m\in \N$, $t\in [0,T]$, $x\in \mathbb{R}^d$ it holds that:
|
||||
This and the fact that $\mathcal{W}^0$ has independent increments ensure that for all $n\in $, $t\in [0,T]$, $x\in \mathbb{R}^d$ it holds that:
|
||||
\begin{align}
|
||||
\left(\E \left[\left| U^0 \left(t,x+\mathcal{W}^0_t\right) - \E \left[U^0 \left(t,x+\mathcal{W}^0_t\right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^{\frac{1}{\mathfrak{p}}} \right]
|
||||
\end{align}
|
||||
|
|
@ -341,7 +341,7 @@ Which in turn yields that:
|
|||
\begin{align}\label{(1.48)}
|
||||
\mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_T \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} \leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p}
|
||||
\end{align}
|
||||
Combining (\ref{(1.46)}), (\ref{(1.47)}), and (\ref{(1.48)}) yields that:
|
||||
Combining \ref{(1.46)}, \ref{(1.47)}, and \ref{(1.48)} yields that:
|
||||
\begin{align}
|
||||
\left( \E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t \right) - u \left( t, x+\mathcal{W}^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} &\leqslant \left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} \nonumber\\
|
||||
&\les\mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s\in[0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p}
|
||||
|
|
@ -379,19 +379,19 @@ Thus we get for all $\mft \in [0,T]$, $x\in \R^d$, $n \in $:
|
|||
This completes the proof of Corollary \ref{cor:1.25.1}.
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}\label{tentpole_1} Let $T,L,p,q, \mathfrak{d} \in [0,\infty), m \in \mathbb{N}$, $\Theta = \bigcup_{n\in \mathbb{N}} \Z^n$, let $g_d\in C(\R^d,\R)$, and assume that $d\in \N$, $t \in [0,T]$, $x = (x_1,x_2,...,x_d)\in \R^d$, $v,w \in \R$ and that $\max \{ |g_d(x)|\} \leqslant Ld^p \left(1+\Sigma^d_{k=1}\left|x_k \right|^q\right)$, let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, let $\mathcal{W}^{d,\theta}: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, $\theta \in \Theta$, be independent standard Brownian motions, assume for every $d\in \N$ that $\left(\mathcal{W}^{d,\theta}\right)_{\theta \in \Theta}$ are independent, let $u_d \in C([0,T] \times \R^d,\R)$, $d \in \N$, satisfy for all $d\in \N$, $t\in [0,T]$, $x \in \R^d$ that $\E \left[g_x \left(x+\mathcal{W}^{d,0}_{T-t} \right)\right] < \infty$ and:
|
||||
\begin{theorem}\label{tentpole_1} Let $T,L,p,q, \mathfrak{d} \in [0,\infty), m \in \mathbb{N}$, $\Theta = \bigcup_{n\in \mathbb{N}} \Z^n$, let $g_d\in C(\R^d,\R)$, and assume that $d\in \N$, $t \in [0,T]$, $x = (x_1,x_2,...,x_d)\in \R^d$, $v,w \in \R$ and that $\max \{ |g_d(x)|\} \leqslant Ld^p \left(1+\Sigma^d_{k=1}\left|x_k \right|\right)$, let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, let $\mathcal{W}^{d,\theta}: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, $\theta \in \Theta$, be independent standard Brownian motions, assume for every $d\in \N$ that $\left(\mathcal{W}^{d,\theta}\right)_{\theta \in \Theta}$ are independent, let $u_d \in C([0,T] \times \R^d,\R)$, $d \in \N$, satisfy for all $d\in \N$, $t\in [0,T]$, $x \in \R^d$ that $\E \left[g_x \left(x+\mathcal{W}^{d,0}_{T-t} \right)\right] < \infty$ and:
|
||||
\begin{align}
|
||||
u_d\left(t,x\right) = \E \left[g_d \left(x + \mathcal{W}^{d}_{T-t}\right)\right]
|
||||
u_d\left(t,x\right) = \E \left[g_d \left(x + \mathcal{W}^{d,0}_{T-t}\right)\right]
|
||||
\end{align}
|
||||
Let $U^{d,\theta}_m: [0,T] \times \R^d \times \Omega \rightarrow \R$, $d \in \N$, $m\in \Z$, $\theta \in \Theta$, satisfy for all, $d\in \N$, $m \in \Z$, $\theta \in \Theta$, $t\in [0,T]$, $x\in \R^d$ that:
|
||||
\begin{align}
|
||||
U^{d}_m(t,x) = \frac{1}{m} \left[\sum^{m}_{k=1} g_d \left(x + \mathcal{W}^{d}_{T-t}\right)\right]
|
||||
U^{d,\theta}_m(t,x) = \frac{1}{m} \left[\sum^{m}_{k=1} g_d \left(x + \mathcal{W}^{d,(\theta, 0,-k)}_{T-t}\right)\right]
|
||||
\end{align}
|
||||
and for every $d,n,m \in \N$ let $\mathfrak{C}_{d,n,m} \in \Z$ be the number of function evaluations of $u_d(0,\cdot)$ and the number of realizations of scalar random variables which are used to compute one realization of $U^{d,0}_m(T,0): \Omega \rightarrow \R$.
|
||||
|
||||
There then exists $c \in \R$, and $\mathfrak{N}:\N \times (0,1] \rightarrow \N$ such that for all $d \in \N$, $\varepsilon \in (0,1]$ it holds that:
|
||||
\begin{align}\label{(2.48)}
|
||||
\sup_{t\in[0,T]} \sup_{x \in [-L,L]^d} \left(\E \left[\left| u_d(t,x) - U^{d,0}_{\mathfrak{N}(d,\epsilon)}\right|^\p\right]\right)^\frac{1}{\p} \leqslant \ve
|
||||
\sup_{t\in[0,T]} \sup_{x \in [-L,L]^d} \left(\E \left[\left| u_d(t,x) - U^{d,0}_{\mathfrak{N}(d,\epsilon)}\right|^\p\right]\right)^\frac{1}{\p} \leqslant \epsilon
|
||||
\end{align}
|
||||
|
||||
and:
|
||||
|
|
@ -399,7 +399,7 @@ and:
|
|||
\mathfrak{C}_{d,\mathfrak{N}(d,\varepsilon), \mathfrak{N}(d,\varepsilon)} \leqslant cd^c\varepsilon^{-(2+\delta)}
|
||||
\end{align}
|
||||
\end{theorem}
|
||||
\begin{proof} Throughout the proof let $\mathfrak{m}_\mathfrak{p} = \fk_p\sqrt{\mathfrak{p} -1}$, $\mathfrak{p} \in [2,\infty)$, let $\mathbb{F}^d_t \subseteq \mathcal{F}$, $d\in \N$, $t\in [0,T]$ satisfy for all $d \in \N$, $t\in [0,T]$ that:
|
||||
\begin{proof} Throughout the proof let $\mathfrak{m}_\mathfrak{p} = \sqrt{\mathfrak{p} -1}$, $\mathfrak{p} \in [2,\infty)$, let $\mathbb{F}^d_t \subseteq \mathcal{F}$, $d\in \N$, $t\in [0,T]$ satisfy for all $d \in \N$, $t\in [0,T]$ that:
|
||||
\begin{align}\label{2.3.29}
|
||||
\mathbb{F}^d_t = \begin{cases}
|
||||
\bigcap_{s\in[t,T]} \sigma \left(\sigma \left(W^{d,0}_r: r \in [0,s]\right) \cup \{A\in \mathcal{F}: \mathbb{P}(A)=0\}\right) & :t<T \\
|
||||
|
|
@ -411,7 +411,7 @@ Observe that (\ref{2.3.29}) guarantees that $\mathbb{F}^d_t \subseteq \mathcal{F
|
|||
\item it holds for all $d\in \N$ that $\{ A \in \mathcal{F}: \mathbb{P}(A) = 0 \} \subseteq \mathbb{F}^d_0$
|
||||
\item it holds for all $d \in \N$, $t\in [0,T]$, that $\mathbb{F}^d_t = \bigcap_{s \in (t,T]}\mathbb{F}^d_s$.
|
||||
\end{enumerate}
|
||||
Combining item (I), item (II), (\ref{2.3.29}) and \cite[Lemma 2.17]{hjw2020} assures us that for all $d \in \N$ it holds that $W^{d,0}:[0, T] \times \Omega \rightarrow \R^d$ is a standard $\left(\Omega, \mathcal{F}, \mathbb{P}, \left(\mathbb{F}^d_t\right)_{t\in [0, T]}\right)$- Brownian motion. In addition $(58)$ ensures that it is the case that for all $d\in N$, $x\in \R^d$ it holds that $[0,T] \times \Omega \ni (t,\omega) \mapsto x + W^{d,0}_t(\omega) \in \R^d$ is an $\left(\mathbb{F}^d_t\right)_{t\in [0,T]}/\mathcal{B}\left(\R^d\right)$-adapted stochastic process with continuous sample paths.
|
||||
Combining item (I), item (II), (\ref{2.3.29}) and \cite[Lemma 2.17]{hjw2020} assures us that for all $d \in \N$ it holds that $W^{d,0}:[0, T] \times \Omega \rightarrow \R^d$ is a standard $\left(\Omega, \mathcal{F}, \mathbb{P}, \left(\mathbb{F}^d_t\right)_{t\in [0, T]}\right)$-Brownian Brownian motion. In addition $(58)$ ensures that it is the case that for all $d\in N$, $x\in \R^d$ it holds that $[0,T] \times \Omega \ni (t,\omega) \mapsto x + W^{d,0}_t(\omega) \in \R^d$ is an $\left(\mathbb{F}^d_t\right)_{t\in [0,T]}/\mathcal{B}\left(\R^d\right)$-adapted stochastic process with continuous sample paths.
|
||||
\medskip
|
||||
|
||||
This and the fact that for all $d\in \N$, $t\in [0,T]$, $x\in \R^d$ it holds that $a_d(t,x) = 0$, and the fact that for all $d\in \N$, $t \in [0,T]$, $x$,$v\in \R^d$ it holds that $b_d(t,x)v = v$ yield that for all $d \in \N$, $x\in \R^d$ it holds that $[0,T] \times \Omega \ni (t,\omega) \mapsto x+W^{d,0}_t(\omega) \in \R^d$ satisfies for all $t\in [0,T]$ it holds $\mathbb{P}$-a.s. that:
|
||||
|
|
@ -456,7 +456,7 @@ Thus (\ref{(2.3.33)}) and (\ref{2.3.34}) together proves (\ref{(2.48)}).
|
|||
|
||||
Note that $\mathfrak{C}_{d,\mathfrak{N}_{d,\epsilon},\mathfrak{N}_{d,\epsilon}}$ is the number of function evaluations of $u_d(0,\cdot)$ and the number of realizations of scalar random variables which are used to compute one realization of $U^{d,0}_{\mathfrak{N}_{d,\epsilon}}(T,0):\Omega \rightarrow \R$. Let $\widetilde{\mathfrak{N}_{d,\ve}}$ be the value of $\mathfrak{N}_{d,\ve}$ that causes equality in $(\ref{2.3.34})$. In such a situation the number of evaluations of $u_d(0,\cdot)$ do not exceed $\widetilde{\mathfrak{N}_{d,\ve}}$. Each evaluation of $u_d(0,\cdot)$ requires at most one realization of scalar random variables. Thus we do not exceed $2\widetilde{\mathfrak{N}_{d,\epsilon}}$. Thus note that:
|
||||
\begin{align}\label{(2.3.35)}
|
||||
\mathfrak{C}_{d,\mathfrak{N}_{d,\ve},\mathfrak{N}_{d,\ve}} \leqslant \left\lceil 2\lb L\mathfrak{m}_\p\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb \ve^{-1}\right \rceil
|
||||
\mathfrak{C}_{d,\mathfrak{N}_{d,\ve},\mathfrak{N}_{d,\ve}} \leqslant 2\lb L\mathfrak{m}_\p\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb \ve^{-1}
|
||||
\end{align}
|
||||
Note that other than $d$ and $\ve$ everything on the right-hand side is constant or fixed. Hence (\ref{(2.3.35)}) can be rendered as:
|
||||
\begin{align}
|
||||
|
|
@ -483,23 +483,3 @@ Note that $\mathfrak{C}_{d,\mathfrak{N}_{d,\epsilon},\mathfrak{N}_{d,\epsilon}}$
|
|||
% \end{align}
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
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|
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|
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|
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|
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|
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@ -1,62 +1,38 @@
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@ -104,10 +65,8 @@
|
|||
\setcounter{ALG@blocknr}{1}
|
||||
\setcounter{ALG@storecount}{0}
|
||||
\setcounter{ALG@tmpcounter}{0}
|
||||
\setcounter{@stackindex}{0}
|
||||
\setcounter{ROWcellindex@}{0}
|
||||
\setcounter{lstnumber}{1}
|
||||
\setcounter{theorem}{36}
|
||||
\setcounter{theorem}{28}
|
||||
\setcounter{corollary}{0}
|
||||
\setcounter{lstlisting}{0}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -1,8 +1,7 @@
|
|||
|
||||
\chapter{Introduction}
|
||||
\chapter{Introduction.}
|
||||
\section{Motivation}
|
||||
|
||||
Artificial neural networks represent a sea change in computing. They have successfully been used in a wide range of applications, from protein-folding in \cite{tsaban_harnessing_2022}, knot theory in \cite{davies_signature_2022}, and extracting data from gravitational waves in \cite{zhao_space-based_2023}.
|
||||
Artificial neural networks represent a sea change in computing. They have successfully been used in a wide range of applications, from protein-folding in \cite{tsaban_harnessing_2022}, knot theory in \cite{davies_signature_2021}, and extracting data from gravitational waves in \cite{zhao_space-based_2023}.
|
||||
\\~\\
|
||||
As neural networks become more ubiquitous, we see that the number of parameters required to train them increases, which poses two problems: accessibility on low-power devices and the amount of energy needed to train these models, see for instance \cite{wu2022sustainable} and \cite{strubell2019energy}. Parameter estimates become increasingly crucial in an increasingly climate-challenged world. That we know strict and precise upper bounds on parameter estimates tells us when training becomes wasteful, in some sense, and when, perhaps, different approaches may be needed.
|
||||
\\~\\
|
||||
|
|
@ -10,24 +9,24 @@ Our goal in this dissertation is threefold:
|
|||
\begin{enumerate}[label = (\roman*)]
|
||||
\item Firstly, we will take something called Multi-Level Picard first introduced in \cite{e_multilevel_2019} and \cite{e_multilevel_2021}, and in particular, the version of Multi-Level Picard that appears in \cite{hutzenthaler_strong_2021}. We show that dropping the drift term and substantially simplifying the process still results in convergence of the method and polynomial bounds for the number of computations required and rather nice properties for the approximations, such as integrability and measurability.
|
||||
\item We will then go on to realize that the solution to a modified version of the heat equation has a solution represented as a stochastic differential equation by Feynman-Kac and further that a version of this can be realized by the modified multi-level Picard technique mentioned in Item (i), with certain simplifying assumptions since we dropped the drift term. A substantial amount of this is inspired by \cite{bhj20} and much earlier work in \cite{karatzas1991brownian} and \cite{da_prato_zabczyk_2002}.
|
||||
\item By far, the most significant part of this dissertation is dedicated to expanding and building upon a framework of neural networks as appears in \cite{grohs2019spacetime}. We modify this definition highly and introduce several new neural network architectures to this framework ($\pwr_n^{q,\ve}$, $\pnm_C^{q,\ve}$, $\tun^d_n$,$\etr^{N,h}$, $\xpn_n^{q,\ve}$, $\csn_n^{q,\ve}$, $\sne_n^{q,\ve}$, $\mathsf{E}^{N,h,q,\ve}_n$,$\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$, $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega}$, and $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega}$, among others) and show, for all these neural networks, that the parameter count grows only polynomially as the accuracy of our model increases, thus beating the curse of dimensionality. This finally paves the way for giving neural network approximations to the techniques realized in Item (ii). We show that it is not too wasteful (defined on the polynomiality of parameter counts) to use neural networks to approximate MLP to approximate a stochastic differential equation equivalent to certain parabolic PDEs as Feynman-Kac necessitates.
|
||||
\item By far, the most significant part of this dissertation is dedicated to expanding and building upon a framework of neural networks as appears in \cite{grohs2019spacetime}. We modify this definition highly and introduce several new neural network architectures to this framework ($\tay, \pwr, \trp, \tun,\etr$, among others) and show, for all these neural networks, that the parameter count grows only polynomially as the accuracy of our model increases, thus beating the curse of dimensionality. This finally paves the way for giving neural network approximations to the techniques realized in Item (ii). We show that it is not too wasteful (defined on the polynomiality of parameter counts) to use neural networks to approximate MLP to approximate a stochastic differential equation equivalent to certain parabolic PDEs as Feynman-Kac necessitates.
|
||||
\\~\\
|
||||
We end this dissertation by proposing two avenues of further research: analytical and algebraic. This framework of understanding neural networks as ordered tuples of ordered pairs may be extended to give neural network approximation of classical PDE approximation techniques such as Runge-Kutta, Adams-Moulton, and Bashforth. We also propose three conjectures about neural networks, as defined in \cite{grohs2019spacetime}. They form a bimodule, and that instantiation is a functor.
|
||||
We end this dissertation by proposing two avenues of further research: analytical and algebraic. This framework of understanding neural networks as ordered tuples of ordered pairs may be extended to give neural network approximation of classical PDE approximation techniques such as Runge-Kutta, Adams-Moulton, and Bashforth. We also propose three conjectures about neural networks, as defined in \cite{grohs2019spacetime}. They form a bimodule, and that realization is a functor.
|
||||
\end{enumerate}
|
||||
This dissertation is broken down into three parts. At the end of each part, we will encounter tent-pole theorems, which will eventually lead to the final neural network approximation outcome. These tentpole theorems are Theorem \ref{tentpole_1}, Theorem \ref{thm:3.21}, and Theorem \ref{ues}. Finally, the culmination of these three theorems is Corollary \ref{cor_ues}, the end product of the dissertation. We hope, you, the reader will enjoy this.
|
||||
This dissertation is broken down into three parts. At the end of each part, we will encounter tent-pole theorems, which will eventually lead to the final neural network approximation outcome. These tentpole theorems are Theorem \ref{tentpole_1}, Theorem \ref{thm:3.21}, and Theorem. Finally, the culmination of these three theorems is Theorem, the end product of the dissertation.
|
||||
|
||||
\section{Notation, Definitions \& Basic notions.}
|
||||
We introduce here basic notations that we will be using throughout this dissertation. Large parts are taken from standard literature inspired by \textit{Matrix Computations} by Golub \& van Loan, \cite{golub2013matrix}, \textit{Probability: Theory \& Examples} by Rick Durrett, \cite{durrett2019probability}, and \textit{Concrete Mathematics} by Knuth, Graham \& Patashnik, \cite{graham_concrete_1994}.
|
||||
We introduce here basic notations that we will be using throughout this dissertation. Large parts are taken from standard literature inspired by \textit{Matrix Computations} by \cite{golub2013matrix}, and \textit{Probability: Theory \& Examples} by Rick \cite{durrett2019probability}.
|
||||
\subsection{Norms and Inner Products}
|
||||
\begin{definition}[Euclidean Norm]
|
||||
Let $\left\|\cdot\right\|_E: \R^d \rightarrow [0,\infty)$ denote the Euclidean norm defined for every $d \in \N$ and for all $x= \{x_1,x_2,\cdots, x_d\}\in \R^d$ as:
|
||||
Let $\left\|\cdot\right\|_E: \R^d \rightarrow [0,\infty)$ denote the Euclidean norm defined for every $d \in \N_0$ and for all $x= \{x_1,x_2,\cdots, x_d\}\in \R^d$ as:
|
||||
\begin{align}
|
||||
\| x\|_E = \lp \sum_{i=1}^d x_i^2 \rp^{\frac{1}{2}}
|
||||
\end{align}
|
||||
For the particular case that $d=1$ and where it is clear from context, we will denote $\| \cdot \|_E$ as $|\cdot |$.
|
||||
\end{definition}
|
||||
\begin{definition}[Max Norm]
|
||||
Let $\left\| \cdot \right\|_{\infty}: \R^d \rightarrow [0,\infty )$ denote the max norm defined for every $d \in \N$ and for all $x = \left\{ x_1,x_2,\cdots,x_d \right\} \in \R^d$ as:
|
||||
Let $\left\| \cdot \right\|_{\infty}: \R^d \rightarrow [0,\infty )$ denote the max norm defined for every $d \in \N_0$ and for all $x = \left\{ x_1,x_2,\cdots,x_d \right\} \in \R^d$ as:
|
||||
\begin{align}
|
||||
\left\| x \right\|_{\infty} = \max_{i \in \{1,2,\cdots,d\}} \left\{\left| x_i \right| \right\}
|
||||
\end{align}
|
||||
|
|
@ -47,7 +46,7 @@ Let $\|\cdot \|_F: \R^{m\times n} \rightarrow [0,\infty)$ denote the Frobenius n
|
|||
\begin{definition}[Euclidean Inner Product]
|
||||
Let $\la \cdot, \cdot \ra: \R^d \times \R^d \rightarrow \R$ denote the Euclidean inner product defined for every $d \in \N$, for all $\R^d \ni x = \{x_1,x_2,...,x_d\}$, and for all $\R^d \ni y = \{y_1,y_2,..., y_d\}$ as:
|
||||
\begin{align}
|
||||
\la x, y \ra = \sum^d_{i=1} \left| x_i y_i \right|
|
||||
\la x, y \ra = \sum^d_{i=1} \lp x_i y_i \rp
|
||||
\end{align}
|
||||
\end{definition}
|
||||
|
||||
|
|
@ -57,7 +56,7 @@ Let $\|\cdot \|_F: \R^{m\times n} \rightarrow [0,\infty)$ denote the Frobenius n
|
|||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\Omega$ is a set of outcomes called the \textbf{sample space}.
|
||||
\item $\mathcal{F}$ is a set of events called the \textbf{event space}, where each event is a set of outcomes from the sample space. More specifically, it is a $\sigma$-algebra on the set $\Omega$.
|
||||
\item A measurable function $\mathbb{P}: \mathcal{F} \rightarrow [0,1]$ assigning each event in the \textbf{event space} a probability. More specifically, $\mathbb{P}$ is a measure on $\Omega$ with the caveat that the measure of the entire space is $1$, i.e., $\mathbb{P}(\Omega) = 1$.
|
||||
\item A measurable function $\mathbb{P}: \mathcal{F} \rightarrow [0,1]$ assigning each event in the \textbf{event space} a probability between $0$ and $1$. More specifically, $\mathbb{P}$ is a measure on $\Omega$ with the caveat that the measure of the entire space is $1$, i.e., $\mathbb{P}(\Omega) = 1$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
|
|
@ -71,12 +70,6 @@ Let $\|\cdot \|_F: \R^{m\times n} \rightarrow [0,\infty)$ denote the Frobenius n
|
|||
\E\lb X \rb=\int_\Omega X d\mathbb{P}
|
||||
\end{align}
|
||||
\end{definition}
|
||||
\begin{definition}[Variance]
|
||||
Given a probability space $\lp \Omega, \cF, \bbP \rp$, the variance of variable $X$, assuming $\E \lb X\rb < \infty$, denoted $\var\lb X\rb$, is the identity given by:
|
||||
\begin{align}
|
||||
\var\lb X \rb = \E\lb X^2\rb - \lp \E\lb X\rb\rp^2
|
||||
\end{align}
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Stochastic Process]
|
||||
A stochastic process is a family of random variables over a fixed probability space $(\Omega, \mathcal{F}, \mathbb{R})$, indexed over a set, usually $\lb 0, T\rb$ for $T\in \lp 0,\infty\rp$.
|
||||
|
|
@ -123,7 +116,7 @@ is adapted to the filtration $\mathbb{F}:= (\mathcal{F}_i )_{i \in [0,T]}$
|
|||
\end{align}
|
||||
\medskip
|
||||
|
||||
A strong solution to the stochastic differential equation (\ref{1.5}) on probability space \\ $(\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t \in [0,T]})$, w.r.t Brownian motion $\mathcal{W}$, w.r.t to initial condition $\mathcal{X}_0 = 0$ is a stochastic process $(\mathcal{X}_t)_{t\in[0,\infty)}$ satisfying that:
|
||||
A strong solution to the stochastic differential equation (\ref{1.5}) on probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t \in [0,T]})$, w.r.t Brownian motion $\mathcal{W}$, w.r.t to initial condition $\mathcal{X}_0 = 0$ is a stochastic process $(\mathcal{X}_t)_{t\in[0,\infty)}$ satisfying that:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\mathcal{X}_t$ is adapted to the filtration $(\mathbb{F}_t)_{t \in [0,T]}$.
|
||||
\item $\mathbb{P}(\mathcal{X}_0 = 0) =1$.
|
||||
|
|
@ -139,11 +132,6 @@ is adapted to the filtration $\mathbb{F}:= (\mathcal{F}_i )_{i \in [0,T]}$
|
|||
Let it be the case that whenever we have two strong solutions $\mathcal{X}$ and $\widetilde{\mathcal{X}}$, w.r.t. process $\mathcal{W}$ and initial condition $\mathcal{X}_0 = 0$, as defined in Definition \ref{1.9}, it is also the case that $\mathbb{P}(\mathcal{X}_t = \widetilde{\mathcal{X}}_t) =1$ for all $t\in [0, T]$. We then say that the pair $(\mu, \sigma)$ exhibits a strong uniqueness property.
|
||||
\end{definition}
|
||||
\subsection{Lipschitz and Related Notions}
|
||||
|
||||
\begin{definition}
|
||||
Given a function $f:\R \rightarrow \R$. We will say that this function is continuous everywhere if the Lebesgue measure of the subsets of the domain where it is not continuous is $0$. We will say that $f\in C_{ae}\left( \R, \R\right)$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Globally Lipschitz Function]\label{def:1.13}
|
||||
Let $d \in \N_0$. For every $d\in \N_0$, we say a function $f: \R^d \rightarrow \R^d$ is (globally) Lipschitz if there exists an $L \in (0,\infty)$ such that for all $x,y \in \R^d$ it is the case that :
|
||||
\begin{align}
|
||||
|
|
@ -295,7 +283,7 @@ for all $\epsilon \in (0,\infty)$.
|
|||
Thus under this notation the component-wise square of $A$ is $\lp \lb A \rb_{*,*}\rp^2$, the component-wise $\sin$ is $\sin\lp \lb A \rb_{*,*}\rp$ and the Hadamard product of $A,B \in \R^{m \times n}$ then becomes $ A \odot B = \ \lb A \rb_{*,*} \times \lb B \rb_{*,*}$.
|
||||
\end{definition}
|
||||
\begin{remark}
|
||||
Where we are dealing with a column vector $x \in \R^{d \times 1}$ and it is evident from the context we may choose to write $f\lp \lb x\rb_* \rp$.
|
||||
Where we are dealing with a row vector $x \in \R^{d \times 1}$ and it is evident from the context we may choose to write $f\lp \lb x\rb_* \rp$.
|
||||
\end{remark}
|
||||
\begin{definition}[The Diagonalization Operator]
|
||||
Let $m_1,m_2,n_1,n_2 \in \N$. Given $A \in \R^{m_1 \times n_1}$ and $B \in \R^{m_2\times n_2}$, we will denote by $\diag\lp A,B\rp$ the matrix:
|
||||
|
|
@ -379,17 +367,17 @@ We say that $f \in \Theta(g(x))$ if there exists $c_1,c_2,x_0 \in \lp 0,\infty\r
|
|||
We will stipulate that when concatenating vectors as $x_1 \frown x_2$, $x_1$ is on top, and $x_2$ is at the bottom.
|
||||
\end{remark}
|
||||
\begin{corollary}\label{sum_of_frown_frown_of_sum}
|
||||
Let $m_1,m_2,n_1,n_2 \in \N$, such that $m_1 = n_1$, $m_2=n_2$, and let $x \in \R^{m_1}$, $y \in \R^{n_1}$, $\fx\in \R^{m_2}$, and $\fy \in \R^{n_2}$. It is then the case that $\lb x \frown \fx\rb+\lb y \frown \fy\rb = \lb x+y\rb\frown \lb \fx +\fy\rb$.
|
||||
Let $m_1,m_2,n_1,n_2 \in \N$ and let $x \in \R^{m_1}$, $y \in \R^{n_1}$, $\fx\in \R^{m_2}$, and $\fy \in \R^{n_2}$. It is then the case that $\lb x \frown \fx\rb+\lb y \frown \fy\rb = \lb x+y\rb\frown \lb \fx +\fy\rb$.
|
||||
\end{corollary}
|
||||
\begin{proof}
|
||||
This follows straightforwardly from the fact that:
|
||||
\begin{align}
|
||||
\lb x \frown \fx \rb + \lb y \frown \fy\rb = \begin{bmatrix}
|
||||
\lb x \frown \fx \rb + \lb y + \fy\rb = \begin{bmatrix}
|
||||
x_1 \\ x_2 \\ \vdots \\ x_{m_1} \\ \fx_1 \\ \fx_2 \\ \vdots \\ \fx_{m_2}
|
||||
\end{bmatrix} + \begin{bmatrix}
|
||||
y_1 \\ y_2 \\ \vdots \\ y_{n_1} \\ \fy_1\\ \fy_2 \\ \vdots \\ \fy_{n_2}
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
x_1+y_1 \\ x_2 + y_2 \\ \vdots \\ x_{m_1}+y_{n_1} \\ \fx_1+\fy_1 \\ \fx_2 + \fy_2 \\ \vdots \\ \fx_{m_2} + \fy_{n_2}
|
||||
x_1+y_1 \\ x_2 + y_2 \\ \vdots \\ x_{m_1+n+1} \\ \fx_1+\fy_1 \\ \fx_2 + \fy_2 \\ \vdots \\ \fx_{m_2} + \fy_{n_2}
|
||||
\end{bmatrix} = \lb x+y\rb\frown \lb \fx +\fy\rb
|
||||
\end{align}
|
||||
\end{proof}
|
||||
|
|
|
|||
Binary file not shown.
|
|
@ -1,8 +0,0 @@
|
|||
|
||||
% Appendix A File
|
||||
|
||||
\refstepcounter{chapter}%
|
||||
\chapter*{\thechapter \quad Appendix A Title}
|
||||
\label{appendixB}
|
||||
|
||||
Your content goes here.
|
||||
|
|
@ -1,8 +0,0 @@
|
|||
|
||||
% Appendix B File
|
||||
|
||||
\refstepcounter{chapter}%
|
||||
\chapter*{\thechapter \quad Appendix B Title}
|
||||
\label{appendixB}
|
||||
|
||||
Your content goes here.
|
||||
|
|
@ -1 +0,0 @@
|
|||
\section{Chapter 1}
|
||||
|
|
@ -1,5 +0,0 @@
|
|||
\section{Concluding Remarks}
|
||||
\thispagestyle{empty}
|
||||
\subsection{Summary}
|
||||
|
||||
\subsection{Future directions}
|
||||
|
|
@ -1,53 +0,0 @@
|
|||
|
||||
@inproceedings{key1,
|
||||
title = "Title of the conference paper",
|
||||
author = "John Author",
|
||||
booktitle = "Name of the Conference Proceedings",
|
||||
address = "Location",
|
||||
month = "Mon.",
|
||||
year = "Year"
|
||||
}
|
||||
|
||||
@article{key2,
|
||||
title = "Title of the Journal paper",
|
||||
author = "Andrew Wiles and Ingrid Daubechies",
|
||||
journal = "Name of the Journal",
|
||||
volume = "vol.",
|
||||
number = "iss.",
|
||||
pages = "page",
|
||||
month = "Mon.",
|
||||
year = "Year"
|
||||
}
|
||||
|
||||
@book{key3,
|
||||
title = "Book Title",
|
||||
author = "John Smith and Pierre Fermat and Clark Maxwell",
|
||||
edition = "3rd",
|
||||
publisher = "Publisher Name",
|
||||
year = "Year",
|
||||
address = "Location"
|
||||
}
|
||||
|
||||
@misc{key4,
|
||||
title = "Title of the reference",
|
||||
howpublished = "Publishing details"
|
||||
}
|
||||
|
||||
@phdthesis{key5,
|
||||
title = "Thesis/Dissertation Title",
|
||||
author = "Author Name",
|
||||
school = "School Name",
|
||||
year = "Year"
|
||||
}
|
||||
|
||||
|
||||
|
||||
@book{guidelines,
|
||||
title = "Thesis and Dissertation Guidelines",
|
||||
author = "KAUST",
|
||||
edition = "1",
|
||||
publisher = "KAUST",
|
||||
year = "2011",
|
||||
address = "Thuwal, Makkah Province, Saudi Arabia"
|
||||
}
|
||||
|
||||
|
|
@ -1,307 +0,0 @@
|
|||
%% UA doctoral thesis template to accompany uamaththesis.cls
|
||||
%% last updated April 2018. Please suggest corrections, changes and
|
||||
%% improvements to Dan Luecking, SCEN 354.
|
||||
|
||||
% Load the class file. This takes the options "chapters", "MS", "MA, and
|
||||
% "PhD". The last 3 are mutually exclusive and PhD is the default.
|
||||
% The chapters option indicates that your thesis has chapter subdivisions.
|
||||
% If it is present, this class loads the LaTeX report class, otherwise
|
||||
% it loads the article class. In either case, it is loaded with the
|
||||
% "12pt" option.
|
||||
%
|
||||
% The MS and MA options indicates that this is a Master's thesis rather
|
||||
% than a doctoral dissertation. This has not been thoroughly tested.
|
||||
% Pretty much all they do is to substitute "thesis" for "dissertation" in
|
||||
% a few commands and text. They also sets the default name of the
|
||||
% degree, if you don't provide one (see \degreename below)
|
||||
%
|
||||
% Any other options are passed directly to the loaded class: article or
|
||||
% report
|
||||
%
|
||||
\documentclass{uamaththesis}
|
||||
|
||||
% Loading packages (amsmath, amssymb, amsthm, setspace, and tocbibind
|
||||
% are already preloaded). Do _not_ use tocloft unless you know how to
|
||||
% customize it to satisfy the UA thesis requirements.
|
||||
%
|
||||
% For more flexibility in number formating, load the enumerate package.
|
||||
% However, make sure you satisfy the Grad school thesis checkers.
|
||||
\usepackage{enumerate}
|
||||
\usepackage{amsmath,amssymb,amsthm,enumerate,graphicx}
|
||||
\usepackage[all]{xy}
|
||||
\usepackage{calligra,mathrsfs}
|
||||
\usepackage[hidelinks]{hyperref}
|
||||
\usepackage{bm,color}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amscd}
|
||||
\usepackage[latin2]{inputenc}
|
||||
\usepackage{t1enc}
|
||||
\usepackage[mathscr]{eucal}
|
||||
\usepackage{indentfirst}
|
||||
\usepackage{graphics}
|
||||
\usepackage{pict2e}
|
||||
\usepackage{epic}
|
||||
\numberwithin{equation}{section}
|
||||
%\usepackage[margin=2.9cm]{geometry}
|
||||
\usepackage{epstopdf}
|
||||
\usepackage{tikz-cd}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{shapes,arrows}
|
||||
\usetikzlibrary{decorations.markings}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{cases, caption}
|
||||
\allowdisplaybreaks
|
||||
|
||||
\tikzset{degil/.style={line width=0.75pt,double distance=2pt,
|
||||
decoration={markings,
|
||||
mark= at position 0.5 with {
|
||||
\node[transform shape] (tempnode) {$\backslash$};
|
||||
%\draw[thick] (tempnode.north east) -- (tempnode.south west);
|
||||
}
|
||||
},
|
||||
postaction={decorate}
|
||||
}
|
||||
}
|
||||
|
||||
\tikzset{
|
||||
commutative diagrams/.cd,
|
||||
arrow style=tikz,
|
||||
diagrams={>=open triangle 45, line width=0.5pt, double distance=2pt}}
|
||||
|
||||
|
||||
% your definitions
|
||||
%\newcommand{\disk}{\mathbb{D}} % for example
|
||||
|
||||
\theoremstyle{question}
|
||||
\newtheorem{question}[theorem]{Question}
|
||||
|
||||
\theoremstyle{setup}
|
||||
\newtheorem{setup}[theorem]{Set-up}
|
||||
|
||||
\theoremstyle{conjecture}
|
||||
\newtheorem{conjecture}[theorem]{Conjecture}
|
||||
|
||||
|
||||
%%% to repeat theorem
|
||||
|
||||
\theoremstyle{ams-restatedtheorem}
|
||||
% the argument of restatedtheorem* shouldn't ever actually be used
|
||||
\newtheorem*{restatedtheorem*}{}
|
||||
\newenvironment{restatement}[2][]{%
|
||||
\ifempty{#1}
|
||||
\begin{restatedtheorem*}[\autoref*{#2}]%
|
||||
\else%
|
||||
\begin{restatedtheorem*}[\autoref*{#2} (#1)]%
|
||||
\fi%
|
||||
}%
|
||||
{\end{restatedtheorem*}}
|
||||
\makeatother
|
||||
|
||||
%%% Shortcuts and symbols
|
||||
|
||||
\newcommand{\widesim}[2][1.5]{
|
||||
\mathrel{\overset{#2}{\scalebox{#1}[1]{$\sim$}}}
|
||||
}
|
||||
|
||||
\newcommand\norm[1]{\left\lVert#1\right\rVert}
|
||||
\newcommand{\calA}{ \mathcal{ A } }
|
||||
\newcommand{\calB}{ \mathcal{ B } }
|
||||
\newcommand{\la}{\langle}
|
||||
\newcommand{\ra}{\rangle}
|
||||
\newcommand{\NN}{\mathbb{N}}
|
||||
\newcommand{\QQ}{\mathbb{Q}}
|
||||
\newcommand{\RR}{\mathbb{R}}
|
||||
\newcommand{\ZZ}{\mathbb{Z}}
|
||||
\newcommand{\CC}{\mathbb{C}}
|
||||
\newcommand{\RP}{\mathbb{RP}}
|
||||
\newcommand{\PP}{\mathbb{P}}
|
||||
\newcommand{\Hom}{\text{Hom}}
|
||||
\newcommand{\bs}{\backslash}
|
||||
\newcommand{\mf}{\mathfrak}
|
||||
\newcommand{\mc}{\mathcal}
|
||||
\newcommand{\grade}{\text{grade}}
|
||||
\newcommand{\depth}{\text{depth}}
|
||||
\newcommand{\height}{\text{height}}
|
||||
|
||||
\newcommand{\sbt}{\,\begin{picture}(-1,1)(-1,-2)\circle*{2}\end{picture}\ }
|
||||
|
||||
\DeclareMathOperator{\Homs}{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}
|
||||
\DeclareSymbolFont{extraup}{U}{zavm}{m}{n}
|
||||
\DeclareMathSymbol{\varheart}{\mathalpha}{extraup}{86}
|
||||
\DeclareMathSymbol{\vardiamond}{\mathalpha}{extraup}{87}
|
||||
\DeclareMathSymbol{\varspade}{\mathalpha}{extraup}{81}
|
||||
|
||||
% Theorem-like environments:
|
||||
% These are predefined, but you may redo any of them with shorter names
|
||||
% if you prefer. (Note: latex does not allow you to redefine a
|
||||
% theorem-like environment, so if you want to change the style of one of
|
||||
% these, use a different name, e.g. prop instead of proposition)
|
||||
% \theoremstyle{plain}
|
||||
%\newtheorem{theorem}{Theorem}[section]
|
||||
%\newtheorem{proposition}[theorem]{Proposition}
|
||||
%\newtheorem{corollary}[theorem]{Corollary}
|
||||
%\newtheorem{lemma}[theorem]{Lemma}
|
||||
% \theoremstyle{definition}
|
||||
%\newtheorem{definition}{Definition}[section]
|
||||
%\newtheorem{example}{Example}[section]
|
||||
% \theoremstyle{remark}
|
||||
%\newtheorem{remark}{Remark}
|
||||
|
||||
% Your thesis title:
|
||||
% You must use title case: generally every word is capitalized
|
||||
% except articles (the, a, an) prepositions (of, to, in, with, for, etc.)
|
||||
% and conjunctions (and, or, but, while, etc.)
|
||||
% Required.
|
||||
\title{THESIS TITLE}
|
||||
|
||||
% Your name as UAConnect knows you:
|
||||
% Required.
|
||||
\author{NAME HERE}
|
||||
|
||||
% information about your bachelors degree or the equivalent.
|
||||
% Required.
|
||||
\bachelorinstitution{UNIVERSITY NAME}
|
||||
\bachelordegree{DEGREE}
|
||||
\bacheloryear{YEAR}
|
||||
%
|
||||
% If you have more than one bachelors degree:
|
||||
%\bachelorinstitutiontwo{Medium State University}
|
||||
%\bachelordegreetwo{Bachelor of Science in Science}
|
||||
%\bacheloryeartwo{2011}
|
||||
% and so on, up to \bachelor...three
|
||||
|
||||
% information about your masters or other post baccalariat degree.
|
||||
% Required if you have one.
|
||||
\masterinstitution{UNIVERSITY NAME}
|
||||
\masterdegree{DEGREE}
|
||||
\masteryear{YEAR}
|
||||
%
|
||||
% If you have more than one masters degree:
|
||||
%\masterinstitutiontwo{University of Alabama}
|
||||
%\masterdegreetwo{Master of Arts in Art}
|
||||
%\masteryeartwo{2015}
|
||||
% and so on up to \master...three
|
||||
% If you have a previous PhD, use the next available \master...
|
||||
% commands for it.
|
||||
%
|
||||
% Name of degree plus month and year of the final approval.
|
||||
% Required.
|
||||
%\degreename{Master of Science in Mathematics}
|
||||
\degreename{Doctor of Philosophy in Mathematics}
|
||||
\date{DATE}
|
||||
|
||||
% Your advisor
|
||||
% Required.
|
||||
% Use the first for masters, the second for PhD.
|
||||
%\thesisdirector{Luigi N. Mario, M.F.A.} % for master's degree.
|
||||
% or
|
||||
\dissertationdirector{ADVISOR NAME.}
|
||||
|
||||
% Your dissertation/thesis committee. Titles used to be required (Dr. or Prof.
|
||||
% unless neither applies). But now they seem to want just the highest degree
|
||||
% earned after the name.
|
||||
% Two required, extras are optional. I have made provision for up to
|
||||
% four
|
||||
\committeememberone{COMMITTEE MEMBER NAME.}
|
||||
\committeemembertwo{COMMITTEE MEMBER NAME.}
|
||||
%\committeememberthree{Dr.\ Mario N. Luigi}
|
||||
%\committeememberfour{Luigi N. Mario, M.F.A.}
|
||||
|
||||
\begin{document}
|
||||
|
||||
% Start of dissertation/thesis. The \frontmatter command turns off page
|
||||
% numbering until the \mainmatter command. This is the style mandated
|
||||
% by the UA dissertation guide. Do not complain to me.
|
||||
% Required:
|
||||
\frontmatter
|
||||
\maketitle
|
||||
|
||||
% The grad school now requires the right margins not be justified.
|
||||
% The \raggedright command is rather inelegant. One can get more
|
||||
% control of "raggedness" using the ragged2e package.
|
||||
% Required if some package you used turns it off:
|
||||
%\raggedright
|
||||
%\parindent 20pt % reset indentation removed by previous command
|
||||
|
||||
% The abstract. Should be less than one page, but this is not forced.
|
||||
% Required.
|
||||
\include{Abstract}
|
||||
|
||||
% Acknowledgements. Usually less than one page
|
||||
% Not required, but I've never seen a thesis without one.
|
||||
\include{Acknowledgment}
|
||||
|
||||
|
||||
% Table of Contents.
|
||||
% Required:
|
||||
\tableofcontents
|
||||
% Other lists if applicable:
|
||||
% \listoftables
|
||||
% etc.
|
||||
%
|
||||
% Signals start of actual thesis. Starts up page numbering.
|
||||
% Required:
|
||||
\mainmatter
|
||||
|
||||
% Introductory section or chapter.
|
||||
% An introduction is not required but very highly recommended. A
|
||||
% thesis consisting of reproduced published articles *must* include
|
||||
% a section titled "Introduction" separate from those articles:
|
||||
% \chapter{Introduction} or
|
||||
% \section{Introduction}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Put the rest of the theses here. Several more sections %
|
||||
% (chapters) containing actual mathematics and proofs. %
|
||||
|
||||
|
||||
\include{Chapter_1}
|
||||
|
||||
\include{Conclusion}
|
||||
|
||||
\include{Appendix_A}
|
||||
|
||||
\include{Appendix_B}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
% If you load a package that changes the behavior of citations and
|
||||
% bibliography formatting, the following two commands may be necessary
|
||||
% (and shouldn't hurt). Adjust the first fraction if the Grad School
|
||||
% hassles you about spacing between references.
|
||||
\singlespacing
|
||||
\setlength{\itemsep}{.75\baselineskip plus .2\baselineskip minus .1\baselineskip}
|
||||
%
|
||||
% If no biblography package is loaded, the references will be formatted
|
||||
% as a section, starting on a new page. If you want it formatted as a
|
||||
% new chapter, let me know, but expect the grad school to complain about
|
||||
% the formating in the table of contents.
|
||||
%
|
||||
% If you prefer the References section to be labelled something else:
|
||||
%\renewcommand{\refname}{Bibliography}% "Works Cited" is another possibility.
|
||||
%
|
||||
% The reference section is required to be listed in the TOC, and an added
|
||||
% package may change that. If so the following may be needed just before
|
||||
% the bibliography:
|
||||
% \clearpage
|
||||
% \addcontentsline{toc}{section}{\refname}%
|
||||
%
|
||||
% The closing "thebibliography" environment can be completely replaced
|
||||
% (if you use BibTeX) by
|
||||
% \bibliographstyle{plain}% for example, or amsplain or whatever.
|
||||
% \bibliography{nameoffile}% name of your .bib data file
|
||||
% If you use natbib package, use a comatible style, e.g., plainnat
|
||||
%
|
||||
\bibliographystyle{plain}
|
||||
\bibliography{reference.bib}
|
||||
|
||||
%
|
||||
% Appendices go after
|
||||
%\appendix
|
||||
% If chapters are used:
|
||||
%\chapter{Some more stuff}% Appendix A
|
||||
% otherwise
|
||||
%\section{Some more stuff}% Appendix A
|
||||
%
|
||||
\end{document}
|
||||
|
|
@ -1,429 +0,0 @@
|
|||
\NeedsTeXFormat{LaTeX2e}
|
||||
\ProvidesClass{uamaththesis}[2018/04/28 Class for University of
|
||||
Arkansas math theses]
|
||||
\newif\ifMS
|
||||
\MSfalse
|
||||
\newif\ifMA
|
||||
\MAfalse
|
||||
\newif\ifPhD
|
||||
\PhDtrue
|
||||
\newif\ifmasters
|
||||
\mastersfalse
|
||||
\newif\ifchapters
|
||||
\chaptersfalse
|
||||
\DeclareOption{chapters}{\chapterstrue}
|
||||
\DeclareOption{MS}{\MStrue\PhDfalse\masterstrue}
|
||||
\DeclareOption{MA}{\MAtrue\PhDfalse\masterstrue}
|
||||
\DeclareOption{PhD}{\PhDtrue}
|
||||
\DeclareOption*{
|
||||
\ifchapters % Set by the class option.
|
||||
\PassOptionsToClass{\CurrentOption}{report}%
|
||||
\else
|
||||
\PassOptionsToClass{\CurrentOption}{article}%
|
||||
\fi
|
||||
}
|
||||
\ProcessOptions
|
||||
\ifchapters
|
||||
\LoadClass[12pt]{report}
|
||||
\def\@makechapterhead#1{%
|
||||
{\parindent \z@ \raggedright \normalfont
|
||||
\ifnum \c@secnumdepth >\m@ne
|
||||
\normalfont\bfseries \@chapapp\space \thechapter
|
||||
\par\nobreak
|
||||
%\vskip 20\p@
|
||||
\fi
|
||||
\interlinepenalty\@M
|
||||
\normalfont \bfseries #1\par\nobreak
|
||||
%\vskip 40\p@
|
||||
}%
|
||||
}
|
||||
\def\@makeschapterhead#1{%
|
||||
{\parindent \z@ \raggedright
|
||||
\normalfont
|
||||
\interlinepenalty\@M
|
||||
\normalfont \bfseries #1\par\nobreak
|
||||
%\vskip 40\p@
|
||||
}%
|
||||
}
|
||||
\else
|
||||
\LoadClass[12pt]{article}
|
||||
\fi
|
||||
|
||||
\renewcommand{\contentsname}{Table of Contents}
|
||||
\providecommand\refname{}
|
||||
% They seem to allow "Bibliography" now, not that it matters much.
|
||||
\renewcommand{\refname}{References}
|
||||
%
|
||||
% Footnotes the same size as regular text
|
||||
\long\def\@footnotetext#1{\insert\footins{%
|
||||
\reset@font\normalsize
|
||||
\interlinepenalty\interfootnotelinepenalty
|
||||
\splittopskip\footnotesep
|
||||
\splitmaxdepth \dp\strutbox \floatingpenalty \@MM
|
||||
\hsize\columnwidth \@parboxrestore
|
||||
\protected@edef\@currentlabel{%
|
||||
\csname p@footnote\endcsname\@thefnmark
|
||||
}%
|
||||
\color@begingroup
|
||||
\@makefntext{\normalsize\selectfont%
|
||||
\rule\z@\footnotesep\ignorespaces#1\@finalstrut\strutbox}%
|
||||
\color@endgroup}}%
|
||||
|
||||
% Section headings same size as normal text
|
||||
\renewcommand\section{\@startsection {section}{1}{\z@}%
|
||||
{-3.5ex \@plus -1ex \@minus -.2ex}%
|
||||
{2.3ex \@plus.2ex}%
|
||||
{\normalfont\normalsize\bfseries}}
|
||||
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
|
||||
{-3.25ex\@plus -1ex \@minus -.2ex}%
|
||||
{1.5ex \@plus .2ex}%
|
||||
{\normalfont\normalsize\scshape}}
|
||||
\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
|
||||
{-3.25ex\@plus -1ex \@minus -.2ex}%
|
||||
{1.5ex \@plus .2ex}%
|
||||
{\normalfont\normalsize\itshape}}
|
||||
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
|
||||
{3.25ex \@plus1ex \@minus.2ex}%
|
||||
{-1em}%
|
||||
{\normalfont\normalsize\bfseries}}
|
||||
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
|
||||
{3.25ex \@plus1ex \@minus .2ex}%
|
||||
{-1em}%
|
||||
{\normalfont\normalsize\scshape}}
|
||||
|
||||
\newif\if@mainmatter
|
||||
\newcommand\frontmatter{%
|
||||
\clearpage
|
||||
\@mainmatterfalse
|
||||
\pagestyle{empty}}
|
||||
\newcommand\mainmatter{%
|
||||
\clearpage
|
||||
\@mainmattertrue
|
||||
\pagenumbering{arabic}%
|
||||
\pagestyle{plain}}
|
||||
\newcommand\backmatter{%
|
||||
\@mainmatterfalse}
|
||||
|
||||
% This is for the math department, after all.
|
||||
\RequirePackage{amsmath,amssymb, amsthm}
|
||||
\RequirePackage{setspace}
|
||||
\RequirePackage[nottoc]{tocbibind}
|
||||
|
||||
% One inch margins, one column text, USpaper
|
||||
\setlength\oddsidemargin{0pt}
|
||||
\setlength\textwidth{6.5in}
|
||||
\setlength\topmargin{0pt}
|
||||
\setlength\headheight{0pt}
|
||||
\setlength\headsep{0pt}
|
||||
\setlength\topskip{12pt}
|
||||
% This gets the page number where it needs to be, because
|
||||
% we want \texheight + 28.8pt (two baselines to the bottom of the
|
||||
% page number) to give 9.25 in. I.e., page number is .75in from edge.
|
||||
% this should be 8.85 (1/8 in larger) but somehow my printer puts the
|
||||
% page number too low with that value.
|
||||
\setlength\textheight{8.85in}
|
||||
\setlength\footskip{28.8pt}
|
||||
|
||||
% No page headers
|
||||
|
||||
% Upright numbers in enumerates, even in theorems. For more
|
||||
% flexibility in number formating, load the enumerate package.
|
||||
\renewcommand\labelenumi{\normalfont\theenumi.}
|
||||
\renewcommand\labelenumii{\normalfont(\theenumii)}
|
||||
\renewcommand\labelenumiii{\normalfont\theenumiii.}
|
||||
\renewcommand\labelenumiv{\normalfont\theenumiv.}
|
||||
|
||||
% Required: single-spaced entries, with double spacing between. Thus,
|
||||
% turn off double spacing and give \itemsep value
|
||||
% 2017: apparently new requirement that the references appear in the TOC
|
||||
\renewenvironment{thebibliography}[1]
|
||||
{\clearpage
|
||||
\singlespacing
|
||||
\section*{\refname}%
|
||||
\addcontentsline{toc}{section}{\refname}
|
||||
\@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%
|
||||
\setlength\itemsep{.75\baselineskip plus .3333\baselineskip minus
|
||||
.1667\baselineskip}
|
||||
\list{\@biblabel{\@arabic\c@enumiv}}%
|
||||
{\settowidth\labelwidth{\@biblabel{#1}}%
|
||||
\leftmargin\labelwidth
|
||||
\advance\leftmargin\labelsep
|
||||
\@openbib@code
|
||||
\usecounter{enumiv}%
|
||||
\let\p@enumiv\@empty
|
||||
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
|
||||
\sloppy
|
||||
\clubpenalty4000
|
||||
\@clubpenalty \clubpenalty
|
||||
\widowpenalty4000%
|
||||
\sfcode`\.\@m}
|
||||
{\def\@noitemerr
|
||||
{\@latex@warning{Empty `thebibliography' environment}}%
|
||||
\endlist}
|
||||
|
||||
\theoremstyle{plain}
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
\newtheorem{proposition}[theorem]{Proposition}
|
||||
\newtheorem{conj}[theorem]{Conjecture}
|
||||
\newtheorem{corollary}[theorem]{Corollary}
|
||||
\newtheorem{lem}[theorem]{Lemma}
|
||||
\theoremstyle{definition}
|
||||
\newtheorem{defn}{Definition}[section]
|
||||
\newtheorem{example}{Example}[section]
|
||||
\theoremstyle{remark}
|
||||
\newtheorem{remark}{Remark}
|
||||
|
||||
\def\degreename#1{\def\@degreename{#1}}
|
||||
\def\@degreename{%
|
||||
\ifmasters
|
||||
\ifMS
|
||||
Master of Science in Mathematics%
|
||||
\else
|
||||
Master of Arts in Secondary Mathematics%
|
||||
\fi
|
||||
\else
|
||||
Doctor of Philosophy in Mathematics%
|
||||
\fi}
|
||||
|
||||
\def\bachelorinstitutionone#1{\def\@bachelorinstitutionone{#1}}
|
||||
\def\bachelordegreeone#1{\def\@bachelordegreeone{#1}}
|
||||
\def\bacheloryearone#1{\def\@bacheloryearone{#1}}
|
||||
|
||||
\def\bachelorinstitutiontwo#1{\def\@bachelorinstitutiontwo{#1}}
|
||||
\def\bachelordegreetwo#1{\def\@bachelordegreetwo{#1}}
|
||||
\def\bacheloryeartwo#1{\def\@bacheloryeartwo{#1}}
|
||||
|
||||
\def\bachelorinstitutionthree#1{\def\@bachelorinstitutionthree{#1}}
|
||||
\def\bachelordegreethree#1{\def\@bachelordegreethree{#1}}
|
||||
\def\bacheloryearthree#1{\def\@bacheloryearthree{#1}}
|
||||
|
||||
\def\masterinstitutionone#1{\def\@masterinstitutionone{#1}}
|
||||
\def\masterinstitutiontwo#1{\def\@masterinstitutiontwo{#1}}
|
||||
\def\masterinstitutionthree#1{\def\@masterinstitutionthree{#1}}
|
||||
|
||||
\def\masterdegreeone#1{\def\@masterdegreeone{#1}}
|
||||
\def\masterdegreetwo#1{\def\@masterdegreetwo{#1}}
|
||||
\def\masterdegreethree#1{\def\@masterdegreethree{#1}}
|
||||
|
||||
\def\masteryearone#1{\def\@masteryearone{#1}}
|
||||
\def\masteryeartwo#1{\def\@masteryeartwo{#1}}
|
||||
\def\masteryearthree#1{\def\@masteryearthree{#1}}
|
||||
|
||||
%Compatability
|
||||
\let\bachelordegree=\bachelordegreeone
|
||||
\let\bachelorinstitution=\bachelorinstitutionone
|
||||
\let\bacheloryear=\bacheloryearone
|
||||
\let\masterdegree=\masterdegreeone
|
||||
\let\masterinstitution=\masterinstitutionone
|
||||
\let\masteryear=\masteryearone
|
||||
|
||||
\def\thesisdirector#1{\def\@thesisdirector{#1}}
|
||||
\def\dissertationdirector#1{\def\@dissertationdirector{#1}}
|
||||
\def\committeememberone#1{\def\@committeememberone{#1}}
|
||||
\def\committeemembertwo#1{\def\@committeemembertwo{#1}}
|
||||
\def\committeememberthree#1{%
|
||||
\def\@committeememberthree{#1}%
|
||||
\def\extracommittee##1##2{##1}%
|
||||
}
|
||||
\def\committeememberfour#1{%
|
||||
\def\@committeememberfour{#1}%
|
||||
\def\extracommittee##1##2{##1##2}%
|
||||
}
|
||||
%initialization
|
||||
\let\@title\@empty
|
||||
\let\@author\@empty
|
||||
\let\@bachelorinstitutionone\@empty
|
||||
\let\@bachelordegreeone\@empty
|
||||
\let\@bacheloryearone\@empty
|
||||
\let\@bachelorinstitutiontwo\@empty
|
||||
\let\@bachelordegreetwo\@empty
|
||||
\let\@bacheloryeartwo\@empty
|
||||
\let\@bachelorinstitutionthree\@empty
|
||||
\let\@bachelordegreethree\@empty
|
||||
\let\@bacheloryearthree\@empty
|
||||
|
||||
\let\@masterinstitutionone\@empty
|
||||
\let\@masteryearone\@empty
|
||||
\let\@masterdegreeone\@empty
|
||||
\let\@masterinstitutiontwo\@empty
|
||||
\let\@masteryeartwo\@empty
|
||||
\let\@masterdegreetwo\@empty
|
||||
\let\@masterinstitutionthree\@empty
|
||||
\let\@masteryearthree\@empty
|
||||
\let\@masterdegreethree\@empty
|
||||
|
||||
\let\@thesisdirector\@empty
|
||||
\let\@dissertationdirector\@empty
|
||||
\let\@committeememberone\@empty
|
||||
\let\@committeemembertwo\@empty
|
||||
\let\@committeememberthree\@empty
|
||||
\let\@committeememberfour\@empty
|
||||
\let\@date\@empty
|
||||
|
||||
\def\maketitle{%
|
||||
\pagestyle{empty}
|
||||
\begingroup
|
||||
\clearpage
|
||||
\singlespacing
|
||||
\begin{centering}
|
||||
\@title
|
||||
|
||||
\nobreak
|
||||
\vspace{.55in minus .15in}
|
||||
A \ifmasters thesis \else dissertation \fi submitted in partial
|
||||
fulfillment\\* of the requirements for the degree of\\*
|
||||
\@degreename\par
|
||||
|
||||
\nobreak
|
||||
\vspace{.55in minus .15in}
|
||||
by
|
||||
|
||||
\nobreak
|
||||
\vspace{.55in minus .15in}
|
||||
\@author\\*
|
||||
\@bachelorinstitutionone\\*
|
||||
\@bachelordegreeone, \@bacheloryearone
|
||||
% more than one bachelor's degree?
|
||||
\ifx\@bachelordegreetwo\@empty\else\\*
|
||||
\@bachelorinstitutiontwo\\*
|
||||
\@bachelordegreetwo, \@bacheloryeartwo
|
||||
\fi
|
||||
\ifx\@bachelordegreethree\@empty\else\\*
|
||||
\@bachelorinstitutionthree\\*
|
||||
\@bachelordegreethree, \@bacheloryearthree
|
||||
\fi
|
||||
% Up to three master's degrees
|
||||
\ifx\@masterdegreeone\@empty\else\\*
|
||||
\@masterinstitutionone\\*
|
||||
\@masterdegreeone, \@masteryearone
|
||||
\fi
|
||||
\ifx\@masterdegreetwo\@empty\else\\*
|
||||
\@masterinstitutiontwo\\*
|
||||
\@masterdegreetwo, \@masteryeartwo
|
||||
\fi
|
||||
\ifx\@masterdegreethree\@empty\else\\*
|
||||
\@masterinstitutionthree\\*
|
||||
\@masterdegreethree, \@masteryearthree
|
||||
\fi
|
||||
|
||||
\nobreak
|
||||
\vspace{.55in minus .15in}
|
||||
\@date\\*
|
||||
University of Arkansas
|
||||
|
||||
\nobreak
|
||||
\end{centering}
|
||||
|
||||
\nobreak
|
||||
\vspace{.55in minus .15in}
|
||||
\noindent This \ifmasters thesis \else dissertation \fi is approved for
|
||||
recommendation to the Graduate Council.
|
||||
|
||||
\nobreak
|
||||
\vspace{.8in minus .15in}
|
||||
\vbox{
|
||||
\noindent \rule{3in}{.4pt}\hfil\break
|
||||
\hbox to 3.24in{ \ifmasters\@thesisdirector\else\@dissertationdirector\fi\hfil}\hfil\break
|
||||
\hbox to 3.24in{ \ifmasters Thesis \else Dissertation \fi Director\hfil}}
|
||||
|
||||
\nobreak
|
||||
\vspace{.8in minus .1in}
|
||||
\vbox{
|
||||
\noindent \rule{3in}{.4pt}\hfil\rule{3in}{.4pt}\hfil\break
|
||||
\hbox to 3.24in{ \@committeememberone\hfil}%
|
||||
\hbox to 3.24in{ \@committeemembertwo\hfil}\hfil
|
||||
\hbox to 3.24in{ Committee Member \hfil}%
|
||||
\hbox to 3.24in{ Committee Member \hfil}}%\par
|
||||
|
||||
\ifx\@committeememberthree\@empty\else
|
||||
\nobreak
|
||||
\vspace{.5in minus.1in}
|
||||
\vbox{
|
||||
\extracommittee%
|
||||
{\noindent \rule{3in}{.4pt}}%
|
||||
{\hspace{.24in}\rule{3in}{.4pt}}\hfil\break
|
||||
\extracommittee%
|
||||
{\hbox to 3.24in{ \@committeememberthree\hfil}}%
|
||||
{\hbox to 3.24in{ \@committeememberfour\hfil}}\hfil\break
|
||||
\extracommittee%
|
||||
{\hbox to 3.24in{ Committee Member \hfil}}%
|
||||
{\hbox to 3.24in{ Committee Member \hfil}}\par
|
||||
}\fi
|
||||
|
||||
\endgroup
|
||||
\let\@title\@empty
|
||||
\let\@author\@empty
|
||||
\let\@degreename\@empty
|
||||
%
|
||||
\let\@bachelorinstitutionone\@empty
|
||||
\let\@bachelordegreeone\@empty
|
||||
\let\@bacheloryearone\@empty
|
||||
\let\@bachelorinstitutiontwo\@empty
|
||||
\let\@bachelordegreetwo\@empty
|
||||
\let\@bacheloryeartwo\@empty
|
||||
\let\@bachelorinstitutionthree\@empty
|
||||
\let\@bachelordegreethree\@empty
|
||||
\let\@bacheloryearthree\@empty
|
||||
%
|
||||
\let\@masterinstitutionone\@empty
|
||||
\let\@masterinstitutiontwo\@empty
|
||||
\let\@masterinstitutionthree\@empty
|
||||
\let\@masteryearone\@empty
|
||||
\let\@masteryeartwo\@empty
|
||||
\let\@masteryearthree\@empty
|
||||
\let\@masterdegreeone\@empty
|
||||
\let\@masterdegreetwo\@empty
|
||||
\let\@masterdegreethree\@empty
|
||||
%
|
||||
\let\@dissertationdirector\@empty
|
||||
\let\@committeememberone\@empty
|
||||
\let\@committeemembertwo\@empty
|
||||
\let\@committeememberthree\@empty
|
||||
\let\@committeememberfour\@empty
|
||||
\let\@date\@empty
|
||||
|
||||
\clearpage
|
||||
\doublespacing
|
||||
}
|
||||
|
||||
\renewenvironment{abstract}%
|
||||
{\clearpage
|
||||
{\noindent \textbf{Abstract} \par}}
|
||||
{\newpage}
|
||||
|
||||
\newenvironment{acknowledgements}%
|
||||
{\clearpage
|
||||
{\noindent \textbf{Acknowledgements} \par}}
|
||||
{\newpage}
|
||||
|
||||
\newenvironment{dedication}%
|
||||
{\clearpage}
|
||||
{\newpage}
|
||||
|
||||
\renewcommand\tableofcontents{%
|
||||
\clearpage
|
||||
\section*{\contentsname
|
||||
\@mkboth{%
|
||||
\MakeUppercase\contentsname}{\MakeUppercase\contentsname}}%
|
||||
\@starttoc{toc}%
|
||||
}
|
||||
\renewcommand\listoftables{%
|
||||
\clearpage
|
||||
\section*{\listtablename
|
||||
\@mkboth{%
|
||||
\MakeUppercase\listtablename}{\MakeUppercase\listtablename}}%
|
||||
\@starttoc{lot}%
|
||||
}
|
||||
\renewcommand\listoffigures{%
|
||||
\clearpage
|
||||
\section*{\listfigurename
|
||||
\@mkboth{%
|
||||
\MakeUppercase\listfigurename}{\MakeUppercase\listfigurename}}%
|
||||
\@starttoc{lof}%
|
||||
}
|
||||
|
||||
\raggedright
|
||||
\parindent 20pt
|
||||
|
||||
\endinput
|
||||
|
||||
|
|
@ -1,12 +1,5 @@
|
|||
\chapter{ANN first approximations}
|
||||
We will give here a few ANN representations of common functions. Specifically we will posit the existence of a $1$-dimensional identity and show that it acts as a compositional idenity for neural networks with fixed end-widths. Thus under composition neural networks with fixed end-widths under composition act as a monoid with $\id_d$ as the compositional identity.
|
||||
|
||||
We will also posit two new neural networks $\trp^h$, and $\etr^{n,h}$ neural networks for approximating trapezoidal rule integration.
|
||||
|
||||
We will then go on to posit the $\nrm_1^d$ and $\mxm^d$ network, taken mainly from \cite[Chapter~3]{bigbook}, our contribution will be to add parameter estimates.
|
||||
|
||||
We will finally go on to show the $\mathsf{MC}^{N,d}_{x,y}$ neural network which will perform the maximum convolution approximation for functions $f:\R^d \rightarrow \R$. Our contribution will be to show that the parameter counts are polynomial on dimension, $d$.
|
||||
\section{ANN Representations for One-Dimensional Identity}
|
||||
\section{ANN Representations for One-Dimensional Identity and some associated properties}
|
||||
|
||||
\begin{definition}[One Dimensional Identity Neural Network]\label{7.2.1}
|
||||
We will denote by $\id_d \in \neu$ the neural network satisfying for all $d \in \N$ that:
|
||||
|
|
@ -59,13 +52,13 @@ We will finally go on to show the $\mathsf{MC}^{N,d}_{x,y}$ neural network which
|
|||
This proves Item (ii)\textemdash(iii). Item (iv) follows straightforwardly from Item (i). This establishes the lemma.
|
||||
\end{proof}
|
||||
\begin{remark}
|
||||
Note here the difference between Definition \ref{def:mathfrak_i} and Definition \ref{7.2.1}.
|
||||
Note here the difference between Definition \ref{actnn} and Definition \ref{7.2.1}.
|
||||
\end{remark}
|
||||
\begin{lemma}[R\textemdash, 2023]\label{id_param}
|
||||
Let $d \in \N$. It then the case that for all $d \in \N$ we have that $\param\lp \id_d\rp = 4d^2+3d$
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
By observation we have that $\param \lp \id_1\rp = 4(1)^2+3(1) = 7$. By induction, suppose that this holds for all natural numbers up to and including $n$, i.e., for all naturals up to and including $n$; it is the case that $\param \lp \id_n\rp = 4n^2+3n$. Note then that $\id_{n+1} = \id_n \boxminus \id_1$. For $W_1$ and $W_2$ of this new network, this adds a combined extra $8n+4$ parameters. For $b_1$ and $b_2$ of this new network, this adds a combined extra $3$ parameters. Thus, we have the following:
|
||||
By observation we have that $\param \lp \id_1\rp = 4(1)^2+3(1) = 7$. By induction, suppose that this holds for all natural numbers up to and including $n$, i.e., for all naturals up to and including $n$; it is the case that $\param \lp id_n\rp = 4n^2+3n$. Note then that $\id_{n+1} = \id_n \boxminus \id_1$. For $W_1$ and $W_2$ of this new network, this adds a combined extra $8n+4$ parameters. For $b_1$ and $b_2$ of this new network, this adds a combined extra $3$ parameters. Thus, we have the following:
|
||||
\begin{align}
|
||||
4n^2+3n + 8n+4 + 3 &= 4(n+1)^2+3(n+1)
|
||||
\end{align}
|
||||
|
|
@ -126,7 +119,7 @@ Let $x \in \R$. Upon instantiation with $\rect$ and $d=1$ we have:
|
|||
\end{align}
|
||||
\textit{Case 1.ii:} Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp, ..., \lp W_L, b_L \rp \rp $. Deriving from Definition \ref{7.2.1} and \ref{5.2.1} we have that:
|
||||
\begin{align}
|
||||
&\id_1\bullet \nu \nonumber\\ &= \lp \lp W_1,b_1\rp,\lp W_2,b_2 \rp,...,\lp W_{L-1},b_{L-1} \rp, \lp \begin{bmatrix}
|
||||
\id_1\bullet \nu &= \lp \lp W_1,b_1\rp,\lp W_2,b_2 \rp,...,\lp W_{L-1},b_{L-1} \rp, \lp \begin{bmatrix}
|
||||
1 \\-1
|
||||
\end{bmatrix} W_L, \begin{bmatrix}
|
||||
1 \\ -1
|
||||
|
|
@ -211,7 +204,7 @@ This, along with Case 1. iii, implies that the uninstantiated first layer is equ
|
|||
Observe that Definitions \ref{5.2.5} and \ref{7.2.1} tells us that:
|
||||
|
||||
\begin{align}
|
||||
&\boxminus^d_{i=1} \id_i\\ &= \lp \lp \overbrace{\begin{bmatrix}
|
||||
\boxminus^d_{i=1} \id_i = \lp \lp \overbrace{\begin{bmatrix}
|
||||
\we_{\id_1,1} \\
|
||||
&&\ddots \\
|
||||
&&& \we_{\id_1,1}
|
||||
|
|
@ -263,7 +256,7 @@ Let $x \in \R^d$. Upon instantiation with $\rect$ we have that:
|
|||
\end{align}
|
||||
\textit{Case 2.ii:} Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp, ..., \lp W_L, b_L \rp \rp $. Deriving from Definition \ref{7.2.1} and \ref{5.2.1} we have that:
|
||||
\begin{align}
|
||||
&\id_d \bullet \nu \nonumber\\ &=\lp \lp W_1,b_1\rp, \lp W_2, b_2 \rp,...,\lp W_{L-1},b_{L-1} \rp, \lp \begin{bmatrix}
|
||||
\id_d \bullet \nu =\lp \lp W_1,b_1\rp, \lp W_2, b_2 \rp,...,\lp W_{L-1},b_{L-1} \rp, \lp \begin{bmatrix}
|
||||
[W_L]_{1,*} \\
|
||||
-[W_L]_{1,*}\\
|
||||
\vdots \\
|
||||
|
|
@ -291,7 +284,7 @@ This, along with Case 2.i implies that the uninstantiated last layer is equivale
|
|||
|
||||
\textit{Case 2.iii:} Let $\nu = \lp \lp W_1,b_1\rp \rp$. Deriving from Definition \ref{7.2.1} and \ref{5.2.1} we have:
|
||||
\begin{align}
|
||||
&\nu \bullet \id_d = \nonumber\\ &\lp \lp \begin{bmatrix}
|
||||
&\nu \bullet \id_d \nonumber\\ &= \lp \lp \begin{bmatrix}
|
||||
\we_{\id_1,1} \\
|
||||
&&\ddots \\
|
||||
&&& \we_{\id_1,1}
|
||||
|
|
@ -628,7 +621,7 @@ This completes the proof.
|
|||
%\end{proof}
|
||||
\section{$\trp^h$, $\etr^{n,h}$ and Neural Network Approximations For the Trapezoidal Rule.}
|
||||
\begin{definition}[The $\trp$ neural network]
|
||||
Let $h \in \lb 0,\infty \rp $. We define the $\trp^h \in \neu$ neural network as:
|
||||
Let $h \in \R_{\ges 0}$. We define the $\trp^h \in \neu$ neural network as:
|
||||
\begin{align}
|
||||
\trp^h \coloneqq \aff_{\lb \frac{h}{2} \: \frac{h}{2}\rb,0}
|
||||
\end{align}
|
||||
|
|
@ -647,7 +640,7 @@ This completes the proof.
|
|||
This a straight-forward consequence of Lemma \ref{5.3.1}
|
||||
\end{proof}
|
||||
\begin{definition}[The $\etr$ neural network]
|
||||
Let $n\in \N$ and $h \in \lb 0,\infty \rp$. We define the neural network $\etr^{n,h} \in \neu$ as:
|
||||
Let $n\in \N$ and $h \in \R_{\ges 0}$. We define the neural network $\etr^{n,h} \in \neu$ as:
|
||||
\begin{align}
|
||||
\etr^{n,h} \coloneqq \aff_{\underbrace{\lb \frac{h}{2} \ h \ h\ ... \ h \ \frac{h}{2}\rb}_{n+1-many},0}
|
||||
\end{align}
|
||||
|
|
@ -668,10 +661,6 @@ This completes the proof.
|
|||
\begin{remark}
|
||||
Let $h \in \lp 0,\infty\rp$. Note then that $\trp^h$ is simply $\etr^{2,h}$.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
For an R implementation, see Listing \ref{Etr}
|
||||
\end{remark}
|
||||
%\begin{lemma}
|
||||
% Let $f \in C \lp \R, \R \rp$, $a\in \R, b \in \lb a,\infty\rp$, $N\in \N$, and let $h = \frac{b-a}{N}$. Assume also that $f$ has first and second derivatives almost everywhere. Let $ x = \lb x_0 \: x_1 \:...\: x_n\rb \in \R^{n+1}$ such that for all $i \in \{0,1,...,n\}$ it is the case that $x_i = x_0+i\cdot h$, as such let it also be the case that $f\lp \lb x \rb_{*,*}\rp = \lb f(x_0)\: f(x_1) \: \cdots f(x_n) \rb$. Let $a = x_0$ and $b = x_n$. It is then the case that:
|
||||
% \begin{align}\label{(9.6.3)}
|
||||
|
|
@ -720,9 +709,8 @@ This completes the proof.
|
|||
% \end{align}
|
||||
% This completes the proof of the lemma.
|
||||
%\end{proof}
|
||||
\section{Maximum Convolution Approximations for Multi-Dimensional Functions}
|
||||
We will present here an approximation scheme for continuous functions called maximum convolution approximation. This derives mainly from Chapter 4 of \cite{bigbook}, and our contribution is mainly to show parameter bounds, and convergence in the case of $1$-D approximation.
|
||||
\subsection{The $\nrm^d_1$ Neural Networks}
|
||||
\section{Linear Interpolation for Multi-Dimensional Functions}
|
||||
\subsection{The $\nrm^d_1$ Networks}
|
||||
\begin{definition}[The $\nrm_1^d$ neural network]
|
||||
We denote by $\lp \nrm_1^d \rp _{d\in \N} \subseteq \neu$ the family of neural networks that satisfy:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
|
|
@ -804,10 +792,6 @@ We will present here an approximation scheme for continuous functions called max
|
|||
This concludes the proof of the lemma.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
For an R implementation, see Listing \ref{Nrm}
|
||||
\end{remark}
|
||||
|
||||
\subsection{The $\mxm^d$ Neural Networks}
|
||||
|
||||
Given $x\in \R$, it is straightforward to find the maximum; $ x$ is the maximum. For $x \in \R^d$ we may find the maximum via network (\ref{9.7.6.1}), i.e. $\mxm^2$. The strategy is to find maxima for half our entries and half repeatedly until we have one maximum. For $x \in \R^d$ where $d$ is even we may stack $d$ copies of $\mxm^2$ to halve, and for $x \in \R^d$ where $d$ is odd and greater than $3$ we may introduce ``padding'' via the $\id_1$ network and thus require $\frac{d-1}{2}$ copies of $\mxm^2$ to halve.
|
||||
|
|
@ -1013,7 +997,7 @@ Given $x\in \R$, it is straightforward to find the maximum; $ x$ is the maximum.
|
|||
\lp\real_{\rect} \lp \mxm^2 \rp \rp \lp x \rp &= \max \{x_1-x_2,0\} + \max\{x_2,0 \} - \max\{ -x_2,0\} \nonumber \\
|
||||
&= \max \{x_1-x_2,0\} + x_2 = \max\{x_1,x_2\}
|
||||
\end{align}
|
||||
Note next that Lemma \ref{idprop}, Lemma \ref{comp_prop}, and \cite[Proposition~2.19]{grohs2019spacetime} then imply for all $d \in \{2,3,4,...\}$, $x = \{x_1,x_2,...,x_d\} \in \R^d$ it holds that $\lp \real_{\rect} \lp \mxm^d \rp \rp \lp x \rp \in C \lp \R^d,\R \rp$. and $\lp \real_{\rect} \lp \mxm^d \rp \rp \lp x \rp = \max\{ x_1,x_2,...,x_d \}$. This establishes Items (iii)\textemdash(iv).
|
||||
Note next that Lemma \ref{idprop}, Lemma \ref{comp_prop}, and \cite[Proposition~2.19]{grohs2019spacetime} then imply for all $d \in \{2,3,4,...\}$, $x = \{x_1,x_2,...,x_d\} \in \R^d$ it holds that $\lp \real_{\rect} \lp \mxm^d \rp \rp \lp x \rp \in C \lp \R^d,\R \rp$. and $\lp \real_{\rect} \lp \mxm^d \rp \rp \lp x \rp = \max\{ x_1,x_2,...,x_d \}$. This establishes Items (iii)-(iv).
|
||||
|
||||
Consider now the fact that Item (ii) implies that the layer architecture forms a geometric series whence we have that the number of bias parameters is bounded by:
|
||||
\begin{align}
|
||||
|
|
@ -1045,11 +1029,7 @@ Given $x\in \R$, it is straightforward to find the maximum; $ x$ is the maximum.
|
|||
Item (vi) is a straightforward consequence of Item (i). This completes the proof of the lemma.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
For an R implementation, see Listing \ref{Mxm}
|
||||
\end{remark}
|
||||
|
||||
\subsection{The $\mathsf{MC}^{N,d}_{x,y}$ Neural Networks}
|
||||
\subsection{The $\mathsf{MC}$ Neural Network and Approximations via Maximum Convolutions }
|
||||
|
||||
Let $f: [a,b] \rightarrow \R$ be a continuous bounded function with Lipschitz constant $L$. Let $x_0 \les x_1 \les \cdots \les x_N$ be a set of sample points within $[a,b]$, with it being possibly the case that that for all $i \in \{0,1,\hdots, N\}$, $x_i \sim \unif([a,b])$. For all $i \in \{0,1,\hdots, N\}$, define a series of functions $f_0,f_1,\hdots f_N: [a,b] \rightarrow \R$, as such:
|
||||
\begin{align}
|
||||
|
|
@ -1063,13 +1043,13 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
\end{align}
|
||||
It is then the case that:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\inn \lp \mathsf{MC}^{N,d}_{x,y} \rp = d$
|
||||
\item $\out\lp \mathsf{MC}^{N,d}_{x,y} \rp = 1$
|
||||
\item $\hid \lp \mathsf{MC}^{N,d}_{x,y} \rp = \left\lceil \log_2 \lp N \rp \right\rceil +1$
|
||||
\item $\wid_1 \lp \mathsf{MC}^{N,d}_{x,y} \rp = 2dN$
|
||||
\item for all $i \in \{ 2,3,...\}$ we have $\wid_i \lp \mathsf{MC}^{N,d}_{x,y} \rp \les 3 \left\lceil \frac{N}{2^{i-1}} \right\rceil$
|
||||
\item it holds for all $x \in \R^d$ that $\lp \real_{\rect} \lp \mathsf{MC}^{N,d}_{x,y} \rp \rp \lp x \rp = \max_{i \in \{1,2,...,N\}} \lp y_i - L \left\| x-x_i \right\|_1\rp$
|
||||
\item it holds that $\param \lp \mathsf{MC}^{N,d}_{x,y} \rp \les \left\lceil \lp \frac{2}{3}d^2+3d\rp \lp 1+\frac{1}{2}^{2\lp \left\lceil \log_2\lp d\rp\right\rceil+1 \rp}\rp + 1 \right\rceil + 7N^2d^2 + 3\left\lceil \frac{N}{2}\right\rceil \cdot 2dN$
|
||||
\item $\inn \lp \mathsf{MC} \rp = d$
|
||||
\item $\out\lp \mathsf{MC} \rp = 1$
|
||||
\item $\hid \lp \mathsf{MC} \rp = \left\lceil \log_2 \lp N \rp \right\rceil +1$
|
||||
\item $\wid_1 \lp \mathsf{MC} \rp = 2dN$
|
||||
\item for all $i \in \{ 2,3,...\}$ we have $\wid_1 \lp \mathsf{MC} \rp \les 3 \left\lceil \frac{N}{2^{i-1}} \right\rceil$
|
||||
\item it holds for all $x \in \R^d$ that $\lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp = \max_{i \in \{1,2,...,N\}} \lp y_i - L \left\| x-x_i \right\|_1\rp$
|
||||
\item it holds that $\param \lp \mathsf{MC} \rp \les \left\lceil \lp \frac{2}{3}d^2+3d\rp \lp 1+\frac{1}{2}^{2\lp \left\lceil \log_2\lp d\rp\right\rceil+1 \rp}\rp + 1 \right\rceil + 7N^2d^2 + 3\left\lceil \frac{N}{2}\right\rceil \cdot 2dN$
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
|
|
@ -1077,7 +1057,7 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
\begin{align}
|
||||
\mathsf{X} = \aff_{-L\mathbb{I}_N,y} \bullet \lp \lb \boxminus_{i=1}^N \mathsf{S}_i \rb \rp \bullet \cpy_{N,d}
|
||||
\end{align}
|
||||
Note that (\ref{9.7.20}) and Lemma \ref{comp_prop} tells us that $\out \lp \R \rp = \out \lp \mxm^N \rp = 1$ and $\inn \lp \mathsf{MC}^{N,d}_{x,y} \rp = \inn \lp \cpy_{N,d} \rp =d $. This proves Items (i)--(ii). Next observe that since it is the case that $\hid \lp \cpy_{N,d} \rp$ and $\hid \lp \nrm^d_1 \rp = 1$, Lemma \ref{comp_prop} then tells us that:
|
||||
Note that (\ref{9.7.20}) and Lemma \ref{comp_prop} tells us that $\out \lp \R \rp = \out \lp \mxm^N \rp = 1$ and $\inn \lp \mathsf{MC} \rp = \inn \lp \cpy_{N,d} \rp =d $. This proves Items (i)--(ii). Next observe that since it is the case that $\hid \lp \cpy_{N,d} \rp$ and $\hid \lp \nrm^d_1 \rp = 1$, Lemma \ref{comp_prop} then tells us that:
|
||||
\begin{align}
|
||||
\hid \lp \mathsf{X} \rp = \hid \lp\aff_{-L\mathbb{I}_N,y} \rp + \hid \lp \boxminus_{i=1}^N \mathsf{S}_i\rp + \hid \lp \cpy_{N,d} \rp = 1
|
||||
\end{align}
|
||||
|
|
@ -1089,19 +1069,19 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
|
||||
Note next that Lemma \ref{comp_prop} and \cite[Proposition~2.20]{grohs2019spacetime} tells us that:
|
||||
\begin{align}\label{8.3.33}
|
||||
\wid_1 \lp \mathsf{MC}^{N,d}_{x,y} \rp = \wid_1 \lp \mathsf{X} \rp = \wid_1 \lp \boxminus^N_{i=1} \mathsf{S}_i\rp = \sum^N_{i=1} \wid_1 \lp \mathsf{S}_i \rp = \sum^N_{i=1} \wid_1 \lp \nrm^d_1 \rp = 2dN
|
||||
\wid_1 \lp \mathsf{MC} \rp = \wid_1 \lp \mathsf{X} \rp = \wid_1 \lp \boxminus^N_{i=1} \mathsf{S}_i\rp = \sum^N_{i=1} \wid_1 \lp \mathsf{S}_i \rp = \sum^N_{i=1} \wid_1 \lp \nrm^d_1 \rp = 2dN
|
||||
\end{align}
|
||||
This establishes Item (iv).
|
||||
|
||||
Next observe that the fact that $\hid \lp \mathsf{X} \rp=1$, Lemma \ref{comp_prop} and Lemma \ref{9.7.4} tells us that for all $i \in \{2,3,...\}$ it is the case that:
|
||||
\begin{align}
|
||||
\wid_i \lp \mathsf{MC}^{N,d}_{x,y} \rp = \wid_{i-1} \lp \mxm^N \rp \les 3 \left\lceil \frac{N}{2^{i-1}} \right\rceil
|
||||
\wid_i \lp \mathsf{MC} \rp = \wid_{i-1} \lp \mxm^N \rp \les 3 \left\lceil \frac{N}{2^{i-1}} \right\rceil
|
||||
\end{align}
|
||||
This establishes Item (v).
|
||||
|
||||
Next observe that Lemma \ref{9.7.2} and Lemma \ref{5.3.3} tells us that for all $x \in \R^d$, $i \in \{1,2,...,N\}$ it holds that:
|
||||
\begin{align}
|
||||
\lp \real_{\rect} \lp \mathsf{MC}^{N,d}_{x,y} \rp \rp \lp x \rp - \lp \real_{\rect}\lp \nrm^d_1 \rp \circ \real_{\rect}\lp \aff_{\mathbb{I}_d,-x_i} \rp \rp \lp x \rp = \left\| x-x_i \right\|_1
|
||||
\lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp - \lp \real_{\rect}\lp \nrm^d_1 \rp \circ \real_{\rect}\lp \aff_{\mathbb{I}_d,-x_i} \rp \rp \lp x \rp = \left\| x-x_i \right\|_1
|
||||
\end{align}
|
||||
This an \cite[Proposition~2.20]{grohs2019spacetime} combined establishes that for all $x \in \R^d$ it holds that:
|
||||
\begin{align}
|
||||
|
|
@ -1114,7 +1094,7 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
\end{align}
|
||||
Then Lemma \ref{comp_prop} and Lemma \ref{9.7.4} tells us that for all $x\in \R^d$ it holds that:
|
||||
\begin{align}
|
||||
\lp \real_{\rect} \lp \mathsf{MC}^{N,d}_{x,y} \rp \rp \lp x \rp &= \lp \real_{\rect}\lp \mxm^N \rp \circ \lp \real_{\rect}\lp \mathsf{X} \rp \rp \rp \lp x \rp \nonumber \\
|
||||
\lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp &= \lp \real_{\rect}\lp \mxm^N \rp \circ \lp \real_{\rect}\lp \mathsf{X} \rp \rp \rp \lp x \rp \nonumber \\
|
||||
&= \lp \real_{\rect}\lp \mxm^N \rp \rp \lp y_1-L \|x-x_1\|_1,y_2-L\|x-x_2\|_1,...,y_N-L\|x-x_N\|_1\rp \nonumber\\
|
||||
&=\max_{i\in \{1,2,...,N\} } \lp y_i - L \|x-x_i\|_1\rp
|
||||
\end{align}
|
||||
|
|
@ -1138,14 +1118,14 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
\end{align}
|
||||
Finally Lemma \ref{comp_prop}, (\ref{8.3.33}), and Lemma \ref{lem:mxm_prop} yields that:
|
||||
\begin{align}
|
||||
\param(\mathsf{MC}^{N,d}_{x,y}) &= \param \lp \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lp \boxminus_{i=1}^N \lb \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \rp \bullet \cpy_{N,d} \rp \nonumber\\
|
||||
\param(\mathsf{MC}) &= \param \lp \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lp \boxminus_{i=1}^N \lb \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \rp \bullet \cpy_{N,d} \rp \nonumber\\
|
||||
&\les \param \lp \mxm^N \bullet \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb \rp \rp \nonumber\\
|
||||
&\les \param \lp \mxm^N \rp + \param \lp \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp \rp + \nonumber\\ &\wid_1\lp \mxm^N\rp \cdot \wid_{\hid \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp} \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp \nonumber \\
|
||||
&\les \left\lceil \lp \frac{2}{3}d^2+3d\rp \lp 1+\frac{1}{2}^{2\lp \left\lceil \log_2\lp d\rp\right\rceil+1 \rp}\rp + 1 \right\rceil + 7N^2d^2 + 3\left\lceil \frac{N}{2}\right\rceil \cdot 2dN
|
||||
\end{align}
|
||||
\end{proof}
|
||||
\begin{remark}
|
||||
We may represent the neural network diagram for $\mxm^d$ below.
|
||||
We may represent the neural network diagram for $\mxm^d$ as:
|
||||
\end{remark}
|
||||
\begin{figure}[h]
|
||||
\begin{center}
|
||||
|
|
@ -1154,7 +1134,7 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-0.9,xscale=0.9]
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
%uncomment if require: \path (0,560); %set diagram left start at 0, and has height of 560
|
||||
|
||||
%Shape: Rectangle [id:dp1438938274656144]
|
||||
|
|
@ -1257,15 +1237,12 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
|
||||
|
||||
\end{center}
|
||||
\caption{Neural network diagram for the $\mathsf{MC}^{N,d}_{x,y}$ network}
|
||||
\caption{Neural network diagramfor the $\mxm$ network}
|
||||
\end{figure}
|
||||
\begin{remark}
|
||||
For an R implementation, see Listing \ref{MC}.
|
||||
\end{remark}
|
||||
|
||||
\subsection{Lipschitz Function Approximations}\label{(9.7.6)}
|
||||
\begin{lemma}%TODO: Should we stipulate compact sets?
|
||||
Let $\lp E,d \rp$ be a metric space. Let $L \in \lb 0,\infty \rp$, $D \subseteq E$, $\varnothing \neq C \subseteq D$. Let $f:D \rightarrow \R$ satisfy for all $x\in D$, $y \in C$ that $\left| f(x) -f(y)\right| \les L d \lp x,y \rp$, and let $F:E \rightarrow \R \cup \{\infty\}$ satisfy for all $x\in E$ that:
|
||||
Let $\lp E,d \rp$ be a metric space. Let $L \in \lb 0,\infty \rp$, $D \subseteq E$, $\emptyset \neq C \subseteq D$. Let $f:D \rightarrow \R$ satisfy for all $x\in D$, $y \in C$ that $\left| f(x) -f(y)\right| \les L d \lp x,y \rp$, and let $F:E \rightarrow \R \cup \{\infty\}$ satisfy for all $x\in E$ that:
|
||||
\begin{align}\label{9.7.30}
|
||||
F\lp x \rp = \sup_{y\in C} \lb f\lp y \rp - Ld\lp x,y \rp \rb
|
||||
\end{align}
|
||||
|
|
@ -1320,7 +1297,7 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
This establishes Item (v) and hence establishes the Lemma.
|
||||
\end{proof}
|
||||
\begin{corollary}\label{9.7.6.1}
|
||||
Let $\lp E,d \rp$ be a metric space, let $L \in \lb 0,\infty \rp$, $\varnothing \neq C \subseteq E$, let $f: E \rightarrow \R$ satisfy for all $x\in E$, $y \in C$ that $\left\| f(x) - f(y) \right| \les Ld \lp x,y \rp$, and let $F:E \rightarrow \R \cup \{\infty\}$ satisfy for all $x\in E$ that:
|
||||
Let $\lp E,d \rp$ be a metric space, let $L \in \lb 0,\infty \rp$, $\emptyset \neq C \subseteq E$, let $f: E \rightarrow \R$ satisfy for all $x\in E$, $y \in C$ that $\left\| f(x) - f(y) \right| \les Ld \lp x,y \rp$, and let $F:E \rightarrow \R \cup \{\infty\}$ satisfy for all $x\in E$ that:
|
||||
\begin{align}
|
||||
F \lp x \rp = \sup_{y\in C} \lb f(y) - Ld \lp x,y \rp\rb
|
||||
\end{align}
|
||||
|
|
@ -1339,14 +1316,14 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
\end{proof}
|
||||
\subsection{Explicit ANN Approximations }
|
||||
|
||||
\begin{lemma}\label{lem:maxconv_accuracy}
|
||||
\begin{lemma}
|
||||
Let $d,N \in \N$, $L \in \lb 0,\infty \rp$. Let $E \subseteq \R^d$. Let $x_1,x_2,...,x_N \in E$, let $f:E \rightarrow \R$ satisfy for all $x_1,y_1 \in E$ that $\left| f(x_1) -f(y_1)\right| \les L \left\| x_1-x_2 \right\|_1$ and let $\mathsf{MC} \in \neu$ and $y = \lp f\lp x_1 \rp, f \lp x_2 \rp,...,f\lp x_N \rp\rp$ satisfy:
|
||||
\begin{align}
|
||||
\mathsf{MC}^{N,d}_{x,y} = \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lb \boxminus^N_{i=1} \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \bullet \cpy_{N,d}
|
||||
\mathsf{MC} = \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lb \boxminus^N_{i=1} \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \bullet \cpy_{N,d}
|
||||
\end{align}
|
||||
It is then the case that:
|
||||
\begin{align}\label{(9.7.42)}
|
||||
\sup_{x\in E} \left| \lp \real_{\rect}\lp \mathsf{MC}^{N,d}_{x,y} \rp \rp \lp x \rp -f\lp x \rp \right| \les 2L \lb \sup _{x\in E} \lp \min_{i\in \{1,2,...,N\}} \left\| x-x_i\right\|_1\rp\rb
|
||||
\sup_{x\in E} \left| \lp \real_{\rect}\lp \mathsf{MC} \rp \rp \lp x \rp -f\lp x \rp \right| \les 2L \lb \sup _{x\in E} \lp \min_{i\in \{1,2,...,N\}} \left\| x-x_i\right\|_1\rp\rb
|
||||
\end{align}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
|
|
@ -1361,24 +1338,22 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
|
|||
Then Lemma \ref{(9.7.5)} tells us that for all $x\in E$ it holds that $F(x) = \lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp$. This combined with (\ref{(9.7.44)}) establishes (\ref{(9.7.42)}).
|
||||
\end{proof}
|
||||
\begin{lemma}
|
||||
Let $d,N \in \N$, $L \in \lb 0,\infty \rp$. Let $\lb a,b\rb \subsetneq \R^d$. Let $x_1,x_2,...,x_N \in \lb a,b\rb$, let $f:\lb a,b\rb \rightarrow \R$ satisfy for all $x_1,x_2 \in \lb a,b\rb$ that $\left| f(x_1) -f(x_2)\right| \les L \left| x_1-x_2 \right|$ and let $\mathsf{MC}^{N,1}_{x,y} \in \neu$ and $y = f\lp \lb x \rb_*\rp$ satisfy:
|
||||
Let $d,N \in \N$, $L \in \lb 0,\infty \rp$. Let $E \subseteq \R^d$. Let $x_1,x_2,...,x_N \in E$, let $f:E \rightarrow \R$ satisfy for all $x_1,x_2 \in E$ that $\left| f(x_1) -f(x_2)\right| \les L \left\| x_1-x_2 \right\|_1$ and let $\mathsf{MC} \in \neu$ and $y = \lp f\lp x_1 \rp, f \lp x_2 \rp,...,f\lp x_N \rp\rp$ satisfy:
|
||||
\begin{align}
|
||||
\mathsf{MC}^{N,1}_{x,y} = \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lb \boxminus^N_{i=1} \nrm^1_1 \bullet \aff_{1,-x_i} \rb \bullet \cpy_{N,1}
|
||||
\mathsf{MC} = \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lb \boxminus^N_{i=1} \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \bullet \cpy_{N,d}
|
||||
\end{align}
|
||||
It is then the case that:
|
||||
\begin{align}
|
||||
\lim_{N \rightarrow \infty} \lb \mathbb{P} \lp \sup_{x\in E} \left| \lp \real_{\rect}\lp \mathsf{MC} \rp \rp \lp x \rp -f\lp x \rp \right| >0 \rp\rb \rightarrow 0
|
||||
\end{align}
|
||||
It is then the case that for approximant $\mathsf{MC}^{N,1}_{x,y}$ that:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\inn \lp \mathsf{MC}^{N,1}_{x,y} \rp = 1$
|
||||
\item $\out\lp \mathsf{MC}^{N,1}_{x,y} \rp = 1$
|
||||
\item $\hid \lp \mathsf{MC}^{N,1}_{x,y} \rp = \left\lceil \log_2 \lp N \rp \right\rceil +1$
|
||||
\item $\wid_1 \lp \mathsf{MC}^{N,1}_{x,y} \rp = 2N$
|
||||
\item for all $i \in \{ 2,3,...\}$ we have $\wid_1 \lp \mathsf{MC}^{N,1}_{x,y} \rp \les 3 \left\lceil \frac{N}{2^{i-1}} \right\rceil$
|
||||
\item it holds for all $x \in \R^d$ that $\lp \real_{\rect} \lp \mathsf{MC}^{N,1}_{x,y} \rp \rp \lp x \rp = \max_{i \in \{1,2,...,N\}} \lp y_i - L \left| x-x_i \right|\rp$
|
||||
\item it holds that $\param \lp \mathsf{MC}^{N,1}_{x,y} \rp \les 6 + 7N^2 + 3\left\lceil \frac{N}{2}\right\rceil \cdot 2N$
|
||||
\item $\sup_{x\in \lb a,b\rb} \left| F(x) - f(x) \right| \les 2L \frac{|a-b|}{N}$
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Items (i)\textemdash(vii) is an assertion of Lemma \ref{lem:mc_prop}. Item (viii) is an assertion of Lemma \ref{lem:maxconv_accuracy} with $d \curvearrowleft 1$.
|
||||
Note that $N$ is chosen uniformly from set $E \subseteq \R^d$. Note next that (\ref{(9.7.44)}) tells us that:
|
||||
\begin{align}
|
||||
&\lim_{N \rightarrow \infty} \lb \mathbb{P} \lp \sup_{x\in E} \left| \lp \real_{\rect}\lp \mathsf{MC} \rp \rp \lp x \rp -f\lp x \rp \right| >0 \rp\rb \nonumber \\
|
||||
&\les
|
||||
\end{align}
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
|
@ -1399,3 +1374,4 @@ It is then the case that for approximant $\mathsf{MC}^{N,1}_{x,y}$ that:
|
|||
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -1,5 +1,4 @@
|
|||
\chapter{ANN Product Approximations and Their Consequences}\label{chp:ann_prod}
|
||||
|
||||
\chapter{ANN Product Approximations}
|
||||
\section{Approximation for Products of Two Real Numbers}
|
||||
We will build up the tools necessary to approximate $e^x$ via neural networks in the framework described in the previous sections. While much of the foundation comes from, e.g., \cite{grohs2019spacetime} way, we will, along the way, encounter neural networks not seen in the literature, such as the $\tay$, $\pwr$, $\tun$, and finally a neural network approximant for $e^x$. For each of these neural networks, we will be concerned with at least the following:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
|
|
@ -8,9 +7,8 @@ We will build up the tools necessary to approximate $e^x$ via neural networks in
|
|||
\item whether their parameter estimates are bounded at most polynomially on the type of accuracy we want, $\ve$.
|
||||
\item The accuracy of our neural networks.
|
||||
\end{enumerate}
|
||||
The sections pertaining to squaring and taking the product of neural networks derive mostly from \cite{yarotsky_error_2017} via \cite{bigbook}.
|
||||
\subsection{The squares of real numbers in $\lb 0,1 \rb$}
|
||||
One of the most important operators we will approximate is the product operator $\times$ for two real numbers. The following sections takes a streamlined version of the proof given in \cite[Section~3.1]{grohs2019spacetime}. In particular we will assert the existence of the neural network $\Phi$ and $\phi_d$ and work our way towards its properties.
|
||||
|
||||
\begin{definition}[The $\mathfrak{i}_d$ Network]\label{def:mathfrak_i}
|
||||
For all $d \in \N$ we will define the following set of neural networks as ``activation neural networks'' denoted $\mathfrak{i}_d$ as:
|
||||
\begin{align}
|
||||
|
|
@ -50,44 +48,18 @@ One of the most important operators we will approximate is the product operator
|
|||
\end{align}
|
||||
Let $\Phi_k \in \neu$, $k\in \N$ satisfy for all $k \in [2,\infty) \cap \N$ that $\Phi_1 = \lp \aff_{C_1,0} \bullet \mathfrak{i}_4 \rp \bullet \aff_{\mymathbb{e}_4,B}$, that for all $d \in \N$, $\mathfrak{i}_d = \lp \lp \mathbb{I}_d, \mymathbb{0}_d \rp, \lp \mathbb{I}_d, \mymathbb{0}_d \rp \rp$ and that:
|
||||
\begin{align}
|
||||
\Phi_k =\lp \aff_{C_k,0}\bullet \mathfrak{i}_4 \rp \bullet \lp \aff_{A_{k-1},B} \bullet \mathfrak{i}_4\rp \bullet \cdots \bullet \lp \aff_{A_1,B} \bullet \mathfrak{i}_4 \rp \bullet \aff_{\mymathbb{e}_4,B} ,
|
||||
\Phi_k =\lp \aff_{C_k,0}\bullet \mathfrak{i}_4 \rp \bullet \lp \aff_{A_{k-1},B} \bullet \mathfrak{i}_4\rp \bullet \cdots \bullet \lp \aff_{A_1,B} \bullet \mathfrak{i}_4 \rp \bullet \aff_{\mymathbb{e}_4,B}
|
||||
\end{align}
|
||||
It is then the case that:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item for all $k \in \N$, $x \in \R$ we have $\real_{\rect}\lp \Phi_k\rp\lp x \rp \in C \lp \R, \R \rp $
|
||||
\item for all $k \in \N$ we have $\lay \lp \Phi_k \rp = \lp 1,4,4,...,4,1 \rp \in \N^{k+2}$
|
||||
\item for all $k \in \N$, $x \in \R \setminus \lb 0,1 \rb $ that $\lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp = \rect \lp x \rp$
|
||||
\item for all $k \in \N$, $x \in \lb 0,1 \rb$, we have $\left| x^2 - \lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp \right| \les 2^{-2k-2}$, and
|
||||
\item for all $k \in \N$, $x \in \lb 0,1 \rb$, we have $\left| x^2 - \lp \real_{\rect} \lp \xi_k \rp \rp \lp x \rp \right| \les 2^{-2k-2}$, and
|
||||
\item for al $k \in \N$ , we have that $\param \lp \Phi_k \rp = 20k-7$
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Firstly note that Lemma \ref{aff_prop}, Lemma \ref{comp_prop}, and Lemma \ref{lem:mathfrak_i}
|
||||
ensure that for all $k \in \N$, $x \in \R$ it is the case that $\real_{\rect}\lp \Phi_k\rp \lp x\rp \in C\lp \R, \R\rp$. This proves Item (i).
|
||||
|
||||
Note next that Lemma \ref{aff_prop}, Lemma \ref{lem:mathfrak_i}, and Lemma \ref{comp_prop} tells us that:
|
||||
\begin{align}
|
||||
\lay \lp \Phi_1 \rp = \lay \lp \aff_{\mymathbb{e}_4},B\rp = \lp 1,4,1\rp
|
||||
\end{align}
|
||||
and for all $k \in \N$ it is the case that:
|
||||
\begin{align}
|
||||
\lay \lp \aff_{A_k,B} \bullet \mathfrak{i}_4\rp = \lp 4,4,4,4\rp
|
||||
\end{align}
|
||||
|
||||
Whence it is straightforward to see that for $\Phi_k$ where $k \in \N \cap \lb 2,\infty \rp$, Lemma \ref{comp_prop} tells us then that:
|
||||
\begin{align}
|
||||
\lay \lp \Phi_k\rp &= \lay \lp \lp \aff_{C_k,0}\bullet \mathfrak{i}_4 \rp \bullet \lp \aff_{A_{k-1},B} \bullet \mathfrak{i}_4\rp \bullet \cdots \bullet \lp \aff_{A_1,B} \bullet \mathfrak{i}_4 \rp \bullet \aff_{\mymathbb{e}_4,B} \rp \nonumber\\
|
||||
&= (1,\overbrace{4) \: \overbrace{( 4}^{merged},4,4,\overbrace{4) \:( 4}^{merged},4,4,\overbrace{4)\: }^{merged}\hdots \overbrace{\: ( 4}^{merged},4,4,\overbrace{4) \:}^{merged} (4}^{k-1 \text{ many}},1)
|
||||
\end{align}
|
||||
This thus finally yields that:
|
||||
\begin{align}
|
||||
\lay \lp \Phi_k\rp = \lp 1,4,4,\hdots, 4,1\rp \in \N^{k+2}
|
||||
\end{align}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Let $g_k: \R \rightarrow \lb 0,1 \rb$, $k \in \N$ be the functions defined as such, satisfying for all $k \in \N$, $x \in \R$ that:
|
||||
\begin{align}\label{(6.0.3)}
|
||||
g_1 \lp x \rp &= \begin{cases}
|
||||
|
|
@ -97,7 +69,7 @@ One of the most important operators we will approximate is the product operator
|
|||
\end{cases} \\
|
||||
g_{k+1} &= g_1(g_{k}) \nonumber
|
||||
\end{align}
|
||||
and let $f_k: \lb 0,1 \rb \rightarrow \lb 0,1 \rb$, $k \in \N_0$ be the functions satisfying for all $k \in \N_0$, $n \in \{0,1,\hdots,2^k-1\}$, $x \in \lb \frac{n}{2^k}, \frac{n+1}{2^k} \rp$ that $f_k(1)=1$ and:
|
||||
and let $f_k: \lb 0,1 \rb \rightarrow \lb 0,1 \rb$, $k \in \N_0$ be the functions satisfying for all $k \in \N_0$, $n \in \{0,1,...,2^k-1\}$, $x \in \lb \frac{n}{2^k}, \frac{n+1}{2^k} \rp$ that $f_k(1)=1$ and:
|
||||
\begin{align}\label{(6.0.4.2)}
|
||||
f_k(x) = \lb \frac{2n+1}{2^k} \rb x-\frac{n^2+n}{2^{2k}}
|
||||
\end{align}
|
||||
|
|
@ -108,7 +80,7 @@ One of the most important operators we will approximate is the product operator
|
|||
\end{bmatrix}= \rect \lp \begin{bmatrix}
|
||||
x \\ x-\frac{1}{2} \\ x-1 \\ x
|
||||
\end{bmatrix} \rp \\
|
||||
r_{k+1} &= \rect \lp A_{k+1}r_k(x) +B \rp \nonumber
|
||||
r_{k+1} &= A_{k+1}r_k(x) \nonumber
|
||||
\end{align}
|
||||
Note that since it is the case that for all $x \in \R$ that $\rect(x) = \max\{x,0\}$, (\ref{(6.0.3)}) and (\ref{(6.0.5)}) shows that it holds for all $x \in \R$ that:
|
||||
\begin{align}\label{6.0.6}
|
||||
|
|
@ -134,7 +106,7 @@ One of the most important operators we will approximate is the product operator
|
|||
\max\{x,0\} & : x \in \R \setminus \lb 0,1\rb
|
||||
\end{cases} \rp
|
||||
\end{align}
|
||||
We prove (\ref{6.0.8}) and (\ref{6.0.9}) by induction. The base base of $k=1$ is proved by (\ref{6.0.6}) and (\ref{6.0.7}) respectively. For the induction step $\N \ni k \rightarrow k+1$ assume there does exist a $k \in \N$ such that for all $x \in \R$ it is the case that:
|
||||
We prove (\ref{6.0.8}) and (\ref{6.0.9}) by induction. The base base of $k=1$ is proved by (\ref{6.0.6}) and (\ref{6.0.7}). For the induction step $\N \ni k \rightarrow k+1$ assume there does exist a $k \in \N$ such that for all $x \in \R$ it is the case that:
|
||||
\begin{align}
|
||||
2r_{1,k}(x) - 4r_{2,k}(x) + 2r_{3,k}(x) = g_k(x)
|
||||
\end{align}
|
||||
|
|
@ -145,7 +117,7 @@ One of the most important operators we will approximate is the product operator
|
|||
\max\{x,0\} &: x \in \R \setminus \lb 0,1 \rb
|
||||
\end{cases}
|
||||
\end{align}
|
||||
Note that then (\ref{(6.0.3)}), (\ref{(6.0.5)}), and (\ref{6.0.6}) then tells us that for all $x \in \R$ it is the case that:
|
||||
Note that then (\ref{(6.0.3)}),(\ref{(6.0.5)}), and (\ref{6.0.6}) then tells us that for all $x \in \R$ it is the case that:
|
||||
\begin{align}\label{6.0.12}
|
||||
g_{k+1}\lp x \rp &= g_1(g_k(x)) = g_1(2r_{1,k}(x)+4r_{2,k}(x) + 2r_{3,k}(x)) \nonumber \\
|
||||
&= 2\rect \lp 2r_{1,k}(x)) + 4r_{2,k} +2r_{3,k}(x) \rp \nonumber \\
|
||||
|
|
@ -188,7 +160,7 @@ One of the most important operators we will approximate is the product operator
|
|||
\end{align}
|
||||
Which then implies for all $k\in \N$, $x \in \lb 0,1\rb$ that it holds that:
|
||||
\begin{align}
|
||||
\left| x^2-\lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp \right| \les 2^{-2k-2}
|
||||
\left\| x^2-\lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp \right\| \les 2^{-2k-2}
|
||||
\end{align}
|
||||
This, in turn, establishes Item (i).
|
||||
|
||||
|
|
@ -232,8 +204,8 @@ One of the most important operators we will approximate is the product operator
|
|||
and let $\Phi \in \neu$ be defined as:
|
||||
\begin{align}
|
||||
\Phi = \begin{cases}\label{def:Phi}
|
||||
\lb \aff_{C_1,0}\bullet \mathfrak{i}_4\rb \bullet \aff_{\mymathbb{e}_4,B} & :M=1 \\
|
||||
\lb \aff_{C_M,0} \bullet \mathfrak{i}_4\rb\bullet \lb \aff_{A_{M-1},0} \bullet \mathfrak{i}_4 \rb \bullet \cdots \bullet \lb \aff_{A_1,B}\bullet \mathfrak{i}_4\rb \bullet \aff_{\mymathbb{e}_4,B} &: M \in \lb 2,\infty \rp \cap \N
|
||||
\lb \aff_{C_1,0}\bullet \mathfrak{i}_4\rb \bullet \aff_{\mymathbb{e}_4,B} & M=1 \\
|
||||
\lb \aff_{C_M,0} \bullet \mathfrak{i}_4\rb\bullet \lb \aff_{A_{M-1},0} \bullet \mathfrak{i}_4 \rb \bullet \cdots \bullet \lb \aff_{A_1,B}\bullet \mathfrak{i}_4\rb \bullet \aff_{\mymathbb{e}_4,B} & M \in \lb 2,\infty \rp \cap \N
|
||||
\end{cases}
|
||||
\end{align}
|
||||
it is then the case that:
|
||||
|
|
@ -249,7 +221,7 @@ One of the most important operators we will approximate is the product operator
|
|||
\begin{proof}
|
||||
Items (i)--(iii) are direct consequences of Lemma \ref{lem:6.1.1}, Items (i)--(iii). Note next the fact that $M = \min \left\{\N \cap \lb \frac{1}{2} \log_2 \lp \ve^{-1}\rp-1\rb,\infty\right\}$ ensures that:
|
||||
\begin{align}
|
||||
&M = \min \left\{ \N \cap \lb \frac{1}{2}\log_2\lp \ve^{-1}\rp-1\rb, \infty\right\}\\ &\ges \min \left\{ \lb\max \left\{ 1,\frac{1}{2}\log_2 \lp\ve^{-1} \rp-1\right\},\infty \rb\right\}\\ &\ges \frac{1}{2}\log_2 \lp \ve^{-1}\rp-1
|
||||
M = \min \left\{ \N \cap \lb \frac{1}{2}\log_2\lp \ve^{-1}\rp-1\rb, \infty\right\} \ges \min \left\{ \lb\max \left\{ 1,\frac{1}{2}\log_2 \lp\ve^{-1} \rp-1\right\},\infty \rb\right\} \ges \frac{1}{2}\log_2 \lp \ve^{-1}\rp-1
|
||||
\end{align}
|
||||
This and Item (v) of Lemma \ref{lem:6.1.1} demonstrate that for all $x\in \lb 0,1\rb$ it then holds that:
|
||||
\begin{align}
|
||||
|
|
@ -278,9 +250,9 @@ One of the most important operators we will approximate is the product operator
|
|||
\end{remark}
|
||||
Now that we have neural networks that perform the squaring operation inside $\lb -1,1\rb$, we may extend to all of $\R$. Note that this neural network representation differs somewhat from the ones in \cite{grohs2019spacetime}.
|
||||
|
||||
\subsection{The $\sqr^{q,\ve}$ Neural Networks and Squares of Real Numbers}
|
||||
\subsection{The $\sqr$ network}
|
||||
\begin{lemma}\label{6.0.3}\label{lem:sqr_network}
|
||||
Let $\delta,\epsilon \in (0,\infty)$, $\alpha \in (0,\infty)$, $q\in (2,\infty)$, $ \Phi \in \neu$ satisfy that $\delta = 2^{\frac{-2}{q-2}}\ve ^{\frac{q}{q-2}}$, $\alpha = \lp \frac{\ve}{2}\rp^{\frac{1}{q-2}}$, $\real_{\rect}\lp\Phi\rp \in C\lp \R,\R\rp$, $\dep(\Phi) \les \max \left\{\frac{1}{2} \log_2(\delta^{-1})+1,2\right\}$, $\param(\Phi) \les \max\left\{10\log_2\lp \delta^{-1}\rp \right.\\\left. -7,13\right\}$, $\sup_{x \in \R \setminus [0,1]} | \lp \real_{\rect} \lp \Phi \rp -\rect(x) \right| =0$, and $\sup_{x\in \lb 0,1\rb} |x^2-\lp \real_{\rect} \lp \Phi \rp \rp \lp x\rp | \les \delta$, let $\Psi \in \neu$ be the neural network given by:
|
||||
Let $\delta,\epsilon \in (0,\infty)$, $\alpha \in (0,\infty)$, $q\in (2,\infty)$, $ \Phi \in \neu$ satisfy that $\delta = 2^{\frac{-2}{q-2}}\ve ^{\frac{q}{q-2}}$, $\alpha = \lp \frac{\ve}{2}\rp^{\frac{1}{q-2}}$, $\real{\rect}\lp\Phi\rp \in C\lp \R,\R\rp$, $\dep(\Phi) \les \max \left\{\frac{1}{2} \log_2(\delta^{-1})+1,2\right\}$, $\param(\Phi) \les \max\left\{10\log_2\lp \delta^{-1}\rp-7,13\right\}$, $\sup_{x \in \R \setminus [0,1]} | \lp \real_{\rect} \lp \Phi \rp -\rect(x) \right| =0$, and $\sup_{x\in \lb 0,1\rb} |x^2-\lp \real_{\rect} \lp \Phi \rp \rp \lp x\rp | \les \delta$, let $\Psi \in \neu$ be the neural network given by:
|
||||
\begin{align}
|
||||
\Psi = \lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{\alpha,0} \rp \bigoplus\lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{-\alpha,0}\rp
|
||||
\end{align}
|
||||
|
|
@ -301,7 +273,7 @@ Now that we have neural networks that perform the squaring operation inside $\lb
|
|||
&= \frac{1}{\alpha^2}\lp \real_{\rect}\lp \Phi \rp \rp \lp \alpha x\rp + \frac{1}{\alpha^2}\lp \real_{\rect} \lp \Phi \rp \rp \lp -\alpha x\rp \nonumber\\
|
||||
&= \frac{1}{\lp \frac{\ve}{2}\rp^{\frac{2}{q-2}}}\lb \lp \real_{\rect}\lp \Phi \rp \rp \lp \lp \frac{\ve}{2}\rp ^{\frac{1}{q-2}}x \rp + \lp \real_{\rect}\lp \Phi \rp \rp \lp -\lp \frac{\ve}{2}\rp^{\frac{1}{q-2}}x\rp \rb
|
||||
\end{align}
|
||||
This and the assumption that $\Phi \in C\lp \R, \R \rp$ along with the assumption that $\sup_{x\in \R \setminus \lb 0,1\rb } \left| \lp \real_{\rect} \lp \Phi \rp \rp \right.\\ \left.\lp x \rp -\rect\lp x\rp \right| =0$ tells us that for all $x\in \R$ it holds that:
|
||||
This and the assumption that $\Phi \in C\lp \R, \R \rp$ along with the assumption that $\sup_{x\in \R \setminus \lb 0,1\rb } | \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp -\rect\lp x\rp | =0$ tells us that for all $x\in \R$ it holds that:
|
||||
\begin{align}
|
||||
\lp \real_{\rect}\lp \Psi \rp \rp \lp 0 \rp &= \lp \frac{\ve}{2}\rp^{\frac{-2}{q-2}}\lb \lp \real_{\rect}\lp \Phi \rp \rp \lp 0 \rp +\lp \real_{\rect} \lp \Phi\rp \rp \lp 0 \rp \rb \nonumber \\
|
||||
&=\lp \frac{\ve}{2}\rp ^{\frac{-2}{q-2}} \lb \rect (0)+\rect(0) \rb \nonumber \\
|
||||
|
|
@ -332,7 +304,7 @@ Now that we have neural networks that perform the squaring operation inside $\lb
|
|||
&= \lp \frac{\ve}{2}+ \frac{\ve}{2} \rp \left| x \right|^q = \ve \left| x \right|^q \les \ve \max \left\{ 1, \left| x \right|^q \right\}
|
||||
\end{align}
|
||||
|
||||
Note that (\ref{6.0.24}), (\ref{6.0.21}) and the fact that $\delta = 2^{\frac{-2}{q-2}}\ve^{\frac{q}{q-2}}$ then tell for all $x \in \\ \lb -\lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}}, \lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}} \rb$ it holds that:
|
||||
Note that (\ref{6.0.24}), (\ref{6.0.21}) and the fact that $\delta = 2^{\frac{-2}{q-2}}\ve^{\frac{q}{q-2}}$ then tell for all $x \in \lb -\lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}}, \lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}} \rb$ it holds that:
|
||||
\begin{equation}
|
||||
\begin{aligned}\label{6.0.26}
|
||||
% &\left| x^2-\lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp \right| \\
|
||||
|
|
@ -404,14 +376,7 @@ This, and the fact that $\delta = 2^{\frac{-2}{q-2}}\ve ^{\frac{q}{q-2}}$ render
|
|||
\centering
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Sqr_properties/experimental_deps.png}
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Sqr_properties/dep_theoretical_upper_limits.png}
|
||||
\caption{Left: $\log_{10}$ of depths for a simulation with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points for $\sqr^{q,\ve}$. Right: The theoretical upper limits over the same range of values}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Sqr_properties/experimental_params.png}
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Sqr_properties/param_theoretical_upper_limits.png}
|
||||
\caption{Left: $\log_{10}$ of params for a simulation with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points for $\sqr^{q,\ve}$. Right: The theoretical upper limits over the same range of values}
|
||||
\caption{Left: $\log_{10}$ of depths for a simulation with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points. Right: The theoretical upper limits over the same range of values}
|
||||
\end{figure}
|
||||
|
||||
% Please add the following required packages to your document preamble:
|
||||
|
|
@ -421,32 +386,16 @@ This, and the fact that $\delta = 2^{\frac{-2}{q-2}}\ve ^{\frac{q}{q-2}}$ render
|
|||
\begin{tabular}{@{}l|llllll@{}}
|
||||
\toprule
|
||||
& Min. & 1\textsuperscript{st} Qu. & Median & Mean & 3\textsuperscript{rd} Qu. & Max. \\ \midrule
|
||||
Experimental $|x^2 - \real_{\rect}(\mathsf{Sqr}^{q,\ve})(x)$ & 0.00000 & 0.08943 & 0.33787 & 3.14893 & 4.67465 & 20.00 \\ \midrule
|
||||
Theoretical $|x^2 - \real_{\rect}(\mathsf{Sqr}^{q,\ve})(x)$ & 0.010 & 1.715 & 10.402 & 48.063 & 45.538 & 1250.00 \\ \midrule
|
||||
\textbf{Forward Difference} & 0.01 & 1.6012 & 9.8655 & 44.9141 & 40.7102 & 1230
|
||||
Experimental $|x^2 - \real_{\rect}(\mathsf{Sqr}^{q,\ve})(x)$ & 0.000003 & 0.089438 & 0.337870 & 3.148933 & 4.674652 & 20.00 \\ \midrule
|
||||
Theoretical $|x^2 - \real_{\rect}(\mathsf{Sqr})^{q,\ve}(x)$ & 0.010 & 1.715 & 10.402 & 48.063 & 45.538 & 1250.00 \\ \midrule
|
||||
Difference & 0.001 & 1.6012 & 9.8655 & 44.9141 & 40.7102 & 1230
|
||||
\end{tabular}
|
||||
\caption{Theoretical upper bounds for $L^1$ error, experimental $L^1$ error and their forward difference, with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points for $\sqr^{q,\ve}$.}
|
||||
\caption{Theoretical upper bounds for $L^1$ error, experimental $L^1$ error and their forward difference, with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points.}
|
||||
\end{table}
|
||||
|
||||
\begin{table}[h]
|
||||
\begin{tabular}{l|llllll}
|
||||
\hline
|
||||
& Min & 1st. Qu & Median & Mean & 3rd Qu & Max. \\ \hline
|
||||
Experimental \\ $|x^2 - \inst_{\rect}\lp \sqr^{q,\ve}\rp(x)|$ & 0.0000 & 0.0894 & 0.3378 & 3.1489 & 4.6746 & 20.0000 \\ \hline
|
||||
Theoretical upper limits for\\ $|x^2 - \mathfrak{I}_{\mathfrak{r}}(\mathsf{Sqr}^{q,\ve})(x)$ & 0.010 & 1.715 & 10.402 & 48.063 & 45.538 & 1250.000 \\ \hline
|
||||
\textbf{Forward Difference} & 0.001 & 1.6012 & 9.8655 & 44.9141 & 40.7102 & 1230 \\ \hline
|
||||
Experimental depths & 2 & 2 & 2 & 2.307 & 2 & 80 \\ \hline
|
||||
Theoretical upper bound on\\ depths & 2 & 2 & 2 & 2.73 & 2 & 91 \\ \hline
|
||||
\textbf{Forward Difference} & 0 & 0 & 0 & 0.423 & 0 & 11 \\ \hline
|
||||
Experimental params & 25 & 25 & 25 & 47.07 & 25 & 5641 \\ \hline
|
||||
Theoretical upper limit on \\ params & 52 & 52 & 52 & 82.22 & 52 & 6353 \\ \hline
|
||||
\textbf{Forward Differnce} & 27 & 27 & 27 & 35.16 & 27 & 712 \\ \hline
|
||||
\end{tabular}
|
||||
\caption{Table showing the experimental and theoretical $1$-norm difference, depths, and parameter counts respectively for $\sqr^{q,\ve}$ with $q\in [2.1,4]$, $\ve \in [0.01,2]$, and $x \in [-5,5]$ all with $50$ mesh-points, and their forward differences.}
|
||||
\end{table}
|
||||
|
||||
\subsection{The $\prd^{q,\ve}$ Neural Networks and Products of Two Real Numbers}
|
||||
We are finally ready to give neural network representations of arbitrary products of real numbers. However, this representation differs somewhat from those found in the literature, especially \cite{grohs2019spacetime}, where parallelization (stacking) is used instead of neural network sums. This will help us calculate $\wid_1$ and the width of the second to last layer for later neural network calculations.
|
||||
\subsection{The $\prd$ network}
|
||||
We are finally ready to give neural network representations of arbitrary products of real numbers. However, this representation differs somewhat from those found in the literature, especially \cite{grohs2019spacetime}, where parallelization (stacking) is used instead of neural network sums. This will help us calculate $\wid_1$ and the width of the second to last layer.
|
||||
\begin{lemma}\label{prd_network}
|
||||
Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, $A_1,A_2,A_3 \in \R^{1\times 2}$, $\Psi \in \neu$ satisfy for all $x\in \R$ that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, $A_1 = \lb 1 \quad 1 \rb$, $A_2 = \lb 1 \quad 0 \rb$, $A_3 = \lb 0 \quad 1 \rb$, $\real_{\rect} \in C\lp \R, \R \rp$, $\lp \real_{\rect} \lp \Psi \rp \rp \lp 0\rp = 0$, $0\les \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \les \delta+|x|^2$, $|x^2-\lp \real_{\rect}\lp \Psi \rp \rp \lp x \rp |\les \delta \max \{1,|x|^q\}$, $\dep\lp \Psi \rp \les \max\{ 1+\frac{1}{q-2}+\frac{q}{2(q-2)}\log_2 \lp \delta^{-1} \rp ,2\}$, and $\param \lp \Psi \rp \les \max\left\{\lb \frac{40q}{q-2} \rb \log_2\lp \delta^{-1} \rp +\frac{80}{q-2}-28,52\right\}$, then:
|
||||
\begin{enumerate}[label=(\roman*)]
|
||||
|
|
@ -527,7 +476,7 @@ We are finally ready to give neural network representations of arbitrary product
|
|||
\end{align}
|
||||
This proves Item (iv).
|
||||
|
||||
By symmetry it holds that $\param \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp \\ = \param \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_2,0} \rp \rp = \param \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_3,0} \rp \rp$ and further that $\lay \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp = \lay \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_2,0} \rp \rp = \lay \lp -\frac{1}{2}\triangleright\lp \Psi \bullet \aff_{A_3,0} \rp \rp$.
|
||||
By symmetry it holds that $\param \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp = \param \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_2,0} \rp \rp = \param \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_3,0} \rp \rp$ and further that $\lay \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp = \lay \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_2,0} \rp \rp = \lay \lp -\frac{1}{2}\triangleright\lp \Psi \bullet \aff_{A_3,0} \rp \rp$.
|
||||
Note also that Corollary \ref{affcor} tells us that for all $i \in \{1,2,3\}$ and $a \in \{ \frac{1}{2},-\frac{1}{2}\}$ it is the case that:
|
||||
\begin{align}
|
||||
\param \lp a \triangleright \lp \Psi \bullet \aff_{A_i,0}\rp \rp = \param \lp \Psi \rp
|
||||
|
|
@ -604,12 +553,12 @@ Observe next that for $q\in \lp 0,\infty\rp$, $\ve \in \lp 0,\infty \rp$, $\Gamm
|
|||
We shall refer to this neural network for a given $q \in \lp 2,\infty \rp$ and given $\ve \in \lp 0,\infty \rp$ from now on as $\prd^{q,\ve}$.
|
||||
\end{remark}
|
||||
\begin{remark}
|
||||
For an \texttt{R} implementation see Listing \ref{Pwr}
|
||||
For an \texttt{R} implementation see Listing \ref{Prd}
|
||||
\end{remark}
|
||||
\begin{remark}
|
||||
Diagrammatically, this can be represented as:
|
||||
\end{remark}
|
||||
\begin{figure}[h]
|
||||
\begin{figure}
|
||||
\begin{center}
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
|
|
@ -665,32 +614,11 @@ Observe next that for $q\in \lp 0,\infty\rp$, $\ve \in \lp 0,\infty \rp$, $\Gamm
|
|||
\end{tikzpicture}
|
||||
|
||||
\end{center}
|
||||
\caption{Neural network diagram of the $\prd^{q,\ve}$ network.}
|
||||
\caption{A neural network diagram of the $\sqr$. }
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Prd_properties/experimental_deps.png}
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Prd_properties/dep_theoretical_upper_limits.png}
|
||||
\caption{Left: $\log_{10}$ of deps for a simulation of $\prd^{q,\ve}$ with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points. Right: The theoretical upper limits over the same range of values.}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Prd_properties/experimental_params.png}
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Prd_properties/param_theoretical_upper_limits.png}
|
||||
\caption{Left: $\log_{10}$ of params for a simulation of $\prd^{q,\ve}$ with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points. Right: The theoretical upper limits over the same range of values.}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width = \linewidth]{/Users/shakilrafi/R-simulations/Sqr_properties/iso.png}
|
||||
\caption{Isosurface plot showing $|x^2 - \sqr^{q,\ve}|$ for $q \in [2.1,4]$, $\ve \in [0.01,2]$, and $x \in [-5,5]$ with 50 mesh-points in each.}
|
||||
\end{figure}
|
||||
|
||||
\section{Higher Approximations}\label{sec_tun}
|
||||
We take inspiration from the $\sm$ neural network to create the $\prd$ neural network. However, we first need to define a special neural network called \textit{tunneling neural network} to stack two neural networks not of the same length effectively.
|
||||
\subsection{The $\tun^d_n$ Neural Networks and Their Properties}
|
||||
\subsection{The $\tun$ Neural Networks and Their Properties}
|
||||
\begin{definition}[R\textemdash,2023, The Tunneling Neural Networks]\label{def:tun}
|
||||
We define the tunneling neural network, denoted as $\tun_n$ for $n\in \N$ by:
|
||||
\begin{align}
|
||||
|
|
@ -760,7 +688,7 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
\end{align}
|
||||
And that by definition of composition:
|
||||
\begin{align}
|
||||
&\param \lp \tun_3 \rp \\ &= \param \lb \lp \lp \begin{bmatrix}
|
||||
\param \lp \tun_3 \rp &= \param \lb \lp \lp \begin{bmatrix}
|
||||
1 \\ -1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0
|
||||
|
|
@ -794,8 +722,8 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
\end{align}
|
||||
Now for the inductive step assume that for all $n\les N\in \N$, it is the case that $\param\lp \tun_n \rp = 7+6(n-2)$. For the inductive step, we then have:
|
||||
\begin{align}
|
||||
&\param \lp \tun_{n+1} \rp = \param \lp \tun_n \bullet \id_1 \rp =\nonumber\\
|
||||
&\param \lb \lp \lp \begin{bmatrix}
|
||||
&\param \lp \tun_{n+1} \rp = \param \lp \tun_n \bullet \id_1 \rp \nonumber\\
|
||||
&=\param \lb \lp \lp \begin{bmatrix}
|
||||
1 \\ -1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0
|
||||
|
|
@ -807,8 +735,8 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
1 & -1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0
|
||||
\end{bmatrix}\rp \rp \bullet \id_1 \rb =\nonumber \\
|
||||
&\param \lb \lp \lp \begin{bmatrix}
|
||||
\end{bmatrix}\rp \rp \bullet \id_1 \rb \nonumber \\
|
||||
&= \param \lb \lp \lp \begin{bmatrix}
|
||||
1 \\ -1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0
|
||||
|
|
@ -879,8 +807,8 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
1 & -1\\ & &\ddots \\ & & & 1 & -1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ \vdots \\ 0
|
||||
\end{bmatrix}\rp \rp \rb =\nonumber \\
|
||||
&\param \lb \lp \lp \begin{bmatrix}
|
||||
\end{bmatrix}\rp \rp \rb \nonumber \\
|
||||
&= \param \lb \lp \lp \begin{bmatrix}
|
||||
1 \\ -1 \\ & \ddots \\ & & 1 \\ & &-1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0 \\\vdots \\ 0\\0
|
||||
|
|
@ -888,7 +816,7 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
1 & -1 \\ -1 & 1 \\ & & \ddots \\ & & & 1 & -1 \\ & & & -1 & 1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0 \\ \vdots \\ 0 \\ 0
|
||||
\end{bmatrix}\rp, \right. \right.\\ & \left.\left. \lp \begin{bmatrix}
|
||||
\end{bmatrix}\rp, \lp \begin{bmatrix}
|
||||
1 &-1 \\ & &\ddots \\ & & & 1 & -1
|
||||
\end{bmatrix},\begin{bmatrix}
|
||||
0 \\ \vdots \\ 0
|
||||
|
|
@ -901,7 +829,7 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
|
||||
\begin{align}
|
||||
& \param\lp \tun^d_{n+1}\rp = \param \lp \tun_n^d \bullet \id_d\rp \nonumber \\
|
||||
= &\param \lb \lp \begin{bmatrix}
|
||||
& = \param \lb \lp \begin{bmatrix}
|
||||
1 \\ -1 \\ & \ddots \\ & & 1 \\ & &-1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0 \\ \vdots \\ 0 \\ 0
|
||||
|
|
@ -909,12 +837,13 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
1 & -1 \\ -1 & 1 \\ & \ddots \\ & & 1 & -1 \\ & & -1 & 1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0 \\ \vdots \\ 0 \\ 0
|
||||
\end{bmatrix} \rp, \hdots, \right. \\ &\left. \lp \begin{bmatrix}
|
||||
\end{bmatrix} \rp, \hdots, \lp \begin{bmatrix}
|
||||
1 &-1 \\ & \ddots \\ & & 1 & -1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ \vdots \\ 0
|
||||
\end{bmatrix}\rp \bullet \id_d \rb \nonumber\\
|
||||
= &\param \lb \lp \begin{bmatrix}
|
||||
\end{bmatrix}\rp \right. \nonumber \\
|
||||
& \left. \bullet \id_d \rb \nonumber\\
|
||||
& = \param \lb \lp \begin{bmatrix}
|
||||
1 \\ -1 \\ & \ddots \\ & & 1 \\ & &-1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0 \\ \vdots \\ 0 \\ 0
|
||||
|
|
@ -922,11 +851,11 @@ We take inspiration from the $\sm$ neural network to create the $\prd$ neural ne
|
|||
1 & -1 \\ -1 & 1 \\ & \ddots \\ & & 1 & -1 \\ & & -1 & 1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0 \\ \vdots \\ 0 \\ 0
|
||||
\end{bmatrix} \rp, \hdots, \right.\\ &\left.\lp \begin{bmatrix}
|
||||
\end{bmatrix} \rp, \hdots, \lp \begin{bmatrix}
|
||||
1 & -1 \\ -1 & 1 \\ & \ddots \\ & & 1 & -1 \\ & & -1 & 1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ 0 \\ \vdots \\ 0 \\ 0
|
||||
\end{bmatrix} \rp, \lp \begin{bmatrix}
|
||||
\end{bmatrix} \rp, \right. \nonumber\\ &\left. \lp \begin{bmatrix}
|
||||
1 &-1 \\ & \ddots \\ & & 1 & -1
|
||||
\end{bmatrix}, \begin{bmatrix}
|
||||
0 \\ \vdots \nonumber\\ 0
|
||||
|
|
@ -943,7 +872,7 @@ This proves Item (iv). Finally, Item (v) is a consequence of Lemma \ref{5.3.2}
|
|||
|
||||
|
||||
|
||||
\subsection{The $\pwr_n^{q,\ve}$ Neural Networks and Their Properties}
|
||||
\subsection{The $\pwr$ Neural Networks and Their Properties}
|
||||
|
||||
\begin{definition}[R\textemdash, 2023, The Power Neural Network]\label{def:pwr}
|
||||
Let $n\in \N$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. We define the power neural networks $\pwr_n^{q,\ve} \in \neu$, denoted for $n\in \N_0$ as:
|
||||
|
|
@ -1085,7 +1014,7 @@ Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for
|
|||
\end{align}
|
||||
This, with the fact that the composition of continuous functions is continuous, the fact the stacking of continuous instantiated neural networks is continuous tells us that $\lp \real_{\rect} \pwr_n \rp \in C \lp \R, \R \rp$ for $n \in \N \cap \lb 2,\infty \rp$. This establishes Item (i).
|
||||
|
||||
Note next that by observation $\dep \lp \pwr_0^{q,\ve} \rp=1$ and by Item (iv) of Lemma \ref{idprop}, it is the case that $\dep\lp \id_1 \rp = 2$. By Lemmas $\ref{dep_cpy}$ and $\ref{depthofcomposition}$ it is also the case that\\ $\dep\lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr^{q,\ve}_{n-1})} \boxminus \pwr^{q,\ve}_{n-1} \rb \bullet \cpy \rp = \dep \lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr^{q,\ve}_{n-1})} \boxminus \pwr^{q,\ve}_{n-1} \rb\rp $. Note also that by Lemma we have that $\dep \lp \tun_{\dep \lp \pwr^{q,\ve}_{n-1}\rp} \boxminus \pwr^{q,\ve}_{n-1}\rp = \dep \lp \pwr^{q,\ve}_{n-1} \rp$.
|
||||
Note next that by observation $\dep \lp \pwr_0^{q,\ve} \rp=1$ and by Item (iv) of Lemma \ref{idprop}, it is the case that $\dep\lp \id_1 \rp = 2$. By Lemmas $\ref{dep_cpy}$ and $\ref{depthofcomposition}$ it is also the case that: $\dep\lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr^{q,\ve}_{n-1})} \boxminus \pwr^{q,\ve}_{n-1} \rb \bullet \cpy \rp = \dep \lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr^{q,\ve}_{n-1})} \boxminus \pwr^{q,\ve}_{n-1} \rb\rp $. Note also that by Lemma we have that $\dep \lp \tun_{\dep \lp \pwr^{q,\ve}_{n-1}\rp} \boxminus \pwr^{q,\ve}_{n-1}\rp = \dep \lp \pwr^{q,\ve}_{n-1} \rp$.
|
||||
This with Lemma \ref{comp_prop} then yields for $n \in \N$ that:
|
||||
\begin{align}
|
||||
\dep \lp \pwr^{q,\ve}_n \rp &= \dep \lp \prd \bullet \lb \tun_{\mathcal{D} \lp \pwr^{q,\ve}_{n-1} \rp } \boxminus \pwr^{q,\ve}_{n-1} \rb \bullet \cpy_{2,1} \rp \nonumber \\
|
||||
|
|
@ -1187,7 +1116,7 @@ Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for
|
|||
\wid_{\hid \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp} \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp = 24+2=26
|
||||
\end{align}
|
||||
|
||||
Finally Lemma \ref{comp_prop}, (\ref{(6.2.34)}), a geometric series argument, and Corollary \ref{cor:sameparal}, also tells us that:
|
||||
Finally Lemma \ref{comp_prop}, (\ref{(6.2.34)}), and Corollary \ref{cor:sameparal}, also tells us that:
|
||||
\begin{align}
|
||||
&\param \lp \pwr_{n}^{q,\ve}\rp\\ &= \param \lp \prd^{q,\ve} \bullet\lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb \bullet \cpy_{2,1}\rp \nonumber \\
|
||||
&= \param \lp \prd^{q,\ve} \bullet \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp \nonumber \\
|
||||
|
|
@ -1199,7 +1128,7 @@ Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for
|
|||
\end{align}
|
||||
|
||||
|
||||
Next note that $\lp \real_{\rect} \lp \pwr_{0,1} \rp\rp \lp x \rp$ is exactly $1$, which implies that for all $x\in \R$ we have that $|x^0-\lp \real_{\rect} \lp \pwr_{0.1}\rp\lp x \rp\rp |=0$. Note also that the instantiations of $\tun_n$ and $\cpy_{2,1}$ are exact. Note next that since $\tun_n$ and $\cpy_{2,1}$ are exact, the only sources of error for $\pwr^{q,\ve}_n$ are $n$ compounding applications of $\prd^{q,\ve}$.
|
||||
Next note that $\lp \real_{\rect} \lp \pwr_{0,1} \rp\rp \lp x \rp$ is exactly $1$, which implies that for all $x\in \R$ we have that $|x^0-\lp \real_{\rect} \lp \pwr_{0.1}\rp\lp x \rp\rp |=0$. Note also that the instantiations of $\tun_n$ and $\cpy_{2,1}$ are exact. Note next that since $\tun_n$ and $\cpy_{2,1}$ are exact, the only sources of error for $\pwr^{q,\ve}_n$ a are $n$ compounding applications of $\prd^{q,\ve}$.
|
||||
|
||||
Note also that by definition, it is the case that:
|
||||
\begin{align}
|
||||
|
|
@ -1243,30 +1172,8 @@ Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for
|
|||
This completes the proof of the lemma.
|
||||
\end{proof}
|
||||
\begin{remark}\label{rem:pwr_gets_deeper}
|
||||
Note each power network $\pwr_n^{q,\ve}$ is at least as deep and parameter-rich as the previous power network $\pwr_{n-1}^{q,\ve}$, one differs from the next by one $\prd^{q, \ve}$ network.
|
||||
Note each power network $\pwr_n^{q,\ve}$ is at least as big as the previous power network $\pwr_{n-1}^{q,\ve}$, one differs from the other by one $\prd^{q, ve}$ network.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Pwr_3_properties/experimental_deps.png}
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Pwr_3_properties/dep_theoretical_upper_limits.png}
|
||||
\caption{Left: $\log_{10}$ of depths for a simulation of $\pwr_3^{q,\ve}$ with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points. Right: The theoretical upper limits over the same range of values}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Pwr_3_properties/experimental_params.png}
|
||||
\includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Pwr_3_properties/param_theoretical_upper_limits.png}
|
||||
\caption{Left: $\log_{10}$ of params for a simulation of $\pwr_3^{q,\ve}$ with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points. Right: The theoretical upper limits over the same range of values}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width = \linewidth]{/Users/shakilrafi/R-simulations/Pwr_3_properties/isosurface.png}
|
||||
\caption{Isosurface plot showing $|x^3 - \real_{\rect}(\pwr^{q,\ve}_3)(x)|$ for $q \in [2.1,4]$, $\ve \in [0.01,2]$, and $x \in [-5,5]$ with 50 mesh-points in each.}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{$\pnm_{n,C}^{q,\ve}$ and Neural Network Polynomials.}
|
||||
|
||||
\begin{definition}[Neural Network Polynomials]
|
||||
|
|
@ -1364,37 +1271,22 @@ Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for
|
|||
% Text Node
|
||||
\draw (525,162.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{n+1,1}$};
|
||||
% Text Node
|
||||
\draw (471.33,198.4) node [anchor=north west][inner sep=0.75pt] {$\vdots$};
|
||||
\draw (471.33,198.4) node [anchor=north west][inner sep=0.75pt] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (83,163.73) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}_{n+1,1}$};
|
||||
% Text Node
|
||||
\draw (230.67,217.7) node [anchor=north west][inner sep=0.75pt] {$\vdots$};
|
||||
\draw (230.67,214.4) node [anchor=north west][inner sep=0.75pt] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (172,198.4) node [anchor=north west][inner sep=0.75pt] {$\vdots$};
|
||||
\draw (172,193.73) node [anchor=north west][inner sep=0.75pt] {$\vdots $};
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\caption{Neural network diagram for an elementary neural network polynomial, with all coefficients being uniformly $1$.}
|
||||
\caption{Neural network diagram for an elementary neural network polynomial.}
|
||||
\end{figure}
|
||||
|
||||
\begin{table}[h]
|
||||
\begin{tabular}{l|llllll}
|
||||
\hline & Min & 1st. Qu & Median & Mean & 3rd Qu & Max. \\ \hline
|
||||
Experimental \\ $|x^3 - \inst_{\rect}\lp \pwr^{q,\ve}\rp(x)|$ & 0.0000 & 0.2053 & 7.2873 & 26.7903 & 45.4275 & 125.00 \\ \hline
|
||||
Experimental depths & 4 & 4 & 4 & 4.92 & 4 & 238 \\ \hline
|
||||
Theoretical upper bound on\\ depths & 4.30 & 17.82 & 23.91 & 25.80 & 29.63 & 548.86 \\ \hline
|
||||
\textbf{Forward Difference} & 0.30 & 13.82 & 19.91 & 20.88 & 25.63 & 310.86 \\ \hline
|
||||
Experimental params & 1483 & 1483 & 1483 & 1546 & 1483 & 5711 \\ \hline
|
||||
Theoretical upper limit on \\ params & 9993 & 9993 & 9993 & 11589 & 9993 & 126843 \\ \hline
|
||||
\textbf{Forward Differnce} & 8510 & 8510 & 8510 & 10043 & 8510 & 121132 \\ \hline
|
||||
\end{tabular}
|
||||
\caption{Table showing the experimental and theoretical $1$-norm difference, depths, and parameter counts respectively for $\pwr_3^{q,\ve}$ with $q\in [2.1,4]$, $\ve \in [0.01,2]$, and $x \in [-5,5]$ all with $50$ mesh-points, and their forward differences.}
|
||||
\end{table}
|
||||
|
||||
|
||||
\begin{lemma}[R\textemdash,2023]\label{6.2.9}\label{nn_poly}\label{mnm_prop}
|
||||
Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$ and let $C = \{ c_1,c_2,\hdots c_n\} \in \R^n$ be a set of real numbers, i.e. the set of coefficients. It is then the case for all $n\in\N_0$ and $x\in \R$ that:
|
||||
Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. It is then the case for all $n\in\N_0$ and $x\in \R$ that:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\real_{\rect} \lp \pnm_{n,C}^{q,\ve}\rp \in C \lp \R, \R \rp $
|
||||
\item $\dep \lp \pnm_{n,C}^{q,\ve} \rp \les \begin{cases}
|
||||
|
|
@ -1422,7 +1314,7 @@ Theoretical upper limit on \\ params & 9993 & 9993 & 9993 & 11589
|
|||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Note that by Lemma \ref{5.6.3}, Lemma \ref{power_prop}, and Lemma \ref{comp_prop} indicate for all $n\in \N_0$ it is the case that:
|
||||
Note that by Lemma \ref{5.6.3}, Lemma \ref{power_prop}, and Lemma \ref{comp_prop} for all $n\in \N_0$ it is the case that:
|
||||
\begin{align}
|
||||
\real_{\rect}\lp \pnm_{n,C}^{q,\ve} \rp &= \real_{\rect} \lp \bigoplus^n_{i=0} \lb c_i \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb \rp \nonumber\\
|
||||
&= \sum^n_{i=1}c_i \real_{\rect}\lp \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve} \rp \nonumber\\
|
||||
|
|
@ -1446,7 +1338,7 @@ Theoretical upper limit on \\ params & 9993 & 9993 & 9993 & 11589
|
|||
\end{align}
|
||||
This then yields us $2$ parameters.
|
||||
|
||||
Note that each neural network summand in $\pnm_n^{q,\ve}$ consists of a combination of $\tun_k$ and $\pwr_k$ for some $k\in \N$. Each $\pwr_k$ has at least as many parameters as a tunneling neural network of that depth, as Lemma \ref{param_pwr_geq_param_tun} tells us. This, finally, with Lemma \ref{aff_effect_on_layer_architecture}, Corollary \ref{affcor}, and Lemma \ref{power_prop} then implies that: \\
|
||||
Note that each neural network summand in $\pnm_n^{q,\ve}$ consists of a combination of $\tun_k$ and $\pwr_k$ for some $k\in \N$. Each $\pwr_k$ has at least as many parameters as a tunneling neural network of that depth, as Lemma \ref{param_pwr_geq_param_tun} tells us. This, finally, with Lemma \ref{aff_effect_on_layer_architecture}, Corollary \ref{affcor}, and Lemma \ref{power_prop} then implies that:
|
||||
\begin{align}
|
||||
\param\lp \pnm^{q,\ve}_{n,C} \rp &= \param \lp \bigoplus^n_{i=0} \lb c_i \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb \rp\nonumber \\
|
||||
&\les \lp n+1 \rp \cdot \param \lp c_i \triangleright \lb \tun_1 \bullet \pwr_n^{q,\ve} \rb\rp \nonumber\\
|
||||
|
|
@ -1493,11 +1385,8 @@ Theoretical upper limit on \\ params & 9993 & 9993 & 9993 & 11589
|
|||
\end{align}
|
||||
This completes the proof of the Lemma.
|
||||
\end{proof}
|
||||
\begin{remark}
|
||||
Note that we will implement this in R as the so-called \texttt{Tay} function. Our implementations of neural network exponentials, cosines, and sines will be instantiations of this \texttt{Tay} function with the appropriate coefficients and exponents being replaced to give the appropriate Taylor expansions.
|
||||
\end{remark}
|
||||
\subsection{$\xpn_n^{q,\ve}$, $\csn_n^{q,\ve}$, $\sne_n^{q,\ve}$, and ANN Approximations of $e^x$, $\cos(x)$, and $\sin(x)$.}
|
||||
Once we have neural network polynomials, we may take the next leap to transcendental functions. For approximating them we will use Taylor expansions which will swiftly give us our approximations for our desired functions. Here, we will explore neural network approximations for three common transcendental functions: $e^x$, $\cos(x)$, and $\sin(x)$.
|
||||
\subsection{$\xpn_n^{q,\ve}$, $\csn_n^{q,\ve}$, $\sne_n^{q,\ve}$, and Neural Network Approximations of $e^x$, $\cos(x)$, and $\sin(x)$.}
|
||||
Once we have neural network polynomials, we may take the next leap to transcendental functions. Here, we will explore neural network approximations for three common transcendental functions: $e^x$, $\cos(x)$, and $\sin(x)$.
|
||||
|
||||
\begin{lemma}
|
||||
Let $\nu_1,\nu_2 \in \neu$, $f,g \in C \lp \R, \R \rp$, and $\ve_1,\ve_2 \in \lp 0 ,\infty \rp$ such that for all $x\in \R$ it holds that $\left| f(x) - \real_{\rect} \lp \nu_1 \rp \right| \les \ve_1 $ and $\left| g(x) - \real_{\rect} \lp \nu_2 \rp \right| \les \ve_2$. It is then the case for all $x \in \R$ that:
|
||||
|
|
@ -1514,7 +1403,7 @@ Once we have neural network polynomials, we may take the next leap to transcende
|
|||
\end{align}
|
||||
\end{proof}
|
||||
\begin{lemma}\label{6.2.8}
|
||||
Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$, $\ve_1,\ve_2,...,\ve_n \in \lp 0,\infty \rp$ and $f_1,f_2,...,f_n \in C\lp \R, \R \rp$ such that for all $i \in \{1,2,...,n\}$, and for all $x\in \R$, it is the case that,\\ $\left| f_i\lp x \rp - \real_{\rect} \lp \nu_i \rp\lp x \rp \right| \les \ve_i$. It is then the case for all $x\in \R$, that:
|
||||
Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$, $\ve_1,\ve_2,...,\ve_n \in \lp 0,\infty \rp$ and $f_1,f_2,...,f_n \in C\lp \R, \R \rp$ such that for all $i \in \{1,2,...,n\}$, and for all $x\in \R$, it is the case that, $\left| f_i\lp x \rp - \real_{\rect} \lp \nu_i \rp\lp x \rp \right| \les \ve_i$. It is then the case for all $x\in \R$, that:
|
||||
\begin{align}
|
||||
\left| \sum^n_{i=1} f_i \lp x \rp -\bigoplus^n_{i=1} \lp \real_{\rect}\lp \nu_i \rp \rp \lp x\rp\right| \les \sum_{i=1}^n \ve_i
|
||||
\end{align}
|
||||
|
|
@ -1523,7 +1412,7 @@ Once we have neural network polynomials, we may take the next leap to transcende
|
|||
This is a consequence of a finite number of applications of (\ref{6.2.14}).
|
||||
\end{proof}
|
||||
\begin{definition}[R\textemdash 2023, $\xpn_n^{q,\ve}$ and the Neural Network Taylor Approximations for $e^x$ around $x=0$]
|
||||
Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, and let $\pwr_n^{q,\ve} \subsetneq \neu$ be as in Lemma \ref{power_prop}. We define, for all $n\in \N_0$, the family of neural networks $\xpn_n^{q,\ve} as$:
|
||||
Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, and let $\pwr_n^{q,\ve}$ be as in Lemma \ref{power_prop}. We define, for all $n\in \N_0$, the family of neural networks $\xpn_n^{q,\ve} as$:
|
||||
\begin{align}
|
||||
\xpn_n^{q,\ve}\coloneqq \bigoplus^n_{i=0} \lb \frac{1}{i!} \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb
|
||||
\end{align}
|
||||
|
|
@ -1606,7 +1495,7 @@ Once we have neural network polynomials, we may take the next leap to transcende
|
|||
2 & :n =0 \\
|
||||
\lp 2n+1\rp\lb 4^{2n+\frac{3}{2}} + \lp \frac{4^{2n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N
|
||||
\end{cases}$ \\~\\
|
||||
\item $\left|\sum^n_{i=0} \frac{(-1)^i}{2i!}x^{2i} - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| \\ \les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| x \lp x^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{2i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{2i-1}^q \rp $\\~\\
|
||||
\item $\left|\sum^n_{i=0} \frac{(-1)^i}{2i!}x^{2i} - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| \les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| x \lp x^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{2i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{2i-1}^q \rp $\\~\\
|
||||
Where $\mathfrak{p}_i$ are the set of functions defined for $i \in \N$ as such:
|
||||
\begin{align}
|
||||
\mathfrak{p}_1 &= \ve+1+|x|^2 \nonumber\\
|
||||
|
|
@ -1663,7 +1552,7 @@ Once we have neural network polynomials, we may take the next leap to transcende
|
|||
\begin{lemma}[R\textemdash, 2023]
|
||||
Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}.$ It is then the case for all $n\in\N_0$ and $x\in [a,b]\subseteq \lb 0,\infty \rp$ that:
|
||||
\begin{align}
|
||||
&\left| \cos\lp x\rp - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right|\\ &\les \sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}\lp \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{n-1}^q \rp + + \frac{|x|^{n+1}}{(n+1)!}\nonumber
|
||||
\left| \cos\lp x\rp - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| \les \sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}\lp \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{n-1}^q \rp + + \frac{|x|^{n+1}}{(n+1)!}\nonumber
|
||||
\end{align}
|
||||
\end{lemma}
|
||||
|
||||
|
|
@ -1736,14 +1625,12 @@ Once we have neural network polynomials, we may take the next leap to transcende
|
|||
&\left| \sin\lp x\rp - \inst_{\rect}\lp \sne_n^{q,\ve}\rp\right| \nonumber\\
|
||||
&= \left| \cos \lp x - \frac{\pi}{2}\rp - \inst_{\rect}\lp \csn_n^{q,\ve}\bullet \aff_{1,-\frac{\pi}{2}}\rp\lp x\rp\right| \nonumber\\
|
||||
&=\left| \cos \lp x-\frac{x}{2}\rp - \inst_{\rect}\csn_n^{q,\ve}\lp x-\frac{\pi}{2} \rp\right| \nonumber \\
|
||||
&\les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| \lp x -\frac{\pi}{2}\rp\lp \lp x -\frac{\pi}{2}\rp^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x-\frac{\pi}{2}\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp \nonumber\\&+ \frac{|x|^{n+1}}{(n+1)!}\nonumber
|
||||
&\les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| \lp x -\frac{\pi}{2}\rp\lp \lp x -\frac{\pi}{2}\rp^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x-\frac{\pi}{2}\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp+ \frac{|x|^{n+1}}{(n+1)!}\nonumber
|
||||
\end{align}
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}\label{rem:pyth_idt}
|
||||
Note that under these neural network architectures the famous Pythagorean identity $\sin^2\lp x\rp + \cos^2 \lp x\rp = 1$, may be rendered approximately, for appropriately fixed $n,q,\ve$ as: $\lb \sqr^{q,\ve}\bullet \csn^{q,\ve}_n \rb \oplus\lb \sqr^{q,\ve}\bullet \sne^{q,\ve}_n\rb \approx 1$. On a similar note, it is the case, with appropriate $n,q,\ve$ that $\real_{\rect}\lp \xpn^{q,\ve}_n \triangleleft \:i \rp\lp \pi \rp \approx -1$
|
||||
|
||||
A full discussion of the associated parameter, depth, and accuracy bounds are beyond the scope of this dissertation, and may be appropriate for future work.
|
||||
\begin{remark}
|
||||
Note that under these neural network architectures the famous Pythagorean identity $\sin^2\lp x\rp + \cos^2 \lp x\rp = 1$, may be rendered approximately, for fixed $n,q,\ve$ as: $\lb \sqr^{q,\ve}\bullet \csn^{q,\ve}_n \rb \oplus\lb \sqr^{q,\ve}\bullet \sne^{q,\ve}_n\rb$. A full discussion of the associated parameter, depth, and accuracy bounds are beyond the scope of this dissertation, and may be appropriate for future work.
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -1,10 +1,10 @@
|
|||
\chapter{ANN representations of Brownian Motion Monte Carlo}
|
||||
%\textbf{This is tentative without any reference to $f$.}
|
||||
We will now take the modified and simplified version of Multi-level Picard introduced in Chapter \ref{chp:2} and show a neural network representation and associated, parameters, depth, and accuracy bounds. However we will also try a different approach in that we will also give a direct neural network representation of the expectation of the stochastic process that Feynman-Kac asserts in Lemma \ref{ues}, and to build up to it we must build the requisite technology in Lemma \ref{mathsfE}, Lemma \ref{UE-prop}, Lemma \ref{UEX}.
|
||||
\begin{lemma}[R\textemdash,2023]
|
||||
\textbf{This is tentative without any reference to $f$.}
|
||||
|
||||
\begin{lemma}[R--,2023]
|
||||
Let $d,M \in \N$, $T \in (0,\infty)$ , $\act \in C(\R,\R)$, $ \Gamma \in \neu$, satisfy that $\real_{\act} \lp \mathsf{G}_d \rp \in C \lp \R^d, \R \rp$, for every $\theta \in \Theta$, let $\mathcal{U}^\theta: [0,T] \rightarrow [0,T]$ and $\mathcal{W}^\theta:[0,T] \rightarrow \R^d$ be functions , for every $\theta \in \Theta$, let $U^\theta: [0,T] \rightarrow \R^d \rightarrow \R$ satisfy satisfy for all $t \in [0,T]$, $x \in \R^d$ that:
|
||||
\begin{align}
|
||||
U^\theta(t,x) = \frac{1}{M} \lb \sum^M_{k=1} \lp \real_{\act} \lp \mathsf{G}_d \rp \rp \lp x+ \mathcal{W}^{\lp \theta,0,-k \rp } \rp \rb
|
||||
U^\theta(t,x) = \frac{1}{M} \lb \sum^M_{k=1} \lp \real_{\act} \lp \Gamma \rp \rp \lp x+ \mathcal{W}^{\lp \theta,0,-k \rp } \rp \rb
|
||||
\end{align}
|
||||
Let $\mathsf{U}^\theta_t \in \neu$ , $\theta \in \Theta$ satisfy for all $\theta \in \Theta$, $t \in [0,T]$ that:
|
||||
\begin{align}
|
||||
|
|
@ -16,12 +16,12 @@ We will now take the modified and simplified version of Multi-level Picard intro
|
|||
\item for all $\theta \in \Theta$, $t \in [0,T]$, that $\dep \lp \mathsf{U}^\theta_t \rp \les \dep (\mathsf{G}_d)$
|
||||
\item for all $\theta \in \Theta$, $t \in [0,T]$ that:
|
||||
\begin{align}
|
||||
\left\| \lay\lp \mathsf{U}^\theta_t \rp \right\|_{\infty} \les \|\lay \lp \mathsf{G}_d \rp \|_{\infty} \lp 1+ \sqrt{2} \rp M
|
||||
\left\| \lay\lp \mathsf{U}^\theta_t \rp \right\|_{\max} \les \|\lay \lp \mathsf{G}_d \rp \|_{\max} \lp 1+ \sqrt{2} \rp M
|
||||
\end{align}
|
||||
\item for all $\theta \in \Theta$, $t \in [0,T]$, $x \in \R^d$ that $U^\theta (t,x) = \lp \real_{\act} \lp \mathbf{U}^\theta_t \rp \rp \lp x \rp $ and
|
||||
\item for all $\theta \in \Theta$, $t \in [0,T]$ that:
|
||||
\begin{align}
|
||||
\param \lp \mathsf{U}^\theta_t \rp \les 2 \dep \lp \mathsf{G}_d \rp \lb \lp 1+\sqrt{2} \rp M \left\| \lay \lp \mathsf{G}_d \rp \right\|_{\infty}\rb^2
|
||||
\param \lp \mathsf{U}^\theta_t \rp \les 2 \dep \lp \mathsf{G}_d \rp \lb \lp 1+\sqrt{2} \rp M \left\| \lay \lp \mathsf{G}_d \rp \right\|_{\max}\rb^2
|
||||
\end{align}
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
|
|
@ -29,9 +29,9 @@ We will now take the modified and simplified version of Multi-level Picard intro
|
|||
\begin{proof}
|
||||
Throughout the proof let $\mathsf{P}^\theta_t \in \neu$, $\theta \in \Theta$, $t \in [0,T]$ satisfy for all $\theta \in \Theta$, $t \in [0,T]$ that:
|
||||
\begin{align}
|
||||
\mathsf{P}^\theta_t = \bigoplus^M_{k=1} \lb \frac{1}{M} \triangleright \lp \mathsf{G}_d \bullet \aff_{\mathbb{I}_d, \mathcal{W}^{\theta}_{T-t}} \rp \rb
|
||||
\mathsf{P}^\theta_t = \bigoplus^M_{k=1} \lb \frac{1}{M} \triangleright \lp \mathsf{G}_d \bullet \aff_{\mathbb{I}_d, \mathcal{W}^{\theta,0,-k}_{T-t}} \rp \rb
|
||||
\end{align}
|
||||
Note the hypothesis that for all $\theta \in \Theta$, $t \in [0,T]$ it holds that $\mathcal{W}^\theta_t \in \R^d$ and Lemma \ref{5.6.5} applied for every $\theta \in \Theta$ $t \in [0,T]$ with $v \curvearrowleft M$, $ c_{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \frac{1}{M} \rp_{i \in \{u,u+1,...,v\}}$, $\lp B_i \rp _{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \mathcal{W}^{\lp \theta, 0 , -k \rp }_{T-t} \rp_{k \in \{1,2,...,M\}}$, $\lp \nu_i \rp_{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \mathsf{G}_d \rp _{i \in \{u,u+1,...,v\}}$, $\mu \curvearrowleft \Phi^\theta_t$ and with the notation of Lemma \ref{5.6.5} tells us that for all $\theta \in \Theta$, $t \in [0,T]$, and $x \in \R^d$ it holds that:
|
||||
Note the hypothesis that for all $\theta \in \Theta$, $t \in [0,T]$ it holds that $\mathcal{W}^\theta_t \in \R^d$ and Lemma \ref{5.6.5} applied for every $\theta \in \Theta$ $t \in [0,T]$ with $v \curvearrowleft M$, $ c_{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \frac{1}{M} \rp_{i \in \{u,u+1,...,v\}}$, $\lp B_i \rp _{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \mathcal{W}^{\lp \theta, 0 , -k \rp }_{T-t} \rp_{k \in \{1,2,...,M\}}$, $\lp \nu_i \rp_{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \mathsf{G}_d \rp _{i \in \{u,u+1,...,v\}}$, $\mu \curvearrowleft \Phi^\theta_t$ and with the notation of Lemma \ref{5.6.5} tells us that for all $\theta \in \Theta$, $t \in [0,T]$, and $x \in \R^d$ it holds that: La lala
|
||||
\begin{align}\label{8.0.6}
|
||||
\lay \lp \mathsf{P}^\theta_t \rp = \lp d, M \wid_1 \lp \G \rp, M\wid_2 \lp \G \rp,...,M\wid_{\dep\lp \G \rp -1}\lp \G \rp ,1\rp = \lay \lp \sP^0_0 \rp \in \N^{\dep \lp \G \rp +1}
|
||||
\end{align}
|
||||
|
|
@ -79,10 +79,9 @@ Items (ii)--(iii) together shows that for all $\theta \in \Theta$, $t \in [0,T]$
|
|||
\end{align}
|
||||
This proves Item (v) and hence the whole lemma.
|
||||
\end{proof}
|
||||
While we realize that the modified Multi-Level {Picard may approximate solutions to non-linear PDEs we may chose a more circuitous route. It is quite possible, now that we have networks $\pwr_n^{q,\ve}$, to approximate polynomials using these networks. Once we have polynomials we may approximate more sophisticated PDEs.
|
||||
\section{The $\mathsf{E}^{N,h,q,\ve}_n$ Neural Networks}
|
||||
\section{The $\mathsf{E}^{N,h,q,\ve}_n$ Neural Network}
|
||||
\begin{lemma}[R\textemdash, 2023]\label{mathsfE}
|
||||
Let $n, N\in \N$ and $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots \: x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{N,h,q,\ve}_{n} \in \neu$ be the neural network given by:
|
||||
Let $n, N\in \N$ and $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{N,h,q,\ve}_{n} \in \neu$ be the neural network given by:
|
||||
\begin{align}
|
||||
\mathsf{E}^{N,h,q,\ve}_n = \xpn_n^{q,\ve} \bullet \etr^{N,h}
|
||||
\end{align}
|
||||
|
|
@ -100,10 +99,9 @@ While we realize that the modified Multi-Level {Picard may approximate solutions
|
|||
\item for all $x = \{x_0,x_1,\hdots, x_N \}\in \R^{N+1}$, where $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ we have that:
|
||||
\begin{align}
|
||||
&\left| \exp \lb \int^b_afdx\rb - \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp\lp f \lp \lb x \rb _{*,*}\rp\rp\right| \nonumber\\
|
||||
&\les \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp \cdot n^2 \cdot \lb \Xi + \frac{\lp b-a\rp^3}{12N^2} f''\lp \xi\rp\rb^{n-1} \nonumber \\
|
||||
&+\sum^n_{i=1} \frac{1}{i!}\lp \left| \Xi \lp \Xi^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp \Xi\rp\rp\right| + \ve + |\Xi|^q + \mathfrak{p}_{i-1}^q \rp + \frac{e^{\xi}\cdot \left| \int^b_a f dx\right|^{n+1}}{(n+1)!}
|
||||
&\les \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp \cdot n^2 \cdot \lb \Xi + \frac{\lp b-a\rp^3}{12N^2} f''\lp \xi\rp\rb^{n-1} + \nonumber \\
|
||||
&\sum^n_{i=1} \frac{1}{i!}\lp \left| \Xi \lp \Xi^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp \Xi\rp\rp\right| + \ve + |\Xi|^q + \mathfrak{p}_{i-1}^q \rp
|
||||
\end{align}
|
||||
Where $\Xi = \real_{\rect} \lp \etr^{N,h}\rp \lp f\lp \lb x\rb_{*,*}\rp\rp$.
|
||||
\item it is the case that $\wid_{\hid \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp} \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp \les 24+2n $
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
|
|
@ -225,7 +223,7 @@ While we realize that the modified Multi-Level {Picard may approximate solutions
|
|||
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=0.9]
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
%uncomment if require: \path (0,477); %set diagram left start at 0, and has a height of 477
|
||||
|
||||
%Shape: Rectangle [id:dp8133807694586985]
|
||||
|
|
@ -334,7 +332,7 @@ While we realize that the modified Multi-Level {Picard may approximate solutions
|
|||
% Text Node
|
||||
\draw (122,262.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (41,250.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}_{n}{}_{,}{}_{1}$};
|
||||
\draw (41,250.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{n}{}_{,}{}_{1}$};
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
|
|
@ -342,10 +340,10 @@ While we realize that the modified Multi-Level {Picard may approximate solutions
|
|||
\end{center}
|
||||
\caption{Diagram of $\mathsf{E}^{N,h,q,\ve}_n$.}
|
||||
\end{figure}
|
||||
\section{The $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$ Neural Networks}
|
||||
\section{The $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$ Neural Network}
|
||||
\begin{lemma}[R\textemdash,2023]\label{UE-prop}
|
||||
|
||||
Let $n, N\in \N$, $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{\exp}_{n,h,q,\ve} \in \neu$ be the neural network given by:
|
||||
Let $n, N,h\in \N$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{\exp}_{n,h,q,\ve} \in \neu$ be the neural network given by:
|
||||
\begin{align}
|
||||
\mathsf{E}^{N,h,q,\ve}_n = \xpn_n^{q,\ve} \bullet \etr^{N,h}
|
||||
\end{align}
|
||||
|
|
@ -383,7 +381,7 @@ Let $n, N\in \N$, $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp
|
|||
|
||||
\begin{center}
|
||||
|
||||
\begin{figure}
|
||||
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
|
|
@ -438,12 +436,10 @@ Let $n, N\in \N$, $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp
|
|||
|
||||
|
||||
\end{tikzpicture}
|
||||
\caption{Neural network diagram for $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$}
|
||||
\end{figure}
|
||||
\end{center}
|
||||
|
||||
\begin{proof}
|
||||
Note that from Lemma \ref{comp_prop}, and Lemma \ref{inst_of_stk}, we have that for $\fx \in \R^{N+1}$, and $x \in \R^d$ it is the case that $\real_{\rect} \lp \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb \rp\lp f\lp \lb \fx\rb_* \frown x\rp\rp = \real_{\rect} \lp \prd^{q,\ve}\rp \circ \real_{\rect}\lp \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb \rp \\ \lp f\lp \lb \fx\rb_*\rp \frown x\rp $. Then Lemma \ref{prd_network} tells us that $\real_{\rect} \lp \prd^{q,\ve}\rp \in C \lp \R^2,\R\rp$. Lemma \ref{mathsfE} tells us that $\real_{\rect }\lp \mathsf{E}^{N,h,q,\ve}_{n} \rp \in C \lp \R^{N+1},\R\rp$ and by hypothesis it is the case that $\real_{\rect} \lp \mathsf{G}_d\rp \in C \lp \R^d,\R\rp $. Thus, by the stacking properties of continuous instantiated networks and the fact that the composition of continuous functions is continuous, we have that $\real_{\rect} \lp \mathsf{UE}^{N, h,q,\ve}_{n,\mathsf{G}_d}\rp \in C \lp \R^{N+1} \times \R^d,\R \rp$.
|
||||
Note that from Lemma \ref{comp_prop}, and Lemma \ref{inst_of_stk}, we have that for $\fx \in \R^{N+1}$, and $x \in \R^d$ it is the case that $\real_{\rect} \lp \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb \rp\lp f\lp \lb \fx\rb_* \frown x\rp\rp = \real_{\rect} \lp \prd^{q,\ve}\rp \circ \real_{\rect}\lp \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb \rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp $. Then Lemma \ref{prd_network} tells us that $\real_{\rect} \lp \prd^{q,\ve}\rp \in C \lp \R^2,\R\rp$. Lemma \ref{mathsfE} tells us that $\real_{\rect }\lp \mathsf{E}^{N,h,q,\ve}_{n} \rp \in C \lp \R^{N+1},\R\rp$ and by hypothesis it is the case that $\real_{\rect} \lp \mathsf{G}_d\rp \in C \lp \R^d,\R\rp $. Thus, by the stacking properties of continuous instantiated networks and the fact that the composition of continuous functions is continuous, we have that $\real_{\rect} \lp \mathsf{UE}^{N, h,q,\ve}_{n,\mathsf{G}_d}\rp \in C \lp \R^{N+1} \times \R^d,\R \rp$.
|
||||
|
||||
Note that by Lemma \ref{comp_prop} it is the case that:
|
||||
\begin{align}
|
||||
|
|
@ -507,10 +503,10 @@ Let $n, N\in \N$, $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp
|
|||
&= 3\ve +2\ve \left| \mathfrak{u}_d\lp x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d\lp x \rp \nonumber
|
||||
\end{align}
|
||||
\end{proof}
|
||||
\section{The $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ Neural Networks}
|
||||
\begin{lemma}[R\textemdash,2023]\label{UEX}
|
||||
\section{The $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ network}\label{UEX}
|
||||
\begin{lemma}[R\textemdash,2023]\label{UE-prop}
|
||||
|
||||
Let $n, N\in \N$, $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots \: x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{\exp}_{n,h,q,\ve} \in \neu$ be the neural network given by:
|
||||
Let $n, N,h\in \N$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{\exp}_{n,h,q,\ve} \in \neu$ be the neural network given by:
|
||||
\begin{align}
|
||||
\mathsf{E}^{N,h,q,\ve}_n = \xpn_n^{q,\ve} \bullet \etr^{N,h}
|
||||
\end{align}
|
||||
|
|
@ -538,8 +534,7 @@ Let $n, N\in \N$, $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp
|
|||
\end{align}
|
||||
\item It is also the case that:
|
||||
\begin{align}
|
||||
&\left| \exp \lp \int^T_t fds\rp \mathfrak{u}_d^T\lp x\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp \right|\nonumber\\
|
||||
&\les 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber
|
||||
&\left| \exp \lp \int^T_t fds\rp \mathfrak{u}_d^T\lp x\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp \right|\nonumber\\ &\les 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber
|
||||
\end{align}
|
||||
Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as:
|
||||
\begin{align}
|
||||
|
|
@ -549,7 +544,7 @@ Let $n, N\in \N$, $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp
|
|||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Note that (\ref{fc-assertion}) is an assertion of Feynman-Kac. Now notice that for $x \in \R^{N+1} \times \R^d$ it is the case that:
|
||||
Note that (\ref{fc-assertion}) is an assertion of Feynman-Kac. LetNow notice that for $x \in \R^{N+1} \times \R^d$ it is the case that:
|
||||
\begin{align}
|
||||
\real_{\rect} \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp x\rp &= \real_{\rect} \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb \rp \lp x\rp\nonumber \\
|
||||
&=\real_{\rect} \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \rp \circ \real_{\rect}\lp \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb\rp \lp x \rp \nonumber
|
||||
|
|
@ -606,6 +601,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} = \mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb
|
||||
\end{align}
|
||||
It is then the case that for all $\fx = \{x_0,x_1,\hdots, x_N\} \in \R^{N+1}$ and $x \in \R^d$ that:
|
||||
\item It is also the case that:
|
||||
\begin{align}
|
||||
&\left| \exp \lp \int^T_t fds\rp \mathfrak{u}_d^T\lp x\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp \right|\nonumber\\ &\les 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber
|
||||
\end{align}
|
||||
|
|
@ -618,21 +614,20 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
\begin{proof}
|
||||
Note that for a fixed $T \in \lp 0,\infty \rp$ it is the case that $u_d\lp t,x \rp \in C^{1,2}\lp \lb 0,T\rb \times \R^d, \R \rp$ projects down to a function $\mathfrak{u}_d^T\lp x\rp \in C^2\lp \R^d, \R\rp$. Furthermore given a probability space $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ and a stochastic process $\mathcal{X}^{d,t,x}: \lb t,T\rb \times \Omega \rightarrow \R^d$, for a fixed outcome space $\omega_i \in \Omega$ it is the case that $\mathcal{X}^{d,t,x}$ projects down to $\mathcal{X}^{d,t,x}_{\omega_i}: \lb t,T\rb \rightarrow \R^d$. Thus given $\alpha_d: \R^d \rightarrow \R$ that is infinitely often differentiable, we get that $\alpha_d\circ \mathcal{X}_{\omega_i}^{d,t,x}: \lb t,T\rb \rightarrow\R$.
|
||||
|
||||
Taken together with Lemma \ref{UE-prop} with $x \curvearrowleft \mathcal{X}^{d,t,x}_{r,\omega}, f \curvearrowleft \alpha_d\circ \mathcal{X}_{\omega_i}^{d,t,x}$, $b \curvearrowleft T$, $a \curvearrowleft t$, and $\mathfrak{u}_d^T\lp x\rp \curvearrowleft u_d \lp T,\mathcal{X}^{d,t,x}_{\omega_i}\rp$, our error term is rendered as:
|
||||
Taken together with Lemma \ref{UE-prop} with $x \curvearrowleft \mathcal{X}^{d,t,x}_{r,\omega}, f \curvearrowleft \alpha_d\circ \mathcal{X}_{\omega_i}^{d,t,x}$, $b \curvearrowleft T$, $a \curvearrowleft t$, and $\mathfrak{u}_d^T\lp x\rp \curvearrowleft u_d \lp T,\mathcal{X}^{d,t,x}_{\omega_i}\rp$, our error term is rendered as is rendered as:
|
||||
\begin{align}
|
||||
&\left| \exp \lp \int^T_t \lp \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds\rp \rp u_d\lp T,\mathcal{X}^{d,t,x}_{\omega_i}\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right|\nonumber\\
|
||||
&\les 3\ve +2\ve \left| u_d\lp T,\mathcal{X}_{r,\omega_i}^{d,t,x}\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}u_d\lp T,\mathcal{X}^{d,t,x}_{r,\omega_i} \rp\nonumber
|
||||
\end{align}
|
||||
This completes the proof of the Lemma.
|
||||
This completes the proof of the lemma.
|
||||
\end{proof}
|
||||
\begin{remark}
|
||||
Diagrammatically, this can be represented as:
|
||||
|
||||
\begin{center}
|
||||
\begin{figure}[h]
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=0.9]
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
%uncomment if require: \path (0,475); %set diagram left start at 0, and has a height of 475
|
||||
|
||||
%Shape: Rectangle [id:dp5014556157804896]
|
||||
|
|
@ -696,16 +691,13 @@ Note that for a fixed $T \in \lp 0,\infty \rp$ it is the case that $u_d\lp t,x \
|
|||
% Text Node
|
||||
\draw (490,225.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mymathbb{0}_{d,d} ,\mathcal{X}^{d,t,x}_{r,\omega_i}}$};
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
\caption{Neural network diagram for $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d, \omega_i}$}
|
||||
\end{figure}
|
||||
\end{center}
|
||||
\end{remark}
|
||||
|
||||
\section{The $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn}$ Neural Networks}
|
||||
|
||||
\section{The $\mathsf{UES}$ network}
|
||||
\begin{lemma}\label{lem:sm_sum}
|
||||
Let $\nu_1,\nu_2,\hdots, \nu_n \in \neu$ such that for all $i \in \{1,2,\hdots, n\}$ it is the cast that $\out\lp \nu_i\rp = 1$, and it is also the case that $\dep \lp \nu_1 \rp = \dep \lp \nu_2 \rp = \cdots =\dep \lp \nu_n\rp$. Let $x_1 \in \R^{\inn\lp \nu_1\rp},x_2 \in \R^{\inn\lp \nu_2\rp},\hdots, x_n \in \R^{\inn\lp \nu_n\rp}$ and $\fx \in \R^{\sum_{i=1}^n \inn \lp \nu_i\rp}$. It is then the case that we have that:
|
||||
Let $\nu_1,\nu_2,\hdots, \nu_n \in \neu$ such that for all $i \in \{1,2,\hdots, n\}$ it is the cast that $\out\lp \nu_i\rp = 1$, and it is also the case that $\dep \lp \nu_1 \rp = \dep \lp \nu_2 \rp = \cdots =\dep \lp \nu_n\rp$. Let $x_1 \in \R^{\inn\lp \nu_1\rp},x_2 \in \R^{\inn\lp \nu_2\rp},\hdots x_n \in \R^{\inn\lp \nu_n\rp}$ and $\fx \in \R^{\sum_{i=1}^n \inn \lp \nu_i\rp}$. It is then the case that we have that:
|
||||
\begin{align}
|
||||
\real_{\rect}\lp \sm_{n,1} \bullet \lb \boxminus_{i=1}^n \nu_i \rb \rp \lp \fx\rp = \sum^n_{i=1} \real_{\rect} \lp \nu_i\rp \lp x_i\rp
|
||||
\end{align}
|
||||
|
|
@ -738,32 +730,18 @@ Note that for a fixed $T \in \lp 0,\infty \rp$ it is the case that $u_d\lp t,x \
|
|||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
We will prove this with induction. This is straight-forward for the case where we have just one neural network where for all $x \in \R^{\inn\lp \nu_1\rp}$ it is the case that $\left\|\inst_{\rect}\lp \nu_1\rp \lp x\rp - f\lp x\rp\right\|_1 \les \ve_1 = \sum_{i=1}^1\ve_i$. Suppose now, that, $\left\|\inst_{\rect}\lp \boxminus_i^n\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^n f_i\rb\lp \fx \rp\right\|_1 \les \sum_{i=1}^n\ve_i$ holds true for all cases upto and including $n$. Consider what happens when we have a triple, a function $f_{n+1}$, a neural network $\nu_{n+1}$, and $\ve_{n+1}\in \lp 0,\infty \rp$ with a maximum error over all $x \in \R^{\inn\lp \nu_1\rp}$ of $| \inst_{\rect}\lp \nu_{n+1}\rp\lp x\rp-f_{n+1}\lp x\rp| \les \ve_{n+1}$. Then Lemma \ref{inst_of_stk}, Corollary \ref{sum_of_frown_frown_of_sum}, and the triangle inequality tells us that:
|
||||
We will prove this with induction. This is straight-forward for the case where we have just one neural network where for all $x \in \R^{\inn\lp \nu_1\rp}$ it is the case that $\left\|\inst_{\rect}\lp \nu_1\rp \lp x\rp - f\lp x\rp\right\|_1 \les \ve_1 = \sum_{i=1}^1\ve_i$. Suppose now, that, $\left\|\inst_{\rect}\lp \boxminus_i^n\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^n f_i\rb\lp \fx \rp\right\|_1 \les \sum_{i=1}^n\ve_i$ holds true for all cases upto and including $n$. Consider what happens when we have a triple, a function $f_{n+1}$, a neural network $\nu_{n+1}$, and $\ve_{n+1}\in \lp 0,\infty \rp$ with a maximum error over all $x \in \R^{\inn\lp \nu_1\rp}$ of $|f_{n+1}\lp x\rp - \inst_{\rect}\lp \nu_{n+1}\rp\lp x\rp | \les \ve_{n+1}$. Then Lemma \ref{inst_of_stk}, Corollary \ref{sum_of_frown_frown_of_sum}, and the triangle inequality tells us that:
|
||||
\begin{align}
|
||||
&\left\|\inst_{\rect}\lp \boxminus_i^{n+1}\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^{n+1} f_i\rb\lp \fx \rp\right\|_1 \nonumber \\
|
||||
&\les \left\|\inst_{\rect}\lp \boxminus_i^n\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^n f_i\rb\lp \fx \rp\right\|_1 + | \inst_{\rect}\lp \nu_{n+1}\rp\lp x\rp-f_{n+1}\lp x\rp| \nonumber \\
|
||||
&\les \left\|\inst_{\rect}\lp \boxminus_i^n\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^n f_i\rb\lp \fx \rp\right\|_1 + |f_{n+1}\lp x\rp - \inst_{\rect}\lp \nu_{n+1}\rp\lp x\rp | \nonumber \\
|
||||
&\les \sum_{i=1}^{n+1}\ve_i
|
||||
\end{align}
|
||||
This proves the inductive case and hence the Lemma.
|
||||
\end{proof}
|
||||
\begin{lemma}\label{var_of_rand}
|
||||
Let $\lp \Omega, \cF, \mathbb{P} \rp$ be a probability space. Let $X_d: \Omega \rightarrow \R_d$ be a random variable. Let $f: \R_d \rightarrow \R$ be a function such that for all $x,\fx \in \R^d$ it is the case that $\left\| f\lp x\rp - f\lp \fx\rp\right\|_E \les \fL\left| x-\fx\right|$. It is then the case that $\var\lb f\lp X_d\rp\rb \les 2\fL^2\var\lb X_d\rb$.
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Let $\fX_d$ be an i.i.d. copy of $X_d$. As such it is the case that $\cov \lp X_d, \fX_d\rp = 0$, whence it is the case that $\var\lb X_d, \fX_d\rb = \var\lb X_d\rb + \var\lb \fX_d\rb = \var[X_d] + \var\lb -\fX_d\rb = \var\lb X_d - \fX_d\rb = 2\var\lb X_d\rb$. Note that $f\lp X_d\rp$ and $f\lp \fX_d\rp$ are also indepentend and thus $\cov\lp f\lp X_d\rp,f\lp \fX_d\rp\rp = 0$, and whence we get that $\var\lb f\lp X_d\rp - f\lp \fX_d\rp\rb = 2\var \lb \fX_d\rb$. This then yields that:
|
||||
\begin{align}
|
||||
2\var \lb f\lp X_d\rp\rb &= \var\lb f\lp X_d\rp - f\lp \fX_d\rp\rb \nonumber\\
|
||||
&= \E \lb \lp f\lp X_d\rp -f\lp \fX_d\rp\rp^2\rb - \lp \E \lb f\lp X_d\rp - f\lp \fX_d\rp\rb\rp^2 \nonumber \\
|
||||
&= \E \lb \lp f\lp X_d\rp -f\lp \fX_d\rp\rp^2\rb \nonumber\\
|
||||
&= \fL^2\cdot \E \lb \lp X_d - \fX_d \rp^2\rb \nonumber\\
|
||||
&= \fL^2\cdot 2 \var \lb X_d\rb \nonumber\\
|
||||
\implies \var \lb f\lp X_d\rp\rb &= \fL^2\cdot \var\lb X_d\rb
|
||||
\end{align}
|
||||
This proves the Lemma.
|
||||
\end{proof}
|
||||
\begin{lemma}[R\textemdash, 2024, Approximants for Brownian Motion]\label{ues}
|
||||
|
||||
Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space. Let $n,N\in \N$, and $h \in \lp 0, \infty \rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $f:[t, T] \rightarrow \R$ be continuous almost everywhere in $\lb t, T \rb$. Let it also be the case that $f = g \circ \fh$, where $\fh: \lb t,T\rb \rightarrow \R^d$, and $g: \R^d \rightarrow \R$. Let $t=t_0 \les t_1\les \cdots \les t_{N-1} \les t_N=T$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{T-t}{N}$, and $t_i = t_0+i\cdot h$ . Let $\mathbf{t} = \lb t_0 \: t_1\: \cdots t_N \rb$ and as such let $f\lp\lb \mathbf{t} \rb_{*,*} \rp = \lb f(t_0) \: f(t_1)\: \cdots \: f(t_N) \rb$. Let $u_d \in C \lp \R^d,\R\rp$, bounded by $\mathfrak{M}_{u,d}$ satisfy for all $d \in \N$, $t \in \lb 0,T\rb$, $x \in \R^d$ that:
|
||||
\begin{lemma}[R\textemdash, 2024, Approximants for Brownian Motion]
|
||||
|
||||
Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space. Let $n,N\in \N$, and $h \in \lp 0, \infty \rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $f:[t, T] \rightarrow \R$ be continuous almost everywhere in $\lb t, T \rb$. Let it also be the case that $f = g \circ \fh$, where $\fh: \lb t,T\rb \rightarrow \R^d$, and $g: \R^d \rightarrow \R$. Let $t=t_0 \les t_1\les \cdots \les t_{N-1} \les t_N=T$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{T-t}{N}$, and $t_i = t_0+i\cdot h$ . Let $\mathbf{t} = \lb t_0 \: t_1\: \cdots t_N \rb$ and as such let $f\lp\lb \mathbf{t} \rb_{*,*} \rp = \lb f(t_0) \: f(t_1)\: \cdots \: f(t_N) \rb$. Let $u_d \in C \lp \R^d,\R\rp$ satisfy for all $d \in \N$, $t \in \lb 0,T\rb$, $x \in \R^d$ that:
|
||||
\begin{align}
|
||||
\lp \frac{\partial}{\partial t} u_d\rp \lp t,x\rp + \lp \nabla_x^2 u_d\rp \lp t,x \rp + \alpha_d\lp x\rp u_d \lp t,x\rp = 0
|
||||
\end{align}
|
||||
|
|
@ -785,7 +763,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
|
||||
Furthermore, let $\mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d}\subsetneq \neu$ be neural networks given by:
|
||||
\begin{align}
|
||||
\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} = \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb
|
||||
\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} = \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n,h,q,\ve} \DDiamond \mathsf{G}_d \rb
|
||||
\end{align}
|
||||
Futhermore, let $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \subsetneq \neu$ be neural networks given by:
|
||||
\begin{align}
|
||||
|
|
@ -798,18 +776,18 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
|
||||
It is then the case that for all $\fX \in \R^{\fn \lp N+1\rp} \times \R^{\fn d}$:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\real_{\rect} \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \rp \lp \fX \rp\in C \lp \R^{\mathfrak{n}\lp N+1 \rp}\times \R^{\mathfrak{n} d}, \R \rp$
|
||||
\item $\real_{\rect} \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \rp \in C \lp \R^{\mathfrak{n}\lp N+1 \rp}\times \R^{\mathfrak{n} d}, \R \rp$
|
||||
\item $\dep \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn}\rp \les \begin{cases}
|
||||
\frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\dep \lp \mathsf{G}_d\rp-1 &:n = 0\\
|
||||
\frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\max\left\{\dep \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp,\dep \lp \mathsf{G}_d\rp\right\}-1 &:n \in \N\\
|
||||
\end{cases}$
|
||||
\item It is also the case that:
|
||||
\begin{align}
|
||||
\param \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp &\les \fn^2 \cdot \lb \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\right. \nonumber\\ &\left. +24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} \rb
|
||||
\item It is also the case that:\begin{align}
|
||||
\param \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp &\les \param \lp \prd^{q,\ve}\rp + 2\lp\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\}\rp^2 \nonumber\\
|
||||
&+ 8 \max\left\{\lp 1+4n\rp, \wid_{\hid \lp \mathsf{G}_d\rp} \lp \mathsf{G}_d\rp \right\}\nonumber
|
||||
\end{align}
|
||||
\item It is also the case that:
|
||||
\begin{align}
|
||||
&\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \lp \fX \rp\right| \nonumber\\
|
||||
&\left| \mathbb{E} \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r}\rp ds\rp u_d\lp T,\mathcal{X}^{d,t,x}_{r,\omega_i}\rp \rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn}\rp\right| \\
|
||||
&\les 3\ve +2\ve \left| \fu^T_d\lp x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\fu^T_d\lp x \rp\nonumber
|
||||
\end{align}
|
||||
Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as:
|
||||
|
|
@ -819,7 +797,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Note that for all $i \in \{ 1,2,\hdots, \mathfrak{n}\}$, Lemma \ref{UEX} tells us that $\real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \\ \in C\lp \R^{N+1} \times \R^d, \R\rp$. Lemma \ref{nn_sum_cont} and Lemma \ref{nn_sum_is_sum_nn}, thus tells us that \\ $\real_{\rect}\lp \lp \bigoplus_{i=1}^{\mathfrak{n}}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rp = \sum_{i=1}^\mathfrak{n}\lb \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp\rb $. The sum of continuous functions is continuous. Note next that $\frac{1}{\mathfrak{n}}\triangleright$ is an affine neural network, and hence, by Lemma \ref{aff_prop}, must be continuous.
|
||||
Note that for all $i \in \{ 1,2,\hdots, \mathfrak{n}\}$, Lemma \ref{UEX} tells us that $\real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \in C\lp \R^{N+1} \times \R^d, \R\rp$. Lemma \ref{nn_sum_cont} and Lemma \ref{nn_sum_is_sum_nn}, thus tells us that $\real_{\rect}\lp \lp \bigoplus_{i=1}^{\mathfrak{n}}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rp = \sum_{i=1}^\mathfrak{n}\lb \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp\rb $. The sum of continuous functions is continuous. Note next that $\frac{1}{\mathfrak{n}}\triangleright$ is an affine neural network, and hence, by Lemma \ref{aff_prop}, must be continuous.
|
||||
|
||||
Then Lemmas \ref{comp_prop}, \ref{5.3.4}, and the fact that by Lemma \ref{UEX} each of the individual stacked $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ neural networks is continuous then ensures us that it must therefore be the case that: $\real_{\rect} \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega,\fn}\rp \in C \lp \R^{\mathfrak{n}\lp N+1 \rp}\times \R^{\mathfrak{n} d}, \R \rp$. This proves Item (i).
|
||||
|
||||
|
|
@ -827,7 +805,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
|
||||
Whence by Lemma \ref{comp_prop} it is the case that $\dep \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn} \rp = \dep \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp$. This then proves Item (ii).
|
||||
|
||||
Next, observe that each of the $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ networks has the same architecture for all $\omega_i \in \Omega$ by construction. Corollary \ref{cor:sameparal} then yields that:
|
||||
Next, observe that each of the $\mathsf{UEX}$ networks has the same architecture by construction. Corollary \ref{cor:sameparal} then yields that:
|
||||
\begin{align}
|
||||
\param \lp \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \les \mathfrak{n}^2\cdot \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp
|
||||
\end{align}
|
||||
|
|
@ -835,262 +813,39 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
\begin{align}
|
||||
&\param \lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \les \param \lp \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \nonumber\\
|
||||
&\les \mathfrak{n}^2\cdot \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \nonumber\\
|
||||
\ \end{align}
|
||||
&\les \fn^2 \cdot \lb \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\right. \nonumber\\ &\left. +24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} \rb
|
||||
\end{align}
|
||||
and therefore that:
|
||||
\begin{align}
|
||||
&\param \lp \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d, \omega_i}\rb\rp \rp \nonumber\\
|
||||
&\les \param \lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \nonumber\\
|
||||
&\les \param \lp \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \nonumber\\
|
||||
&\les \mathfrak{n}^2\cdot \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \nonumber \\
|
||||
&\les \fn^2 \cdot \lb \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\right. \nonumber\\ &\left. + 24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} \rb
|
||||
&\les \fn^2 \cdot \lb \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\right. \nonumber\\ &\left. +24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} \rb
|
||||
\end{align}
|
||||
Observe that the absolute homogeneity condition for norms, the fact that the Brownian motions are independent of each other, Lemma \ref{lem:sm_sum}, the fact that $\mathfrak{n}\in \N$, the fact that the upper limit of error remains bounded by the same bound for all $\omega_i \in \Omega$, and Lemma \ref{sum_of_errors_of_stacking}, then yields us:
|
||||
Now observe that by the triangle inequality, we have that:
|
||||
\begin{align}
|
||||
&\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp\right|\nonumber \\
|
||||
&=\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rb\right| \nonumber \\
|
||||
&\left| \E \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp ds\rp u_d^T\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \nonumber \\
|
||||
&=\left| \E \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds\rp u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp \rb - \inst_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb\right| \nonumber\\
|
||||
&\les \left| \E \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds\rp u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb \right|\nonumber \\
|
||||
&+\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb\right| \nonumber \\
|
||||
\end{align}
|
||||
Observe that by the triangle inequality, the absolute homogeneity condition for norms, the fact that the Brownian motions are independent of each other, Lemma \ref{lem:sm_sum}, the fact that $\mathfrak{n}\in \N$, the fact that the upper limit of error remains bounded by the same bound for all $\omega_i \in \Omega$, and Lemma \ref{sum_of_errors_of_stacking}, then renders the second summand as:
|
||||
\begin{align}
|
||||
&\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rb\right| \nonumber \\
|
||||
&\les \left|\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1} \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lp \real_{\rect}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb \right| \nonumber \\
|
||||
&\les \cancel{\frac{1}{\mathfrak{n}} \sum^{\mathfrak{n}}_{i=1}}\left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \right| \nonumber\\
|
||||
&= \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right| \nonumber \\
|
||||
&\les 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber
|
||||
&\les \cancel{\frac{1}{\mathfrak{n}} \sum^{\mathfrak{n}}_{i=1}}\left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\right| \nonumber\\
|
||||
&\les \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\right| \nonumber\\
|
||||
\end{align}
|
||||
\end{proof}
|
||||
% Now observe that by the triangle inequality, we have that:
|
||||
% \begin{align}
|
||||
% &\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \label{big_eqn_lhs} \\
|
||||
% &=\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \inst_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb\right| \nonumber\\
|
||||
% &\les \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\label{big_eqn_rhs_summand_1} \\
|
||||
% &+\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb\right| \label{big_eqn_lhs_summand_2}
|
||||
% \end{align}
|
||||
|
||||
% Observe that by the triangle inequality, the absolute homogeneity condition for norms, the fact that the Brownian motions are independent of each other, Lemma \ref{lem:sm_sum}, the fact that $\mathfrak{n}\in \N$, the fact that the upper limit of error remains bounded by the same bound for all $\omega_i \in \Omega$, and Lemma \ref{sum_of_errors_of_stacking}, then renders the second summand, (\ref{big_eqn_lhs_summand_2}), as:
|
||||
% \begin{align}
|
||||
% &\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rb\right| \nonumber \\
|
||||
% &\les \left|\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1} \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lp \real_{\rect}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb \right| \nonumber \\
|
||||
% &\les \cancel{\frac{1}{\mathfrak{n}} \sum^{\mathfrak{n}}_{i=1}}\left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \right| \nonumber\\
|
||||
% &\les \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right| \nonumber
|
||||
% \end{align}
|
||||
\begin{corollary}\label{cor_ues}
|
||||
Let $N,n,\fn \in \N$, $h,\ve \in \lp 0,\infty\rp$, $q\in\lp 2,\infty\rp$, given $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \subsetneq \neu $, it is then the case that:
|
||||
\begin{align}
|
||||
&\lp \E\lb \left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb \right.\right.\right.\nonumber\\ &\left. \left.\left.-\frac{1}{\mathfrak{n}}\lp \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rp \right|^2\rb\rp^{\frac{1}{2}} \nonumber \\
|
||||
&\les \frac{\fk_2}{\fn^{\frac{1}{2}}} \lp \E \lb \left| \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \right|^2\rb \rp^{\frac{1}{2}}
|
||||
\end{align}
|
||||
\end{corollary}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{proof}
|
||||
Note that $\E \lb \cX^{d,tx}_{r,\Omega}\rb < \infty$, and $\fu^T$ being bounded yields that $\E \lb \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb < \infty$, and also that $\E \lb \alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rb < \infty$. Thus we also see that $\E \lb \int^T_t\alpha_d\circ \cX^{d,t,x}_{r,\Omega} ds\rb < \infty$, and thus $\E \lb \exp \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp\rb < \infty$. Thus together these two facts, along with the fact that the two factors are independent by \cite[Corollary~2.5]{hutzenthaler_overcoming_2020}, then assert that $\E \lb \exp \lp \int^T_t \alpha_d\circ \cX^{d,t,x}_{r,\Omega}\rp \cdot \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb < \infty$.
|
||||
|
||||
Note that \cite[Corollary~3.8]{hutzenthaler_strong_2021} tells us that:
|
||||
\begin{align}\label{kk_application}
|
||||
&\lp \E\lb \left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb \right.\right.\right.\nonumber\\ &\left. \left.\left.-\frac{1}{\mathfrak{n}}\lp \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rp \right|^2\rb\rp^{\frac{1}{2}} \nonumber \\
|
||||
&\les \frac{\fk_2}{\fn^{\frac{1}{2}}} \lp \E \lb \left| \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \right|^2\rb \rp^{\frac{1}{2}} \nonumber \\
|
||||
&\les \frac{\fk}{\fn^\frac{1}{2}}\lp \exp \lp \lp T-t\rp \mathfrak{M}_{\alpha,d}\rp \mathfrak{M}_{u,d}\rp
|
||||
\end{align}
|
||||
This, combined with Lyapunov's Inequality for Expectation and the Triangle Inequality yields that:
|
||||
\begin{align}
|
||||
&\E\lb \left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb \right.\right.\nonumber\\ &\left.\left.-\frac{1}{\mathfrak{n}}\lp \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rp \right|\rb \nonumber \\
|
||||
&\les \frac{\fk}{\fn^\frac{1}{2}}\lp \exp \lp \lp T-t\rp \mathfrak{M}_{\alpha,d}\rp \mathfrak{M}_{u,d}\rp
|
||||
\end{align}
|
||||
Finally, combined with, the linearity of expectation, and the fact that the expectation of a deterministic constant and a deterministic function is, respectively, the constant and function itself, and the triangle inequality then yields that:
|
||||
\begin{align}
|
||||
&\E\lb \left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb -\real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega ,\fn}\rp\right|\rb \nonumber \\
|
||||
&\les \frac{\fk}{\fn^\frac{1}{2}}\lp \exp \lp \lp T-t\rp \mathfrak{M}_{\alpha,d}\rp \mathfrak{M}_{u,d}\rp \nonumber \\
|
||||
&+ 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp \nonumber
|
||||
\end{align}
|
||||
This completes the proof of the corollary.
|
||||
% For the purposes of this proof let it be the case that $\ff: [0,T] \rightarrow \R$ is the function represented for all $t \in \lb 0,T \rb$ as:
|
||||
% \begin{align}
|
||||
% \ff\lp t\rp = \int^T_{T-t} \alpha_d\circ \cX^{d,t,x}_{r,\Omega} ds
|
||||
% \end{align}
|
||||
% In which case we haved that $\ff\lp 0\rp = 0$, and thus, stipulating $g\lp x\rp = \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp$ we may define $u\lp t,x\rp$ as the function given by:
|
||||
% \begin{align}
|
||||
% u\lp t,x\rp &= \exp \lp \ff\lp t\rp\rp \cdot g\lp x\rp \nonumber\\
|
||||
% &= \lb \exp\lp \ff\lp 0\rp\rp + \int_0^s \ff'\lp s\rp\cdot \exp \lp \ff\lp s\rp\rp ds\rb \cdot g\lp x\rp\nonumber \\
|
||||
% &= g\lp x\rp + \int_0^s \ff'\lp s\rp \cdot \exp\lp \ff\lp s\rp\rp \cdot g\lp x\rp ds \nonumber\\
|
||||
% &= g\lp x\rp + \int^s_0 \ff'\lp s\rp\cdot u\lp s,x \rp ds \nonumber \\
|
||||
% &= g\lp x\rp+ \int^s_0 \fF \lp s,x, u\lp s,x \rp\rp ds
|
||||
% \end{align}
|
||||
% Then \cite[Corollary~2.5]{hutzenthaler_strong_2021} with $f \curvearrowleft \fF$, $u \curvearrowleft u$, $x+ \cW_{s-t} \curvearrowleft \cX^{d,t,s}_{r,\Omega}$, and tells us that with $q \curvearrowleft 2$ that:
|
||||
% \begin{align}
|
||||
% & \lp \E \lb \left| \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \right|^2\rb \rp^{\frac{1}{2}} \nonumber\\
|
||||
% &\les \fL \lp T+1\rp \exp \lp LT\rp \lb \sup_{s\in \lb 0,T\rb} \lp \E \lb \lp 1+\left\| x + \cW_s\right\|^p\rp^2\rb\rp^{\frac{1}{2}}\rb
|
||||
% \end{align}
|
||||
% Together with (\ref{kk_application}) we then get that:
|
||||
% \begin{align}
|
||||
% &\lp \E\lb \left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb \right.\right.\right.\nonumber\\ &\left. \left.\left.-\frac{1}{\mathfrak{n}}\lp \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rp \right|^2\rb\rp^{\frac{1}{2}} \nonumber \\
|
||||
% &\les \frac{\fk_p }{n^{\frac{1}{2}}} \cdot \fL \lp T+1\rp \exp \lp LT\rp \lb \sup_{s\in \lb 0,T\rb} \lp \E \lb \lp 1+\left\| x + \cW_s\right\|^p\rp^2\rb\rp^{\frac{1}{2}}\rb
|
||||
% \end{align}
|
||||
\end{proof}
|
||||
% Note that Taylor's theorem states that:
|
||||
% \begin{align}
|
||||
% \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp = 1 + \int^T_t \alpha_d \circ \cX ^{d,t,x}_{r,\Omega}ds + \frac{1}{2}\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds + \fR_3
|
||||
% \end{align}
|
||||
% Where $\fR_3$ is the Lagrange form of the reamainder. Thus $\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot u\lp T,\cX^{d,t,x}_{r,\Omega}\rp$ is rendered as:
|
||||
% \begin{align}
|
||||
% &\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \\
|
||||
% &= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 \\
|
||||
% &+\fR_3 \cdot \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp
|
||||
% \end{align}
|
||||
% \end{proof}
|
||||
% Jensen's Inequality, the fact that $\fu^T$ does not depend on time, and the linearity of integrals gives us:
|
||||
% \begin{align}
|
||||
% &= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } ds + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp^2 \nonumber\\
|
||||
% &+\fR_3 \cdot \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \nonumber\\
|
||||
% &\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \frac{1}{2}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\cdot \lp \frac{1}{T-t}\int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds\rp^2 \\ &+ \fR_3\nonumber\\
|
||||
% &\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int_t^T\fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \int^T_t \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega }\rp \cdot \lp \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds\\ &+ \fR_3\nonumber \\
|
||||
% &= \fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp + \int^T_t \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} + \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp\alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 ds + \fR_3\nonumber
|
||||
% \end{align}
|
||||
% Thus \cite[Lemma~2.3]{hutzenthaler_strong_2021} with $f \curvearrowleft \fu^T$ tells us that:
|
||||
% \begin{align}
|
||||
% \E
|
||||
% \end{align}
|
||||
% \begin{proof}
|
||||
% Note that $\fu^T$ is deterministic, and $\cX^{d,t,x}_{r,\Omega}$ is a $d$-vector of random variables, where $\mu = \mymathbb{0}_d$, and $\Sigma = \mathbb{I}_d$. Whence we have that:
|
||||
% \begin{align}
|
||||
% \var \lb \fu^T\lp x\rp\rb &= \lb \nabla \fu^T \lp x\rp\rb^\intercal \cdot \mathbb{I}_d \cdot \nabla \fu^T\lp x\rp + \frac{1}{2}\cdot \Trace\lp \Hess_x^2 \lp f\rp\lp x\rp\rp \nonumber \\
|
||||
% &= \lb \nabla \fu^T\lp x\rp \rb_*^2 + \frac{1}{2}\cdot \Trace\lp \Hess_x^2\lp f\rp\lp x\rp\rp
|
||||
% \end{align}
|
||||
% We will call the right hand side of the equation above as $\fU.$
|
||||
%
|
||||
% For the second factor in our product consider the following:
|
||||
% \begin{align}
|
||||
% \cY^{d,t}_{x,s} = \int_t^T\alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds
|
||||
% \end{align}
|
||||
% Whose Reimann sum, with $\Delta t = \frac{T-t}{n}$ and $t_k = t+k\Delta t$, and Lemma \ref{var_of_rand} is thus rendered as:
|
||||
% \begin{align}
|
||||
% \cY_n &= \Delta t \lb \sum^{n-1}_{k=0} \alpha \circ \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rb \nonumber\\
|
||||
% \var\lb \cY_n \rb &= \var \lb \Delta_t\sum^{n-1}_{k=0}\alpha \circ \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rb \nonumber\\
|
||||
% &= \lp\Delta t\rp^2 \sum^{n-1}_{k=0}\lb \var \lb \alpha \circ \cX^{d,t,x}_{r,\Omega}\lp t_k\rp \rb\rb \nonumber\\
|
||||
% &\les \lp \Delta t\rp^2 \sum^{n-1}_{k=0}\lb \fL^2\cdot \var\lp \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rp\rb \nonumber\\
|
||||
% &=\lp \fL\Delta t\rp^2 \sum^{n-1}_{k=0}\lb \var \lp \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rp \rb
|
||||
% \end{align}
|
||||
% \textbf{Alternatively}:
|
||||
% \begin{align}
|
||||
% &\var \lb \int_t^T\alpha \circ \cX\rb \\
|
||||
% &=\E \lb \lp \int^T_t \alpha \circ \cX \rp^2\rb - \lp \E \lb \int^T_t \alpha \circ \cX \rb\rp^2 \\
|
||||
% &=\E \lb \int^T_t\lp \alpha \circ \cX \rp^2\rb - \lp \int_t^T \E \lb \alpha \circ \cX \rb\rp^2 \\
|
||||
% &=
|
||||
% \end{align}
|
||||
%
|
||||
% Note that since $\alpha_d$ is Lipschitz with constant $\fL$ we may say that for $\fX^x_t = \cX_t -x$ that:
|
||||
% \begin{align}
|
||||
% \left| \alpha_d\circ \fX^x_t -\alpha_d \circ \fX^x_0 \right| &\les \fL \cdot\left|\fX^x_t - \fX^x_0\right| \nonumber\\
|
||||
% \implies \left| \alpha_d \circ \fX^x_t - \alpha_d\lp 0\rp\right| &\les \fL \left| \fX^x_t-0\right| \nonumber \\
|
||||
% \implies \alpha_d \circ \fX^x_t &\les \alpha_d\lp 0\rp + \fL t
|
||||
% \end{align}
|
||||
% Thus it is the case that:
|
||||
% \begin{align}
|
||||
% \left| \E \lb \int^T_t \alpha_d \circ \fX_s^t ds \rb\right| &\les \left| \E \lb \int^T_t \alpha_d \lp 0\rp + \fL s ds\rb\right| \nonumber\\
|
||||
% &\les \left| \E \lb \int^T_t\alpha_d\lp 0\rp ds +\int^T_t \fL s ds\rb\right| \nonumber\\
|
||||
% &\les |\alpha_d\lp 0\rp |\lp T-t\rp + \fL \lp \frac{T^2-t^2}{2} \rp
|
||||
% \end{align}
|
||||
% We will call the right hand side as $\mathfrak{E}$.
|
||||
%
|
||||
% And it is also the case that:
|
||||
% \begin{align}
|
||||
% \left| \E \lb \lp \int^T_t \alpha_d \circ \fX^x_t \rp^2\rb\right| &\les \left| \E \lb \iint_{s,\fs=t}^T \lp \alpha_d \circ \fX^x_s\rp\lp \alpha_d \circ \fX^x_\fs\rp\rb dsd\fs\right| \nonumber\\
|
||||
% &\les |\alpha_d\lp 0\rp|^2\lp T-t\rp^2 +2\fL |\alpha_d\lp 0\rp |\lp T-t\rp\lp \frac{T^2-t^2}{2}\rp + \fL^2\lp \frac{T^2-t^2}{2}\rp \nonumber
|
||||
% \end{align}
|
||||
% Thus it is the case that:
|
||||
% \begin{align}
|
||||
% \var\lp \int_t^T\alpha_d \circ \fX^x_t\rp &\les |\alpha_d\lp 0\rp|^2\lp T-t\rp^2 +2\fL |\alpha_d\lp 0\rp |\lp T-t\rp\lp \frac{T^2-t^2}{2}\rp + \fL^2\lp \frac{T^2-t^2}{2}\rp \nonumber\\
|
||||
% &+ |\alpha_d\lp 0\rp |\lp T-t\rp + \fL \lp \frac{T^2-t^2}{2} \rp \nonumber
|
||||
% \end{align}
|
||||
% Denote the right hand side of the equation above as $\fV$. The the variance vecomes:
|
||||
%
|
||||
%
|
||||
% \end{proof}
|
||||
% \begin{corollary}
|
||||
% We may see that
|
||||
% \end{corollary}
|
||||
|
||||
% This renders (\ref{big_eqn_lhs}) as:
|
||||
% \begin{align}
|
||||
% &\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \nonumber \\
|
||||
% &\les \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right| \nonumber \\
|
||||
% &+\left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right|
|
||||
% \end{align}
|
||||
% Taking the expectation on both sides of this inequality, and applying the linearity and monotonicity of expectation yields:
|
||||
% \begin{align}
|
||||
% &\E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right|\rb \label{big_eqn_stage_2_lhs}\\
|
||||
% &\les \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\rb \label{big_eqn_stage_2_rhs_1} \\
|
||||
% &+\E\lb \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right|\rb \label{big_eqn_stage_2_rhs_2}
|
||||
% \end{align}
|
||||
% Consider now, the Lyapunov inequality applied to (\ref{big_eqn_stage_2_rhs_1}), which renders it as:
|
||||
% \begin{align}
|
||||
% &\E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\rb \nonumber\\
|
||||
% &\les \lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|^2\rb \rp^{\frac{1}{2}} \label{where_grohs_will be applied}
|
||||
% \end{align}
|
||||
% Then, \cite[Corollary~2.6]{grohsetal} applied to (\ref{where_grohs_will be applied}), then yields that:
|
||||
% \begin{align}
|
||||
% &\lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|^2\rb \rp^{\frac{1}{2}} \nonumber\\
|
||||
% &\les 2\sqrt{\frac{1}{\fn}} \lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb \right|^2\rb \rp^{\frac{1}{2}}
|
||||
% \end{align}
|
||||
% Looking back at (\ref{big_eqn_stage_2_rhs_2}), we see that the monotonicity and linearity of expectation tells us that:
|
||||
% \begin{align}
|
||||
% &\E\lb \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right|\rb \\
|
||||
% &\les \E \lb 3\ve +2\ve \left| \fu^T_d\lp x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\fu^T_d\lp x \rp\rb \\
|
||||
% &\les 3\ve +2\ve \cdot\E\lb \left| \fu^T_d\lp x\rp\right|^q\rb + 2\ve \cdot \E \lb \left| \exp \lp \int^b_afdx\rp\right|^q\rb + \ve \cdot\E \lb \left| \exp \lp \int^b_a f dx\rp - \mathfrak{e}\right|^q\rb -\fe\cdot \E \lb \fu_d^T \lp x\rp\rb \nonumber\\
|
||||
% \end{align}
|
||||
% Note that:
|
||||
% \begin{align}
|
||||
% \E\lb \mathcal{X}^{d,t,x}_s\rb &= \E\lb x + \int^t_s \sqrt{2} d\mathcal{W}^d_r\rb \nonumber\\
|
||||
% &\les x + \sqrt{2}\cdot\E \lb \int^t_s d\mathcal{W}^d_r \rb \\
|
||||
% &= x + \sqrt{2}\cdot \E \lb \mathcal{W}^d_{t-s}\rb \\
|
||||
% &= x
|
||||
% \end{align}
|
||||
% Consider now:
|
||||
% \begin{align}
|
||||
% \va \lb \cX^{d,t,x}_s\rb &= \va \lb x + \int^t_s \sqrt{2}d\cW^d_r\rb \nonumber \\
|
||||
% &= \E \lb\lp x+\int^t_s\sqrt{2}d\cW^d_r - \E \lb x+\int^t_s\sqrt{2}d\cW^d_r\rb\rp^2\rb \nonumber\\
|
||||
% &=\E \lb\lp x+\int^t_s\sqrt{2}d\cW^d_r -x\rp^2\rb \nonumber \\
|
||||
% &=2\cdot \E\lb \lp \int^t_s d\cW_r^d\rp^2\rb \nonumber\\
|
||||
% &=2\cdot \E \lb \lp \cW^d_{t-s}\rp^2\rb
|
||||
% \end{align}
|
||||
%
|
||||
%Note now that:
|
||||
%\begin{align}
|
||||
% \va \lb \cW^d_{t-s}\rb &= \E \lb \lp \cW_{t-s}^d\rp^2\rb - \E \lb \cW^d_{t-s}\rb^2 \nonumber \\
|
||||
% \E\lb \lp \cW^d_{t-s}\rp^2\rb &= \lp t-s \rp\mathbb{I}_d \\
|
||||
% 2\cdot \E\lb \lp \cW^d_{t-s}\rp^2\rb &= 2\lp t-s\rp\mathbb{I}_d
|
||||
%\end{align}
|
||||
%Now note that since $\cW^d_r$ are standard Brownian motions, and their expectation and variance are $\mymathbb{0}_d$ and $\mathbb{I}_d$ respectively. Whence it is the case that the probability density function for $\cW_{t-s}^d$ is:
|
||||
%\begin{align}
|
||||
% \ff_{\cW^d_{t-s}} \lp x\rp= \lp 2\pi\rp^{-\frac{d}{2}}\lp t-s\rp^{-\frac{1}{2}}\exp \lp \frac{-1}{2(t-s)}\mymathbb{e}_{1,d}\cdot \lb x \rb_*^2\rp
|
||||
%\end{align}
|
||||
%However $\cX^{d,t,x}_s$ is a shifted normal distribution, specifically shifted by $x$. Its p.d.f. is thus:
|
||||
%\begin{align}
|
||||
% \ff_{\cX^{d,t,x}_s}\lp \scrX \rp = \lp 2\pi\rp^{-\frac{d}{2}}\lp t-s\rp^{-\frac{1}{2}}\exp \lp \frac{-1}{2(t-s)}\mymathbb{e}_{1,d}\cdot \lb \scrX +x\rb_*^2\rp
|
||||
%\end{align}
|
||||
%The Law of the Unconscious Statistician then says that:
|
||||
%\begin{align}
|
||||
% \E \lb \fu^T_d \lp \cX^{d,t,x}_s\rp\rb = \int_{\R^d}\fu^T_d\lp \scrX\rp\cdot \ff_{\cX^{d,t,x}_s}\lp \scrX\rp d\scrX
|
||||
%\end{align}
|
||||
%And further that:
|
||||
%\begin{align}
|
||||
% \E \lb \alpha_d \lp \cX^{d,t,x}_s\rp\rb = \int_{\R^d} \alpha_d\lp \scrX\rp\cdot \ff_{\cX^{d,t,x}_s}\lp \scrX\rp d\scrX
|
||||
%\end{align}
|
||||
%
|
||||
%\textcolor{red}{\textbf{Need to re-examine $\fu^T_d, \alpha_d$}}
|
||||
%Note that It\^o's Lemma allows us to conclude that:
|
||||
%\begin{align}
|
||||
% d\:\alpha_d \lp \cX^{d,t,x}_s\rp = \alpha_d^{'}\lp \cX^{d,t,x}_s\rp d\cX_t+\alpha_d^{''}\lp \cX_t\rp dt
|
||||
%\end{align}
|
||||
%
|
||||
%Now note this that Fubini's theorem states that:
|
||||
%\begin{align}\label{fubinis_to_integral}
|
||||
% \E \lb \int^T_t \alpha_d \circ \cX^{d,t,x}_s ds\rb = \int^T_t \E \lb \alpha_d\circ \cX^{d,t,x}_s\rb ds
|
||||
%\end{align}
|
||||
%
|
||||
%
|
||||
|
||||
|
||||
% Note that by Lemma \ref{iobm} each of the $\mathcal{X}^{d,t,x}_{r,\omega_i}$ are pairwise independent of each other for all $i \in \{1,2,\hdots,\mathfrak{n}\}$. Note also that by Definition \ref{def:brown_motion} it is the case, for all $\omega_i \in \Omega$ that $\mathcal{X}^{d,t,x}_{T,\omega_i} \sim \norm \lp \mymathbb{0}_d, \diag_d(T) \rp$
|
||||
|
||||
Note for the first summand that it is in $\mathcal{O}\lp \frac{1}{\sqrt{\mathfrak{n}}}\rp$. Notice that both $f$ and $\fu^T_d$ are continuous functions for $d\in \N$. Note also that $F:[t,T] \rightarrow \R$ defined as:
|
||||
\begin{align}
|
||||
F(\fx) \coloneqq \int_t^\ft f\lp\fx\rp dx
|
||||
\end{align}
|
||||
is continuous on $\lb t,T\rb$. Thus , notice that \cite[Theorem~2.1]{rio_moment_2009} with $k$
|
||||
|
||||
\end{proof}
|
||||
\begin{remark}
|
||||
Note that diagrammatically, this can be represented as in figure below.
|
||||
\begin{figure}[h]
|
||||
|
|
@ -1099,7 +854,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=0.9]
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
%uncomment if require: \path (0,475); %set diagram left start at 0, and has a height of 475
|
||||
|
||||
%Shape: Rectangle [id:dp5014556157804896]
|
||||
|
|
@ -1202,7 +957,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
\draw [shift={(17.22,237)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
|
||||
% Text Node
|
||||
\draw (428.28,22.2) node [anchor=north west][inner sep=0.75pt] {$\mathsf{E}^{N,h,q,\ve}_{n}$};
|
||||
\draw (428.28,22.2) node [anchor=north west][inner sep=0.75pt] {$\mathsf{E}_{N,n,h,q,\varepsilon }^{\exp ,f}$};
|
||||
% Text Node
|
||||
\draw (444.46,108.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{G}_d$};
|
||||
% Text Node
|
||||
|
|
@ -1218,9 +973,9 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
% Text Node
|
||||
\draw (535.1,28.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}_{1}^{N+1}$};
|
||||
% Text Node
|
||||
\draw (534.54,108.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mymathbb{0}}{}_{_{d}{}_{,}{}_{d} ,\mathcal{X}}$};
|
||||
\draw (534.54,108.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mathbb{0}}{}_{_{d}{}_{,}{}_{d} ,\mathcal{X}}$};
|
||||
% Text Node
|
||||
\draw (426.15,340.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{E}^{N,h,q,\ve}_{n}$};
|
||||
\draw (426.15,340.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{E}_{N,n,h,q,\varepsilon }^{\exp ,f}$};
|
||||
% Text Node
|
||||
\draw (442.34,426.8) node [anchor=north west][inner sep=0.75pt] {$\mathsf{G}_d$};
|
||||
% Text Node
|
||||
|
|
@ -1236,17 +991,17 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
% Text Node
|
||||
\draw (532.97,346.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}_{1}^{N+1}$};
|
||||
% Text Node
|
||||
\draw (532.41,426.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mymathbb{0}}{}_{_{d}{}_{,}{}_{d} ,\mathcal{X}}$};
|
||||
\draw (532.41,426.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mathbb{0}}{}_{_{d}{}_{,}{}_{d} ,\mathcal{X}}$};
|
||||
% Text Node
|
||||
\draw (444,215.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (553,215.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
\draw (553,216.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (336,215.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (619,215.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
\draw (619,214.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (234,215.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
\draw (234,217.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $};
|
||||
% Text Node
|
||||
\draw (132,221.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}$};
|
||||
% Text Node
|
||||
|
|
@ -1259,9 +1014,12 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\caption{Neural network diagram for the $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn}$ network.}
|
||||
\caption{Neural network diagram for the $\mathsf{UES}$ network.}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
\begin{remark}
|
||||
It may be helpful to think of this as a very crude form of ensembling.
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
||||
|
|
@ -1288,6 +1046,5 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -9,12 +9,12 @@ Parts of this code have been released on \texttt{CRAN} under the package name \t
|
|||
|
||||
\lstinputlisting[language = R, style = rstyle, label = activations, caption = {R code for activation functions ReLU and Sigmoid}]{"/Users/shakilrafi/R-simulations/activations.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = instantiation, caption = {R code for intanitation}]{"/Users/shakilrafi/R-simulations/instantiation.R"}
|
||||
\lstinputlisting[language = R, style = rstyle, label = instantiation, caption = {R code for realizations}]{"/Users/shakilrafi/R-simulations/instantiation.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = stk, caption = {R code for parallelizing two neural networks}]{"/Users/shakilrafi/R-simulations/stacking.R"}
|
||||
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = affn, caption = {R code for affine neural networks}]{"/Users/shakilrafi/R-simulations/Aff.R"}
|
||||
\lstinputlisting[language = R, style = rstyle, label = Aff, caption = {R code for affine neural networks}]{"/Users/shakilrafi/R-simulations/Aff.R"}
|
||||
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = comp, caption = {R code for composition of two neural networks}]{"/Users/shakilrafi/R-simulations/comp.R"}
|
||||
|
|
@ -41,6 +41,9 @@ Parts of this code have been released on \texttt{CRAN} under the package name \t
|
|||
\lstinputlisting[language = R, style = rstyle, label = Phi, caption = {R code for $\Phi$}]{"/Users/shakilrafi/R-simulations/Phi.R"}
|
||||
|
||||
|
||||
\includegraphics{"/Users/shakilrafi/R-simulations/Phi_properties/Phi_diff_contour.png"}
|
||||
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Phi_properties, caption = {R code for simulations involving $\Phi$}]{"/Users/shakilrafi/R-simulations/Phi_properties.R"}
|
||||
|
||||
|
||||
|
|
@ -48,35 +51,17 @@ Parts of this code have been released on \texttt{CRAN} under the package name \t
|
|||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Sqr_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Sqr_properties.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Prd, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Sqr_properties.R"}
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Pwr.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr, caption = {R code for $\pwr^{q,\ve}$ networks}]{"/Users/shakilrafi/R-simulations/Pwr.R"}
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Pwr_3_properties.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\pwr_3^{q,\ve}$}]{"/Users/shakilrafi/R-simulations/Pwr_3_properties.R"}
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Nrm.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Nrm, caption = {R code simulations involving $\nrm^d_1$}]{"/Users/shakilrafi/R-simulations/Nrm.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Mxm, caption = {R code simulations involving $\mxm_d$}]{"/Users/shakilrafi/R-simulations/Mxm.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Tay, caption = {R code simulations involving $\tay$, note that this implementation is different from how it is presented in the exposition. We chose to explicitly define the $\tay$ network, and let neural network exponentials, cosines, and sines be instantiations of this network with various different coefficients.}]{"/Users/shakilrafi/R-simulations/Tay.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Mxm, caption = {R code simulations for $\csn_n^{q,\ve}$}]{"/Users/shakilrafi/R-simulations/Csn.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Mxm, caption = {R code simulations for $\sne_n^{q,\ve}$}]{"/Users/shakilrafi/R-simulations/Sne.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Etr, caption = {R code simulations involving $\etr$}]{"/Users/shakilrafi/R-simulations/Etr.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = MC, caption = {R code simulations involving $\etr$}]{"/Users/shakilrafi/R-simulations/MC.R"}
|
||||
|
||||
\newpage
|
||||
|
||||
\begin{center}
|
||||
\textbf{Vita}
|
||||
\end{center}
|
||||
|
||||
The author was born in November 1\textsuperscript{st}, 1992 in the city of Dhaka in the heart of Bangladesh. He grew up in the large city with a childhood that included setting things on fire, and very occasionally focusing on mathematics. He failed to achieve his childhood goal of becoming an astronomer however when he entered college at Troy University in 2011 and realized it would involve cold nights outside, and so chose mathematics instead. He has continued his pursuits in mathematics and is now a graduate student at the University of Arkansas trying to graduate.
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Mxm.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\tay$}]{"/Users/shakilrafi/R-simulations/Tay.R"}
|
||||
|
||||
\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\etr$}]{"/Users/shakilrafi/R-simulations/Etr.R"}
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -1,3 +0,0 @@
|
|||
%\nocite{*}
|
||||
|
||||
|
||||
|
|
@ -26,7 +26,7 @@ Substituting (\ref{4.0.1}) and (\ref{4.0.2}) into (\ref{3.3.20}) renders (\ref{
|
|||
v(t,x) &= \E \lb v\lp T, \mathcal{X}_T^{t,x} \rp \rb + \int ^T_t \E \lb f \lp s, \mathcal{X}^{t,x}_s, v \lp s, \mathcal{X}^{t,x}_s \rp \rp ds\rb \nonumber\\
|
||||
v\lp t,x \rp &= \E \lb g\lp \mathcal{X}^{t,x}_T \rp \rb+ \int^T_t \E \lb \lp F \lp v \rp \rp \lp s,\mathcal{X}^{t,x}_s\rp \rb ds\nonumber
|
||||
\end{align}
|
||||
\label{def:1.18}\label{Setting 1.1} Let $d,m \in \mathbb{N}$, $T, \mathfrak{L},p \in [0,\infty)$, $\mathfrak{p} \in [2,\infty)$ $\mathfrak{m} = \mathfrak{k}_{\mathfrak{p}}\sqrt{\mathfrak{p}-1}$, $\Theta = \bigcup_{n\in \mathbb{N}}\mathbb{Z}^n$, \\ $f \in C\lp \lb 0,T \rb \times \R^d \times \R \rp $, $g \in C(\mathbb{R}^d,\mathbb{R})$, let $F: C \lp \lb 0,T \rb \times \R^d, \R \rp \rightarrow C \lp \lb 0,T \rb \times \R^d, \R \rp$ assume for all $t \in [0,T],x\in \mathbb{R}^d$ that:
|
||||
\label{def:1.18}\label{Setting 1.1} Let $d,m \in \mathbb{N}$, $T, \mathfrak{L},p \in [0,\infty)$, $\mathfrak{p} \in [2,\infty)$ $\mathfrak{m} = \mathfrak{k}_{\mathfrak{p}}\sqrt{\mathfrak{p}-1}$, $\Theta = \bigcup_{n\in \mathbb{N}}\mathbb{Z}^n$, $f \in C\lp \lb 0,T \rb \times \R^d \times \R \rp $, $g \in C(\mathbb{R}^d,\mathbb{R})$, let $F: C \lp \lb 0,T \rb \times \R^d, \R \rp \rightarrow C \lp \lb 0,T \rb \times \R^d, \R \rp$ assume for all $t \in [0,T],x\in \mathbb{R}^d$ that:
|
||||
\begin{align}\label{(1.12)}
|
||||
\lv f\lp t,x,w \rp - f\lp t,x,\mathfrak{w} \rp \rv \leqslant L \lv w - \mathfrak{w} \rv &&\max\left\{\lv f \lp t,x,0 \rp \rv, \lv g(x) \rv \right\} \leqslant \mathfrak{L} \lp 1+\|x\|_E^p \rp
|
||||
\end{align}
|
||||
|
|
|
|||
|
|
@ -1,47 +1,46 @@
|
|||
\chapter{Conclusions and Further Research}
|
||||
|
||||
We will present three avenues of further research and related work on parameter estimates here. We will present these as a series of recommendations and conjectures to further extend this framework for understanding neural networks.
|
||||
We will present three avenues of further research and related work on parameter estimates here.
|
||||
|
||||
\section{Further operations}
|
||||
\section{Further operations and further kinds of neural networks}
|
||||
|
||||
Note, for instance, that several classical operations are done on neural networks that have yet to be accounted for in this framework and talked about in the literature. We will discuss one of them \textit{dropout} and discuss how they may be brought into this framework.
|
||||
Note, for instance, that several classical operations are done on neural networks that have yet to be accounted for in this framework and talked about in the literature. We will discuss two of them \textit{dropout} and \textit{dilation} and provide lemmas that may be useful to future research.
|
||||
|
||||
\subsection{Mergers and Dropout}
|
||||
|
||||
Overfitting presents an important challenge for all machine learning models, including deep learning. There exists a technique called \textit{dropout} introduced in \cite{srivastava_dropout_2014} that seeks to ameliorate this situation.
|
||||
\begin{definition}[Hadamard Product]
|
||||
Let $m,n \in \N$. Let $A,B \in \R^{m \times n}$. For all $i \in \{ 1,2,\hdots,m\}$ and $j \in \{ 1,2,\hdots,n\}$ define the Hadamard product $\odot: \R^{m\times n} \times \R^{m \times n} \rightarrow \R^{m \times n}$ as:
|
||||
\begin{align}
|
||||
A \odot B \coloneqq \lb A \odot B \rb _{i,j} = \lb A \rb_{i,j} \times \lb B \rb_{i,j} \quad \forall i,j
|
||||
\end{align}
|
||||
\end{definition}
|
||||
We will define the dropout operator introduced in \cite{srivastava_dropout_2014}, and explained further in \cite{Goodfellow-et-al-2016}.
|
||||
|
||||
|
||||
\begin{definition}[Instantiation with dropout]
|
||||
Let $\nu \in \neu$, $L,n \in \N$, $p \in \lp 0,1\rp$, $\lay \lp \nu\rp = \lp l_0,l_1,\hdots, \l_L\rp$, and that $\nu = \lp \lp W_1,b_1\rp, \lp W_2,b_2\rp, \hdots , \lp W_L,b_L\rp \rp$. Let it be the case that for each $n\in \N$, $\rho_n = \{ x_1,x_2,\hdots,x_n\} \in \R^n$ where for each $i \in \{1,2,\hdots,n\}$ it is the case that $x_i \sim \bern(p)$. We will then denote $\real_{\act}^{D,p} \lp \nu \rp \in C\lp \R^{\inn\lp \nu\rp},\R^{\out\lp \nu \rp}\rp$, the continuous function given by:
|
||||
\begin{definition}[Scalar product of weights]
|
||||
Let $\nu \in \neu$, $L\in \N$, $i,j,k \in \N$, and $c\in \R$. Assume also that $\lay \lp \nu\rp = \lp l_0,l_1,l_2,\hdots, l_L\rp$. Assume then that the neural network is given by $\nu = \lp \lp W_1,b_1\rp, \lp W_2,b_2\rp,\hdots, \lp W_L,b_L\rp\rp$. We will denote by $c\circledast^{i,j}\nu$ as the neural network which, for $i \in \N \cap \lb 1,L-1\rb$, $j \in \N \cap \lb 1,l_i\rb$, is given by $c \circledast^{i,j} \nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2\rp, \hdots,\lp \tilde{W}_i,b_i \rp,\lp \tilde{W}_{i+1},b_{i+1}\rp,\hdots \lp W_L,b_L\rp\rp$ where it is the case that:
|
||||
\begin{align}
|
||||
\real_{\act}^{D,p}\lp \nu \rp = \rho_{l_L}\odot \act \lp W_l\lp \rho_{l_{L-1}} \odot \act \lp W_{L-1}\lp \hdots\rp + b_{L-1}\rp\rp + b_L\rp
|
||||
\tilde{W}_i = \lp \mymathbb{k}^{j,j,c-1}_{l_i,l_{i}} + \mathbb{I}_{l_i}\rp W_i
|
||||
\end{align}
|
||||
\end{definition}
|
||||
Dropout is an example of \textit{ensemble learning}, a form of learning where versions of our model (e.g. random forests or neural networks) are made (e.g. by dropout for neural networks or by enforcing a maximum depth to the trees in our forest), and a weighted average of the predictions of our different models is taken to be the predictive model. That such a model can work, and indeed work well, is the subject of \cite{schapire_strength_1990}.
|
||||
\begin{definition}[The Dropout Operator]
|
||||
Let $\nu \in \neu$, $L\in \N$, $i_1,i_2,\hdots, i_k,j,k \in \N$, and $c_1,c_2,\hdots,c_k\in \R$. Assume also that $\lay \lp \nu\rp = \lp l_0,l_1,l_2,\hdots, l_L\rp$. Assume then that the neural network is given by $\nu = \lp \lp W_1,b_1\rp, \lp W_2,b_2\rp,\hdots, \lp W_L,b_L\rp\rp$. We will denote by $\dropout_n^{\unif}\lp \nu \rp$ the neural network that is given by:
|
||||
\begin{align}
|
||||
0\circledast^{i_1,j_1} \lp 0 \circledast^{i_2,j_2}\lp \hdots 0\circledast^{i_n,j_n}\nu \hdots \rp\rp
|
||||
\end{align}
|
||||
Where for each $k \in \{1,2,\hdots,n \}$ it is the case that $i \sim \unif \{ 1,L-1\}$ and $j\sim \unif\{1,l_j\} $
|
||||
\end{definition}
|
||||
|
||||
We will also define the dropout operator introduced in \cite{srivastava_dropout_2014}.
|
||||
|
||||
|
||||
\begin{definition}[Realization with dropout]
|
||||
Let $\nu \in \neu$, $L,n \in \N$, $p \in \lp 0,1\rp$, $\lay \lp \nu\rp = \lp l_0,l_1,\hdots, \l_L\rp$, and that $\neu = \lp \lp W_1,b_1\rp, \lp W_2,b_2\rp, \hdots , \lp W_L,b_L\rp \rp$. Let it be the case that for each $n\in \N$, $\rho_n = \{ x_1,x_2,\hdots,x_n\} \in \R^n$ where for each $i \in \{1,2,\hdots,n\}$ it is the case that $x_i \sim \bern(p)$. We will then denote $\real_{\rect}^{D} \lp \nu \rp \in C\lp \R^{\inn\lp \nu\rp},\R^{\out\lp \nu \rp}\rp$, the continuous function given by:
|
||||
\begin{align}
|
||||
\real_{\rect}^D\lp \nu \rp = \rho_{l_L}\odot \rect \lp W_l\lp \rho_{l_{L-1}} \odot \rect \lp W_{L-1}\lp \hdots\rp + b_{L-1}\rp\rp + b_L\rp
|
||||
\end{align}
|
||||
\end{definition}
|
||||
|
||||
|
||||
\section{Further Approximants}
|
||||
|
||||
In theory the approximation schemes given in the case of $\xpn_n^{q,\ve}, \csn_n^{q,\ve}$, and $\sne_n^{q,\ve}$ given in the previous sections, could be used to approximate more transcendental functions, and identities such as alluded to in Remark \ref{rem:pyth_idt}. Indeed, recent attempts have been made to approximate backwards and forward Euler methods as in \cite{grohs2019spacetime}. In fact, this architecture was originally envisioned to approximate, Multi-Level Picard iterations, as seen in \cite{ackermann2023deep}. These neural network methods have been proven to beat the curse of dimensionality in the sense that the size of these networks (parameter and depth counts) grow only polynomially with respect to the desired accuracy. In practice, it remains to be seen whether for larger dimensions, the increased number of operations and architectures to contend with do not make up for the polynomial increase in parameter and depths, especially when it comes to computaiton time.
|
||||
|
||||
In a similar vein, these architectures have so far lacked a consistent implementation in a widely available programming language. Part of the dissertation work has been focused on implementing these architectures as an $\texttt{R}$ package, available at \texttt{CRAN}.
|
||||
|
||||
\section{Algebraic Properties of this Framework}
|
||||
|
||||
It is quite straightforward to see that the instantiation operation has sufficiently functorial properties, at the very least, when instantiating with the identity function. More specifically consider the category \texttt{Mat} whose objects are natural numbers, $m,n$, and whose arrows $m \xleftarrow{A} n$ are matrices $A \in \R^{m\times n}$, i.e. a continuous function between vector spaces $\R^n$ and $\R^m$ respectively. Consider as well the set of neural networks $\nu \subsetneq \neu$ where $\inn\lp \nu \rp = n$ and $\out\lp \nu \rp = m$.
|
||||
\\
|
||||
In such a case, note that the instantiation operation preserves the axiom of functoriality, namely that composition is respected under instantiation. Note also that we have alluded to the fact that under neural network composition, with $\id$ (the appropriate one for our dimension) behaves like a monoid under instantiation.
|
||||
|
||||
Note for example that a neural network analog for derivatives, one that respects the chain rule under instantiation already exist in the literature, e.g. \cite{nn_diff}. Thus there is a growing and rather rich and growing set of algebraic operations that are and have been proposed for neural networks.
|
||||
|
||||
|
||||
Taken together, these facts seem to imply that a further exploration of the algebraic properties of this artificial neural network framework could present a fruitful avenue of future study. Much remains to be studied.
|
||||
|
||||
This completes this Dissertation.
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -1,60 +0,0 @@
|
|||
\relax
|
||||
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|
||||
\providecommand\hyper@newdestlabel[2]{}
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||||
\@setckpt{front_matter}{
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}
|
||||
|
|
@ -1,161 +0,0 @@
|
|||
\begin{singlespace}
|
||||
\begin{center}
|
||||
Analysis and Construction of Artificial Neural Networks for the Heat Equations, and Their Associated Parameters, Depths, and Accuracies.
|
||||
\end{center}
|
||||
\vspace{0.5cm}
|
||||
\begin{center}
|
||||
A dissertation submitted in partial fulfillment \\
|
||||
of the requirements for the degree of \\
|
||||
Doctor of Philosophy in Mathematics
|
||||
\end{center}
|
||||
\vspace{1cm}
|
||||
\begin{center}
|
||||
by
|
||||
\end{center}
|
||||
\vspace{0.5cm}
|
||||
\begin{center}
|
||||
Shakil Ahmed Rafi \\
|
||||
Troy University \\
|
||||
Bachelor of Science in Mathematics, 2015 \\
|
||||
University of Arkansas \\
|
||||
Master of Science in Mathematics, 2019
|
||||
\end{center}
|
||||
\vspace{0.5cm}
|
||||
\begin{center}
|
||||
May 2024 \\
|
||||
University of Arkansas
|
||||
\end{center}
|
||||
\vspace{0.5cm}
|
||||
|
||||
This dissertation is approved for recommendation to the Graduate Council.
|
||||
|
||||
\vspace{1.5cm}
|
||||
\begin{center}
|
||||
\noindent\hspace*{0cm}\rule{7cm}{0.7pt} \\
|
||||
Joshua Lee Padgett, Ph.D.\\
|
||||
Dissertation Director, \textit{ex-officio}
|
||||
\end{center}
|
||||
\vspace{1cm}
|
||||
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{center}
|
||||
\noindent\hspace*{0cm}\rule{7cm}{0.7pt} \\
|
||||
Ukash Nakarmi, Ph.D.\\
|
||||
Committee Member
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
\begin{raggedleft}
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{center}
|
||||
\noindent\hspace*{0cm}\rule{7cm}{0.7pt} \\
|
||||
Jiahui Chen, Ph.D.\\
|
||||
Committee Member
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
\end{raggedleft}
|
||||
|
||||
\vspace{0.5cm}
|
||||
\begin{center}
|
||||
\noindent\hspace*{0cm}\rule{7cm}{0.7pt} \\
|
||||
Tulin Kaman, Ph.D.\\
|
||||
Committee Member
|
||||
\end{center}
|
||||
\vspace{1cm}
|
||||
\end{singlespace}
|
||||
|
||||
\newpage
|
||||
\begin{center}
|
||||
\textbf{Abstract}
|
||||
\end{center}
|
||||
This dissertation seeks to explore a certain calculus for artificial neural networks. Specifically we will be looking at versions of the heat equation, and exploring strategies on how to approximate them.
|
||||
\\~\\
|
||||
Our strategy towards the beginning will be to take a technique called Multi-Level Picard (MLP), and present a simplified version of it showing that it converges to a solution of the equation $\lp \frac{\partial}{\partial t}u_d\rp\lp t,x\rp = \lp \nabla^2_x u_d\rp\lp t,x\rp$.
|
||||
\\~\\
|
||||
We will then take a small detour exploring the viscosity super-solution properties of solutions to such equations. It is here that we will first encounter Feynman-Kac, and see that solutions to these equations can be expressed the expected value of a certain stochastic integral.
|
||||
\\~\\
|
||||
The final and last part of the dissertation will be dedicated to expanding a certain neural network framework. We will build on this framework by introducing new operations, namely raising to a power, and use this to build out neural network polynomials. This opens the gateway for approximating transcendental functions such as $\exp\lp x\rp,\sin\lp x\rp$, and $\cos\lp x\rp$. This, coupled with a trapezoidal rule mechanism for integration allows us to approximate expressions of the form $\exp \lp \int_a^b \square dt\rp$.
|
||||
\\~\\
|
||||
We will, in the last chapter, look at how the technology of neural networks developed in the previous two chapters work towards approximating the expression that Feynman-Kac asserts must be the solution to these modified heat equations. We will then end by giving approximate bounds for the error in the Monte Carlo method. All the while we will maintain that the parameter estimates and depth estimates remain polynomial on $\frac{1}{\ve}$.
|
||||
\\~\\
|
||||
As an added bonus we will also look at the simplified MLP technque from the previous chapters of this dissertation and show that yes, they can indeed be approximated with artificial neural networks, and that yes, they can be done so with neural networks whose parameters and depth counts grow only polynomially on $\frac{1}{\ve}$.
|
||||
\\~\\
|
||||
Our appendix will contain code listings of these neural network operations, some of the architectures, and some small scale simulation results.
|
||||
|
||||
\newpage
|
||||
|
||||
\begin{center}
|
||||
\vspace*{\fill}
|
||||
\copyright\: 2024 by Shakil Ahmed Rafi \\
|
||||
All Rights Reserved.
|
||||
\vspace*{\fill}
|
||||
\end{center}
|
||||
|
||||
\newpage
|
||||
\begin{center}
|
||||
\textbf{Acknowledgements}
|
||||
\end{center}
|
||||
I would like to acknowledge my advisor Dr. Joshua Padgett who has been instrumental in me Ph.D. journey. I am incredibly thankful for him taking the time out of his busy schedule to meet with me over the weekends and helping me finish my dissertation. Without his help, guidance, and patience I would never have been where I am today. You not only taught me mathematics, but also how to be a mathematician. Thank you.
|
||||
\\~\\
|
||||
I would also like to thank my department, and everyone there, including, but not limited to Dr. Andrew Raich, for his incredible patience and helpful guidance throughout the years. I would also like to thank Dr. Ukash Nakarmi for the excellent collaboartions I've had. I would also like to thank Egan Meaux for all the little things he does to keep the department going.
|
||||
\\~\\
|
||||
I would like to acknowledge Marufa Mumu for believing in me when I didn't. You really made the last few months of writing this dissertation, less painful.
|
||||
\\~\\
|
||||
I would like to acknowledge my cat, a beautiful Turkish Angora, Tommy. He was pretty useless, but stroking his fur made me stress a little less.
|
||||
\\~\\
|
||||
I would like to acknowledge my office-mate Eric Walker, without whom I would never have realized that rage and spite are equally as valid motivators as encouragement and praise.
|
||||
\\~\\
|
||||
Finally, I would like to thank Valetta Ventures, Inc. and their product Texifier. It is marvel of software engineering and made the process of creating this dissertation much less painful than it already was.
|
||||
|
||||
\newpage
|
||||
|
||||
\begin{center}
|
||||
\textbf{Dedication}\\
|
||||
To my grandparents, \\
|
||||
M.A. Hye, M.A., \& Nilufar Hye\\
|
||||
who would've love to see this but can't; \\
|
||||
to my parents, \\
|
||||
Kamal Uddin Ahmed, M.A. \& Shahnaz Parveen, M.A.,\\
|
||||
who kept faith in me, always; \\
|
||||
and finally to my brothers, \\
|
||||
Wakil Ahmed Shabi, BBA \& Nabbil Ahmed Sami, B.Eng., \\
|
||||
for whom I have been somewhat imperfect a role model.\\
|
||||
|
||||
|
||||
|
||||
|
||||
\vspace*{\fill}
|
||||
\end{center}
|
||||
|
||||
\newpage
|
||||
|
||||
\begin{center}
|
||||
\textbf{Epigraph}\\~\\
|
||||
\textit{Read, in the name of your Lord}\\
|
||||
\textemdash Surah Al-Alaq:\:1\\~\\
|
||||
\textit{The conquest of nature must be achieved with number and measure.} \\
|
||||
\textemdash Ren\'e Descartes \\
|
||||
\vspace*{\fill}
|
||||
\end{center}
|
||||
|
||||
\newpage
|
||||
|
||||
\tableofcontents
|
||||
|
||||
\listoffigures
|
||||
|
||||
\listoftables
|
||||
|
||||
\newpage
|
||||
\textbf{List of Published Papers} \\~\\
|
||||
Parts of Chapter \ref{chp:ann_prod} have been made into a paper as \textit{An Algebraic Framework for Understanding Fully Connected Feedforward Artificial Neural Networks, and Their Associated Parameter, Depth, and Accuracy Properties} by Rafi S., Padgett, J.L., and Nakarmi, U. and is currently undergoing review for publication for ICML 2024 at Vienna, Austria.
|
||||
\\~\\
|
||||
Parts of the simulation codebase have been submitted for review as \textit{nnR: Neural Networks Made Algebraic} at \textit{The R Journal}. They have further been published as a package \texttt{nnR} currently available on \texttt{CRAN}.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
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|
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|
||||
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|
||||
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|
||||
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|
||||
\newlabel{9.7.33}{{9.7.44}{165}{Lipschitz function approximations}{equation.9.7.44}{}}
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\@input{ann_rep_brownian_motion_monte_carlo.aux}
|
||||
\citation{*}
|
||||
\bibdata{Dissertation.bib}
|
||||
\bibcite{bass_2011}{{1}{2011}{{Bass}}{{}}}
|
||||
\bibcite{Beck_2021}{{2}{2021a}{{Beck et~al.}}{{}}}
|
||||
\bibcite{BHJ21}{{3}{2021b}{{Beck et~al.}}{{}}}
|
||||
\bibcite{bhj20}{{4}{2021c}{{Beck et~al.}}{{}}}
|
||||
\bibcite{crandall_lions}{{5}{1992}{{Crandall et~al.}}{{}}}
|
||||
\bibcite{da_prato_zabczyk_2002}{{6}{2002}{{Da~Prato and Zabczyk}}{{}}}
|
||||
\bibcite{durrett2019probability}{{7}{2019}{{Durrett}}{{}}}
|
||||
\bibcite{golub2013matrix}{{8}{2013}{{Golub and Van~Loan}}{{}}}
|
||||
\bibcite{grohsetal}{{9}{2018}{{Grohs et~al.}}{{}}}
|
||||
\bibcite{grohs2019spacetime}{{10}{2023}{{Grohs et~al.}}{{}}}
|
||||
\bibcite{Grohs_2022}{{11}{2022}{{Grohs et~al.}}{{}}}
|
||||
\bibcite{Gyngy1996ExistenceOS}{{12}{1996}{{Gy{\"o}ngy and Krylov}}{{}}}
|
||||
\bibcite{hutzenthaler_overcoming_2020}{{13}{2020a}{{Hutzenthaler et~al.}}{{}}}
|
||||
\bibcite{hutzenthaler_strong_2021}{{14}{2021}{{Hutzenthaler et~al.}}{{}}}
|
||||
\bibcite{hjw2020}{{15}{2020b}{{Hutzenthaler et~al.}}{{}}}
|
||||
\bibcite{Ito1942a}{{16}{1942a}{{It\^o}}{{}}}
|
||||
\bibcite{Ito1946}{{17}{1942b}{{It\^o}}{{}}}
|
||||
\bibcite{karatzas1991brownian}{{18}{1991}{{Karatzas and Shreve}}{{}}}
|
||||
\bibcite{rio_moment_2009}{{19}{2009}{{Rio}}{{}}}
|
||||
\bibstyle{apalike}
|
||||
\@writefile{toc}{\contentsline {chapter}{Appendices}{180}{section*.3}\protected@file@percent }
|
||||
\newlabel{pythoncode}{{1}{181}{Python Code}{lstlisting.a..1}{}}
|
||||
\newlabel{pythoncode@cref}{{[lstlisting][1][0]1}{[1][181][]181}}
|
||||
\@writefile{lol}{\contentsline {lstlisting}{\numberline {1}Python Code}{181}{lstlisting.a..1}\protected@file@percent }
|
||||
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|
||||
\newlabel{pythoncode@cref}{{[lstlisting][2][0]2}{[1][183][]183}}
|
||||
\@writefile{lol}{\contentsline {lstlisting}{\numberline {2}Python Code}{183}{lstlisting.a..2}\protected@file@percent }
|
||||
\gdef \@abspage@last{185}
|
||||
|
|
@ -0,0 +1,120 @@
|
|||
\begin{thebibliography}{}
|
||||
|
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||||
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||||
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||||
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||||
\newblock On existence and uniqueness properties for solutions of stochastic
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|
||||
\newblock {\em Discrete \& Continuous Dynamical Systems - B}, 26(9):4927.
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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semilinear parabolic partial differential equations.
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||||
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||||
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||||
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||||
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|
||||
dimensionality in the numerical approximation of Black-Scholes partial
|
||||
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||||
\newblock Papers 1809.02362, arXiv.org.
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||||
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||||
\bibitem[Grohs et~al., 2023]{grohs2019spacetime}
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||||
Grohs, P., Hornung, F., Jentzen, A., and Zimmermann, P. (2023).
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||||
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||||
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||||
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||||
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||||
\bibitem[Grohs et~al., 2022]{Grohs_2022}
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||||
Grohs, P., Jentzen, A., and Salimova, D. (2022).
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||||
\newblock Deep neural network approximations for solutions of {PDEs} based on
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||||
monte carlo algorithms.
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||||
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||||
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||||
\bibitem[Gy{\"o}ngy and Krylov, 1996]{Gyngy1996ExistenceOS}
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||||
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approximations.
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\newblock {\em Probability Theory and Related Fields}, 105:143--158.
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||||
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||||
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||||
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||||
\newblock Overcoming the curse of dimensionality in the numerical approximation
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||||
of semilinear parabolic partial differential equations.
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\newblock {\em Proceedings of the Royal Society A: Mathematical, Physical and
|
||||
Engineering Sciences}, 476(2244):20190630.
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||||
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||||
\bibitem[Hutzenthaler et~al., 2021]{hutzenthaler_strong_2021}
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||||
Hutzenthaler, M., Jentzen, A., Kuckuck, B., and Padgett, J.~L. (2021).
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||||
\newblock Strong {$L^p$}-error analysis of nonlinear {Monte} {Carlo}
|
||||
approximations for high-dimensional semilinear partial differential
|
||||
equations.
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||||
\newblock Technical Report arXiv:2110.08297, arXiv.
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||||
\newblock arXiv:2110.08297 [cs, math] type: article.
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||||
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||||
\bibitem[Hutzenthaler et~al., 2020b]{hjw2020}
|
||||
Hutzenthaler, M., Jentzen, A., and von Wurstemberger~Wurstemberger (2020b).
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||||
\newblock {Overcoming the curse of dimensionality in the approximative pricing
|
||||
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||||
\newblock {\em Electronic Journal of Probability}, 25(none):1 -- 73.
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||||
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||||
\bibitem[It\^o, 1942a]{Ito1942a}
|
||||
It\^o, K. (1942a).
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||||
\newblock Differential equations determining {Markov} processes (original in
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||||
{Japanese}).
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||||
\newblock {\em Zenkoku Shijo Sugaku Danwakai}, 244(1077):1352--1400.
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||||
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||||
\bibitem[It\^o, 1942b]{Ito1946}
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||||
It\^o, K. (1942b).
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||||
\newblock On a stochastic integral equation.
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||||
\newblock {\em Proc. Imperial Acad. Tokyo}, 244(1077):1352--1400.
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||||
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||||
\bibitem[Karatzas and Shreve, 1991]{karatzas1991brownian}
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||||
Karatzas, I. and Shreve, S. (1991).
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||||
\newblock {\em Brownian Motion and Stochastic Calculus}.
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||||
\newblock Graduate Texts in Mathematics (113) (Book 113). Springer New York.
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||||
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||||
\bibitem[Rio, 2009]{rio_moment_2009}
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||||
Rio, E. (2009).
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||||
\newblock Moment {Inequalities} for {Sums} of {Dependent} {Random} {Variables}
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||||
under {Projective} {Conditions}.
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||||
\newblock {\em J Theor Probab}, 22(1):146--163.
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||||
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||||
\end{thebibliography}
|
||||
|
|
@ -55,16 +55,18 @@ journal={Zenkoku Shijo Sugaku Danwakai},
|
|||
year="1942",
|
||||
volume="244",
|
||||
number="1077",
|
||||
pages={1352{\textendash}1400}
|
||||
pages={1352-1400},
|
||||
URL="https://cir.nii.ac.jp/crid/1573105975386021120"
|
||||
}
|
||||
@article{Ito1946,
|
||||
author={It\^o, K.},
|
||||
title={On a stochastic integral equation},
|
||||
journal={Proc. Imperial Acad. Tokyo},
|
||||
year={1942},
|
||||
volume={244},
|
||||
number={1077},
|
||||
pages={1352{\textendash}1400
|
||||
year="1942",
|
||||
volume="244",
|
||||
number="1077",
|
||||
pages="1352-1400",
|
||||
URL="https://cir.nii.ac.jp/crid/1573105975386021120"
|
||||
}
|
||||
|
||||
@inbook{bass_2011, place={Cambridge}, series={Cambridge Series in Statistical and Probabilistic Mathematics}, title={Brownian Motion}, DOI={10.1017/CBO9780511997044.004}, booktitle={Stochastic Processes}, publisher={Cambridge University Press}, author={Bass, Richard F.}, year={2011}, pages={6–12}, collection={Cambridge Series in Statistical and Probabilistic Mathematics}}
|
||||
|
|
@ -193,7 +195,7 @@ type: article},
|
|||
author = {Crandall, Michael G. and Ishii, Hitoshi and Lions, Pierre-Louis},
|
||||
year = {1992},
|
||||
keywords = {comparison theorems, dynamic programming, elliptic equations, fully nonlinear equations, generalized solutions, Hamilton-Jacobi equations, maximum principles, nonlinear boundary value problems, parabolic equations, partial differential equations, Perron’s method, Viscosity solutions},
|
||||
pages = {1{\textendash}67},
|
||||
pages = {1--67},
|
||||
file = {Full Text PDF:files/129/Crandall et al. - 1992 - User’s guide to viscosity solutions of second orde.pdf:application/pdf},
|
||||
}
|
||||
|
||||
|
|
@ -215,7 +217,7 @@ place={Cambridge}, series={London Mathematical Society Lecture Note Series}, tit
|
|||
month = mar,
|
||||
year = {2009},
|
||||
keywords = {60 F 05, 60 F 17, Martingale, Moment inequality, Projective criteria, Rosenthal inequality, Stationary sequences},
|
||||
pages = {146{\textendash}163},
|
||||
pages = {146--163},
|
||||
}
|
||||
@book{golub2013matrix,
|
||||
title={Matrix Computations},
|
||||
|
|
@ -229,13 +231,13 @@ place={Cambridge}, series={London Mathematical Society Lecture Note Series}, tit
|
|||
}
|
||||
|
||||
@article{hjw2020,
|
||||
author = {Martin Hutzenthaler and Arnulf Jentzen and von Wurstemberger},
|
||||
author = {Martin Hutzenthaler and Arnulf Jentzen and von Wurstemberger Wurstemberger},
|
||||
title = {{Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks}},
|
||||
volume = {25},
|
||||
journal = {Electronic Journal of Probability},
|
||||
number = {none},
|
||||
publisher = {Institute of Mathematical Statistics and Bernoulli Society},
|
||||
pages = {1{\textendash}73},
|
||||
pages = {1 -- 73},
|
||||
keywords = {curse of dimensionality, high-dimensional PDEs, multilevel Picard method, semilinear KolmogorovPDEs, Semilinear PDEs},
|
||||
year = {2020},
|
||||
doi = {10.1214/20-EJP423},
|
||||
|
|
@ -244,7 +246,7 @@ URL = {https://doi.org/10.1214/20-EJP423}
|
|||
|
||||
@article{bhj20,
|
||||
author = {Beck, Christian and Hutzenthaler, Martin and Jentzen, Arnulf},
|
||||
title = {On nonlinear {Feynman}–{Kac} formulas for viscosity solutions of semilinear parabolic partial differential equations},
|
||||
title = {On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations},
|
||||
journal = {Stochastics and Dynamics},
|
||||
volume = {21},
|
||||
number = {08},
|
||||
|
|
@ -292,12 +294,13 @@ Publisher: Nature Publishing Group},
|
|||
file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/EKLDKE65/Tsaban et al. - 2022 - Harnessing protein folding neural networks for pep.pdf:application/pdf},
|
||||
}
|
||||
|
||||
@article{davies_signature_2022,
|
||||
title = {The signature and cusp geometry of hyperbolic knots},
|
||||
journal = {Geometry and Topology},
|
||||
author = {Davies, A and Juhasz, A and Lackenby, M and Tomasev, N},
|
||||
year = {2022},
|
||||
note = {Publisher: Mathematical Sciences Publishers},
|
||||
@article{davies_signature_2021,
|
||||
title={The signature and cusp geometry of hyperbolic knots},
|
||||
author={Alex Davies and Andr'as Juh'asz and Marc Lackenby and Nenad Tomasev},
|
||||
journal={ArXiv},
|
||||
year={2021},
|
||||
volume={abs/2111.15323},
|
||||
url={https://api.semanticscholar.org/CorpusID:244729717}
|
||||
}
|
||||
|
||||
@article{zhao_space-based_2023,
|
||||
|
|
@ -318,7 +321,7 @@ Publisher: Nature Publishing Group},
|
|||
note = {Number: 1
|
||||
Publisher: Nature Publishing Group},
|
||||
keywords = {Astronomy and planetary science, Computational science},
|
||||
pages = {1{\textendash}12},
|
||||
pages = {1--12},
|
||||
file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/JCCM78TZ/Zhao et al. - 2023 - Space-based gravitational wave signal detection an.pdf:application/pdf},
|
||||
}
|
||||
@misc{wu2022sustainable,
|
||||
|
|
@ -422,7 +425,7 @@ title = {Xception: Deep Learning with Depthwise Separable Convolutions},
|
|||
year = {2017},
|
||||
volume = {},
|
||||
issn = {1063-6919},
|
||||
pages = {1800{\textendash}1807},
|
||||
pages = {1800-1807},
|
||||
abstract = {We present an interpretation of Inception modules in convolutional neural networks as being an intermediate step in-between regular convolution and the depthwise separable convolution operation (a depthwise convolution followed by a pointwise convolution). In this light, a depthwise separable convolution can be understood as an Inception module with a maximally large number of towers. This observation leads us to propose a novel deep convolutional neural network architecture inspired by Inception, where Inception modules have been replaced with depthwise separable convolutions. We show that this architecture, dubbed Xception, slightly outperforms Inception V3 on the ImageNet dataset (which Inception V3 was designed for), and significantly outperforms Inception V3 on a larger image classification dataset comprising 350 million images and 17,000 classes. Since the Xception architecture has the same number of parameters as Inception V3, the performance gains are not due to increased capacity but rather to a more efficient use of model parameters.},
|
||||
keywords = {computer architecture;correlation;convolutional codes;google;biological neural networks},
|
||||
doi = {10.1109/CVPR.2017.195},
|
||||
|
|
@ -462,7 +465,7 @@ month = {jul}
|
|||
year = {2018},
|
||||
pmid = {30245431},
|
||||
keywords = {Curse of dimension, Deep neural networks, Function approximation, Metric entropy, Neural Networks, Computer, Piecewise smooth functions, Sparse connectivity},
|
||||
pages = {296{\textendash}330},
|
||||
pages = {296--330},
|
||||
file = {Submitted Version:/Users/shakilrafi/Zotero/storage/UL4GLF59/Petersen and Voigtlaender - 2018 - Optimal approximation of piecewise smooth function.pdf:application/pdf},
|
||||
}
|
||||
|
||||
|
|
@ -570,6 +573,40 @@ archivePrefix = {arXiv},
|
|||
primaryClass={cs.LG}
|
||||
}
|
||||
|
||||
@inproceedings{vaswani2,
|
||||
author = {Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N and Kaiser, \L ukasz and Polosukhin, Illia},
|
||||
booktitle = {Advances in Neural Information Processing Systems},
|
||||
editor = {I. Guyon and U. Von Luxburg and S. Bengio and H. Wallach and R. Fergus and S. Vishwanathan and R. Garnett},
|
||||
pages = {},
|
||||
publisher = {Curran Associates, Inc.},
|
||||
title = {Attention is All you Need},
|
||||
url = {https://proceedings.neurips.cc/paper_files/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf},
|
||||
volume = {30},
|
||||
year = {2017}
|
||||
}
|
||||
|
||||
@article{arik2,
|
||||
title = {{TabNet}: {Attentive} {Interpretable} {Tabular} {Learning}},
|
||||
volume = {35},
|
||||
copyright = {Copyright (c) 2021 Association for the Advancement of Artificial Intelligence},
|
||||
issn = {2374-3468},
|
||||
shorttitle = {{TabNet}},
|
||||
url = {https://ojs.aaai.org/index.php/AAAI/article/view/16826},
|
||||
doi = {10.1609/aaai.v35i8.16826},
|
||||
abstract = {We propose a novel high-performance and interpretable canonical deep tabular data learning architecture, TabNet. TabNet uses sequential attention to choose which features to reason from at each decision step, enabling interpretability and more efficient learning as the learning capacity is used for the most salient features. We demonstrate that TabNet outperforms other variants on a wide range of non-performance-saturated tabular datasets and yields interpretable feature attributions plus insights into its global behavior. Finally, we demonstrate self-supervised learning for tabular data, significantly improving performance when unlabeled data is abundant.},
|
||||
language = {en},
|
||||
number = {8},
|
||||
urldate = {2024-02-01},
|
||||
journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
|
||||
author = {Arik, Sercan \"O and Pfister, Tomas},
|
||||
month = may,
|
||||
year = {2021},
|
||||
note = {Number: 8},
|
||||
keywords = {Unsupervised \& Self-Supervised Learning},
|
||||
pages = {6679--6687},
|
||||
file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/7MTMXR4G/Arik and Pfister - 2021 - TabNet Attentive Interpretable Tabular Learning.pdf:application/pdf},
|
||||
}
|
||||
|
||||
@Manual{dplyr,
|
||||
title = {dplyr: A Grammar of Data Manipulation},
|
||||
author = {Hadley Wickham and Romain François and Lionel Henry and Kirill Müller and Davis Vaughan},
|
||||
|
|
@ -578,19 +615,6 @@ archivePrefix = {arXiv},
|
|||
url = {https://dplyr.tidyverse.org},
|
||||
}
|
||||
|
||||
@INPROCEEDINGS{nn_diff,
|
||||
author={Berner, Julius and Elbrächter, Dennis and Grohs, Philipp and Jentzen, Arnulf},
|
||||
booktitle={2019 13th International conference on Sampling Theory and Applications (SampTA)},
|
||||
title={Towards a regularity theory for ReLU networks – chain rule and global error estimates},
|
||||
year={2019},
|
||||
volume={},
|
||||
number={},
|
||||
pages={1\textemdash5},
|
||||
keywords={Neural networks;Standards;Approximation methods;Machine learning;Partial differential equations;Level set},
|
||||
doi={10.1109/SampTA45681.2019.9031005}}
|
||||
|
||||
|
||||
|
||||
@Book{ggplot2,
|
||||
author = {Hadley Wickham},
|
||||
title = {ggplot2: Elegant Graphics for Data Analysis},
|
||||
|
|
@ -624,33 +648,7 @@ archivePrefix = {arXiv},
|
|||
|
||||
@Manual{nnR-package,
title = {nnR: Neural Networks Made Algebraic},
author = {Shakil Rafi and Joshua Lee Padgett},
year = {2024},
note = {R package version 0.1.0},
url = {https://github.com/2shakilrafi/nnR/},
}
|
||||
|
||||
@misc{ackermann2023deep,
|
||||
title={Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense},
|
||||
author={Julia Ackermann and Arnulf Jentzen and Thomas Kruse and Benno Kuckuck and Joshua Lee Padgett},
|
||||
year={2023},
|
||||
eprint={2309.13722},
|
||||
archivePrefix={arXiv},
|
||||
primaryClass={math.NA}
|
||||
|
||||
@book{graham_concrete_1994,
|
||||
address = {Upper Saddle River, NJ},
|
||||
edition = {2nd edition},
|
||||
title = {Concrete {Mathematics}: {A} {Foundation} for {Computer} {Science}},
|
||||
isbn = {978-0-201-55802-9},
|
||||
shorttitle = {Concrete {Mathematics}},
|
||||
abstract = {This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.Major topics include:SumsRecurrencesInteger functionsElementary number theoryBinomial coefficientsGenerating functionsDiscrete probabilityAsymptotic methodsThis second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.},
|
||||
language = {English},
|
||||
publisher = {Addison-Wesley Professional},
|
||||
author = {Graham, Ronald and Knuth, Donald and Patashnik, Oren},
|
||||
month = feb,
|
||||
year = {1994},
|
||||
}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
@software{Rafi_nnR_2024,
|
||||
@software{Rafi_nnR_2024,
|
||||
author = {Rafi, Shakil},
|
||||
license = {GPL-3.0},
|
||||
month = feb,
|
||||
|
|
@ -660,177 +658,6 @@ version = {0.10},
|
|||
year = {2024}
|
||||
}
|
||||
|
||||
@article{https://doi.org/10.1002/cnm.3535,
|
||||
author = {Rego, Bruno V. and Weiss, Dar and Bersi, Matthew R. and Humphrey, Jay D.},
|
||||
title = {Uncertainty quantification in subject-specific estimation of local vessel mechanical properties},
|
||||
journal = {International Journal for Numerical Methods in Biomedical Engineering},
|
||||
volume = {37},
|
||||
number = {12},
|
||||
pages = {e3535},
|
||||
keywords = {digital image correlation, image-based modeling, subject-specific model, uncertainty quantification},
|
||||
doi = {https://doi.org/10.1002/cnm.3535},
|
||||
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.3535},
|
||||
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/cnm.3535},
|
||||
abstract = {Abstract Quantitative estimation of local mechanical properties remains critically important in the ongoing effort to elucidate how blood vessels establish, maintain, or lose mechanical homeostasis. Recent advances based on panoramic digital image correlation (pDIC) have made high-fidelity 3D reconstructions of small-animal (e.g., murine) vessels possible when imaged in a variety of quasi-statically loaded configurations. While we have previously developed and validated inverse modeling approaches to translate pDIC-measured surface deformations into biomechanical metrics of interest, our workflow did not heretofore include a methodology to quantify uncertainties associated with local point estimates of mechanical properties. This limitation has compromised our ability to infer biomechanical properties on a subject-specific basis, such as whether stiffness differs significantly between multiple material locations on the same vessel or whether stiffness differs significantly between multiple vessels at a corresponding material location. In the present study, we have integrated a novel uncertainty quantification and propagation pipeline within our inverse modeling approach, relying on empirical and analytic Bayesian techniques. To demonstrate the approach, we present illustrative results for the ascending thoracic aorta from three mouse models, quantifying uncertainties in constitutive model parameters as well as circumferential and axial tangent stiffness. Our extended workflow not only allows parameter uncertainties to be systematically reported, but also facilitates both subject-specific and group-level statistical analyses of the mechanics of the vessel wall.},
|
||||
year = {2021}
|
||||
}
|
||||
|
||||
@article{schapire_strength_1990,
|
||||
title = {The strength of weak learnability},
|
||||
volume = {5},
|
||||
issn = {1573-0565},
|
||||
url = {https://doi.org/10.1007/BF00116037},
|
||||
doi = {10.1007/BF00116037},
|
||||
abstract = {This paper addresses the problem of improving the accuracy of an hypothesis output by a learning algorithm in the distribution-free (PAC) learning model. A concept class islearnable (orstrongly learnable) if, given access to a source of examples of the unknown concept, the learner with high probability is able to output an hypothesis that is correct on all but an arbitrarily small fraction of the instances. The concept class isweakly learnable if the learner can produce an hypothesis that performs only slightly better than random guessing. In this paper, it is shown that these two notions of learnability are equivalent.},
|
||||
language = {en},
|
||||
number = {2},
|
||||
urldate = {2024-03-06},
|
||||
journal = {Mach Learn},
|
||||
author = {Schapire, Robert E.},
|
||||
month = jun,
|
||||
year = {1990},
|
||||
keywords = {learnability theory, learning from examples, Machine learning, PAC learning, polynomial-time identification},
|
||||
pages = {197{\textendash}227}
|
||||
}
|
||||
|
||||
|
||||
@article{schwab_deep_2019,
|
||||
title = {Deep learning in high dimension: {Neural} network expression rates for generalized polynomial chaos expansions in {UQ}},
|
||||
volume = {17},
|
||||
issn = {0219-5305},
|
||||
shorttitle = {Deep learning in high dimension},
|
||||
url = {https://www.worldscientific.com/doi/abs/10.1142/S0219530518500203},
|
||||
doi = {10.1142/S0219530518500203},
|
||||
abstract = {We estimate the expressive power of certain deep neural networks (DNNs for short) on a class of countably-parametric, holomorphic maps
|
||||
𝑢:𝑈→ℝ
|
||||
𝑢
|
||||
:
|
||||
𝑈
|
||||
→
|
||||
ℝ
|
||||
on the parameter domain
|
||||
𝑈=
|
||||
[−1,1]
|
||||
ℕ
|
||||
𝑈
|
||||
=
|
||||
[
|
||||
−
|
||||
1
|
||||
,
|
||||
1
|
||||
]
|
||||
ℕ
|
||||
. Dimension-independent rates of best
|
||||
𝑛
|
||||
𝑛
|
||||
-term truncations of generalized polynomial chaos (gpc for short) approximations depend only on the summability exponent of the sequence of their gpc expansion coefficients. So-called
|
||||
(𝑏,𝜀)
|
||||
(
|
||||
𝑏
|
||||
,
|
||||
𝜀
|
||||
)
|
||||
-holomorphic maps
|
||||
𝑢
|
||||
𝑢
|
||||
, with
|
||||
𝑏∈
|
||||
ℓ
|
||||
𝑝
|
||||
𝑏
|
||||
∈
|
||||
ℓ
|
||||
𝑝
|
||||
for some
|
||||
𝑝∈(0,1)
|
||||
𝑝
|
||||
∈
|
||||
(
|
||||
0
|
||||
,
|
||||
1
|
||||
)
|
||||
, are known to allow gpc expansions with coefficient sequences in
|
||||
ℓ
|
||||
𝑝
|
||||
ℓ
|
||||
𝑝
|
||||
. Such maps arise for example as response surfaces of parametric PDEs, with applications in PDE uncertainty quantification (UQ) for many mathematical models in engineering and the sciences. Up to logarithmic terms, we establish the dimension independent approximation rate
|
||||
𝑠=1/𝑝−1
|
||||
𝑠
|
||||
=
|
||||
1
|
||||
/
|
||||
𝑝
|
||||
−
|
||||
1
|
||||
for these functions in terms of the total number
|
||||
𝑁
|
||||
𝑁
|
||||
of units and weights in the DNN. It follows that certain DNN architectures can overcome the curse of dimensionality when expressing possibly countably-parametric, real-valued maps with a certain degree of sparsity in the sequences of their gpc expansion coefficients. We also obtain rates of expressive power of DNNs for countably-parametric maps
|
||||
𝑢:𝑈→𝑉
|
||||
𝑢
|
||||
:
|
||||
𝑈
|
||||
→
|
||||
𝑉
|
||||
, where
|
||||
𝑉
|
||||
𝑉
|
||||
is the Hilbert space
|
||||
𝐻
|
||||
1
|
||||
0
|
||||
([0,1])
|
||||
𝐻
|
||||
0
|
||||
1
|
||||
(
|
||||
[
|
||||
0
|
||||
,
|
||||
1
|
||||
]
|
||||
)
|
||||
.},
|
||||
number = {01},
|
||||
urldate = {2024-03-07},
|
||||
journal = {Anal. Appl.},
|
||||
author = {Schwab, Christoph and Zech, Jakob},
|
||||
month = jan,
|
||||
year = {2019},
|
||||
note = {Publisher: World Scientific Publishing Co.},
|
||||
keywords = {deep networks, Generalized polynomial chaos, sparse grids, uncertainty quantification},
|
||||
pages = {19--55},
|
||||
}
|
||||
|
||||
@book{Goodfellow-et-al-2016,
|
||||
title={Deep Learning},
|
||||
author={Ian Goodfellow and Yoshua Bengio and Aaron Courville},
|
||||
publisher={MIT Press},
|
||||
note={\url{http://www.deeplearningbook.org}},
|
||||
year={2016}
|
||||
}
|
||||
|
||||
@article{yarotsky_error_2017,
|
||||
title = {Error bounds for approximations with deep {ReLU} networks},
|
||||
volume = {94},
|
||||
issn = {0893-6080},
|
||||
url = {https://www.sciencedirect.com/science/article/pii/S0893608017301545},
|
||||
doi = {10.1016/j.neunet.2017.07.002},
|
||||
abstract = {We study expressive power of shallow and deep neural networks with piece-wise linear activation functions. We establish new rigorous upper and lower bounds for the network complexity in the setting of approximations in Sobolev spaces. In particular, we prove that deep ReLU networks more efficiently approximate smooth functions than shallow networks. In the case of approximations of 1D Lipschitz functions we describe adaptive depth-6 network architectures more efficient than the standard shallow architecture.},
|
||||
urldate = {2024-03-22},
|
||||
journal = {Neural Networks},
|
||||
author = {Yarotsky, Dmitry},
|
||||
month = oct,
|
||||
year = {2017},
|
||||
keywords = {Approximation complexity, Deep ReLU networks},
|
||||
pages = {103--114},
|
||||
file = {ScienceDirect Snapshot:/Users/shakilrafi/Zotero/storage/4HS3Z6ZE/S0893608017301545.html:text/html;Submitted Version:/Users/shakilrafi/Zotero/storage/C6KQ6BFJ/Yarotsky - 2017 - Error bounds for approximations with deep ReLU net.pdf:application/pdf},
|
||||
}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -1,48 +1,55 @@
|
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Capacity: max_strings=200000, hash_size=200000, hash_prime=170003
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|
@ -1,11 +1,13 @@
|
|||
\include{preamble}
|
||||
\include{commands}
|
||||
|
||||
\title{Artificial Neural Networks Applied to Stochastic Monte Carlo as a Way to Approximate Modified Heat Equations, and Their Associated Parameters.}
|
||||
\author{Shakil Rafi}
|
||||
\begin{document}
|
||||
\pagenumbering{gobble}
|
||||
\include{front_matter}
|
||||
\maketitle
|
||||
|
||||
\tableofcontents
|
||||
|
||||
\pagenumbering{arabic}
|
||||
\part{On Convergence of Brownian Motion Monte Carlo}
|
||||
|
||||
\include{Introduction}
|
||||
|
|
@ -14,7 +16,7 @@
|
|||
|
||||
\include{u_visc_sol}
|
||||
|
||||
%\include{brownian_motion_monte_carlo_non_linear_case}
|
||||
\include{brownian_motion_monte_carlo_non_linear_case}
|
||||
|
||||
\part{A Structural Description of Artificial Neural Networks}
|
||||
|
||||
|
|
@ -22,24 +24,21 @@
|
|||
|
||||
\include{ann_product}
|
||||
|
||||
%\include{modified_mlp_associated_nn}
|
||||
\include{modified_mlp_associated_nn}
|
||||
|
||||
\include{ann_first_approximations}
|
||||
|
||||
\part{Artificial Neural Networks for $u$ and Brownian motions}
|
||||
\part{A deep-learning solution for $u$ and Brownian motions}
|
||||
|
||||
\include{ann_rep_brownian_motion_monte_carlo}
|
||||
|
||||
\include{conclusions-further-research}
|
||||
|
||||
\chapter{Bibliography and Code Listings}
|
||||
%\nocite{*}
|
||||
\nocite{*}
|
||||
\singlespacing
|
||||
\bibliography{main.bib}
|
||||
\bibliographystyle{apa}
|
||||
|
||||
\include{appendices}
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,58 @@
|
|||
\contentsline {part}{I\hspace {1em}On Convergence of Brownian Motion Monte Carlo}{4}{part.1}%
|
||||
\contentsline {chapter}{\numberline {1}Introduction.}{5}{chapter.1}%
|
||||
\contentsline {section}{\numberline {1.1}Notation, Definitions \& Basic notions.}{5}{section.1.1}%
|
||||
\contentsline {subsection}{\numberline {1.1.1}Norms and Inner Product}{5}{subsection.1.1.1}%
|
||||
\contentsline {subsection}{\numberline {1.1.2}Probability Space and Brownian Motion}{6}{subsection.1.1.2}%
|
||||
\contentsline {subsection}{\numberline {1.1.3}Lipschitz and Related Notions}{9}{subsection.1.1.3}%
|
||||
\contentsline {subsection}{\numberline {1.1.4}Kolmogorov Equations}{10}{subsection.1.1.4}%
|
||||
\contentsline {subsection}{\numberline {1.1.5}Linear Algebra Notation and Definitions}{12}{subsection.1.1.5}%
|
||||
\contentsline {subsection}{\numberline {1.1.6}$O$-type notation and function growth}{13}{subsection.1.1.6}%
|
||||
\contentsline {subsection}{\numberline {1.1.7}The Iverson Bracket}{15}{subsection.1.1.7}%
|
||||
\contentsline {chapter}{\numberline {2}Brownian Motion Monte Carlo}{16}{chapter.2}%
|
||||
\contentsline {section}{\numberline {2.1}Brownian Motion Preliminaries}{16}{section.2.1}%
|
||||
\contentsline {section}{\numberline {2.2}Monte Carlo Approximations}{20}{section.2.2}%
|
||||
\contentsline {section}{\numberline {2.3}Bounds and Covnvergence}{21}{section.2.3}%
|
||||
\contentsline {chapter}{\numberline {3}That $u$ is a viscosity solution}{30}{chapter.3}%
|
||||
\contentsline {section}{\numberline {3.1}Some Preliminaries}{30}{section.3.1}%
|
||||
\contentsline {section}{\numberline {3.2}Viscosity Solutions}{34}{section.3.2}%
|
||||
\contentsline {section}{\numberline {3.3}Solutions, characterization, and computational bounds to the Kolmogorov backward equations}{53}{section.3.3}%
|
||||
\contentsline {chapter}{\numberline {4}Brownian motion Monte Carlo of the non-linear case}{59}{chapter.4}%
|
||||
\contentsline {part}{II\hspace {1em}A Structural Description of Artificial Neural Networks}{61}{part.2}%
|
||||
\contentsline {chapter}{\numberline {5}Introduction and Basic Notions about Neural Networks}{62}{chapter.5}%
|
||||
\contentsline {section}{\numberline {5.1}The Basic Definition of ANNs}{62}{section.5.1}%
|
||||
\contentsline {section}{\numberline {5.2}Composition and extensions of ANNs}{66}{section.5.2}%
|
||||
\contentsline {subsection}{\numberline {5.2.1}Composition}{66}{subsection.5.2.1}%
|
||||
\contentsline {subsection}{\numberline {5.2.2}Extensions}{68}{subsection.5.2.2}%
|
||||
\contentsline {section}{\numberline {5.3}Parallelization of ANNs}{68}{section.5.3}%
|
||||
\contentsline {section}{\numberline {5.4}Affine Linear Transformations as ANNs}{72}{section.5.4}%
|
||||
\contentsline {section}{\numberline {5.5}Sums of ANNs}{75}{section.5.5}%
|
||||
\contentsline {subsection}{\numberline {5.5.1}Neural Network Sum Properties}{76}{subsection.5.5.1}%
|
||||
\contentsline {section}{\numberline {5.6}Linear Combinations of ANNs}{83}{section.5.6}%
|
||||
\contentsline {section}{\numberline {5.7}Neural Network Diagrams}{93}{section.5.7}%
|
||||
\contentsline {chapter}{\numberline {6}ANN Product Approximations}{95}{chapter.6}%
|
||||
\contentsline {section}{\numberline {6.1}Approximation for simple products}{95}{section.6.1}%
|
||||
\contentsline {subsection}{\numberline {6.1.1}The $\prd $ network}{106}{subsection.6.1.1}%
|
||||
\contentsline {section}{\numberline {6.2}Higher Approximations}{111}{section.6.2}%
|
||||
\contentsline {subsection}{\numberline {6.2.1}The $\tun $ Neural Network}{112}{subsection.6.2.1}%
|
||||
\contentsline {subsection}{\numberline {6.2.2}The $\pwr $ Neural Networks}{114}{subsection.6.2.2}%
|
||||
\contentsline {subsection}{\numberline {6.2.3}The $\tay $ neural network}{123}{subsection.6.2.3}%
|
||||
\contentsline {subsection}{\numberline {6.2.4}Neural network approximations for $e^x$.}{128}{subsection.6.2.4}%
|
||||
\contentsline {chapter}{\numberline {7}A modified Multi-Level Picard and associated neural network}{129}{chapter.7}%
|
||||
\contentsline {chapter}{\numberline {8}Some categorical ideas about neural networks}{132}{chapter.8}%
|
||||
\contentsline {chapter}{\numberline {9}ANN first approximations}{133}{chapter.9}%
|
||||
\contentsline {section}{\numberline {9.1}Activation Function as Neural Networks}{133}{section.9.1}%
|
||||
\contentsline {section}{\numberline {9.2}ANN Representations for One-Dimensional Identity}{134}{section.9.2}%
|
||||
\contentsline {section}{\numberline {9.3}Modulus of Continuity}{142}{section.9.3}%
|
||||
\contentsline {section}{\numberline {9.4}Linear Interpolation of real-valued functions}{143}{section.9.4}%
|
||||
\contentsline {subsection}{\numberline {9.4.1}The Linear Interpolation Operator}{143}{subsection.9.4.1}%
|
||||
\contentsline {subsection}{\numberline {9.4.2}Neural Networks to approximate the $\lin $ operator}{144}{subsection.9.4.2}%
|
||||
\contentsline {section}{\numberline {9.5}Neural network approximation of 1-dimensional functions.}{148}{section.9.5}%
|
||||
\contentsline {section}{\numberline {9.6}$\trp ^h$ and neural network approximations for the trapezoidal rule.}{151}{section.9.6}%
|
||||
\contentsline {section}{\numberline {9.7}Linear interpolation for multi-dimensional functions}{154}{section.9.7}%
|
||||
\contentsline {subsection}{\numberline {9.7.1}The $\nrm ^d_1$ and $\mxm ^d$ networks}{154}{subsection.9.7.1}%
|
||||
\contentsline {subsection}{\numberline {9.7.2}The $\mxm ^d$ neural network and maximum convolutions }{160}{subsection.9.7.2}%
|
||||
\contentsline {subsection}{\numberline {9.7.3}Lipschitz function approximations}{164}{subsection.9.7.3}%
|
||||
\contentsline {subsection}{\numberline {9.7.4}Explicit ANN approximations }{167}{subsection.9.7.4}%
|
||||
\contentsline {part}{III\hspace {1em}A deep-learning solution for $u$ and Brownian motions}{169}{part.3}%
|
||||
\contentsline {chapter}{\numberline {10}ANN representations of Brownian Motion Monte Carlo}{170}{chapter.10}%
|
||||
\contentsline {chapter}{Appendices}{180}{section*.3}%
|
||||
|
|
@ -33,7 +33,7 @@ We now look at neural networks in the context of multi-level Picard iterations.
|
|||
\begin{lemma}
|
||||
Let $\Theta = \lp \bigcup^{n\in \N} \Z^n \rp$, $d,M \in \N$, $T\in \lp 0,\infty \rp$, $f \in C \lp \R, \R \rp$, $g,\in C \lp \R^d, \R \rp$, $\mathsf{F}, \mathsf{G} \in \neu$ satisfy that $\real_{\rect} \lp \mathsf{F} \rp = f$ and $\real_{\rect} \lp \mathsf{G} \rp = g$, let $\mathfrak{u}^\theta \in \lb 0,1 \rb$, $\theta \in \Theta$, and $\mathcal{U}^\theta: \lb 0,T \rb \rightarrow \lb 0,T \rb$, $\theta \in \Theta$, satisfy for all $t \in \lb 0,T \rb$, $theta \in \Theta$ that $\mathcal{U}^\theta_t = t+(T-t)\mathfrak{u}^\theta$, let $\mathcal{W}^\theta: \lb 0,T \rb \rightarrow \R^d$, $\theta \in \Theta$, for every $\theta \in \Theta$, $t\in \lb 0,T\rb$, $s \in \lb t,T\rb$, let $\mathcal{Y}^\theta_{t,s} \in \R$ satisfy $\mathcal{Y}^\theta_{t,s} = \mathcal{W}^\theta_s - \mathcal{W}^\theta_t$ and let $\mathcal{U}^\theta_n: \lb 0,T\rb \times \R^d \rightarrow \R$, $n\in \N_0$, $\theta \in \Theta$, satisfy for all $\theta \in \Theta$, $n\in \N_0$, $t\in \lb 0,T\rb$, $x\in \R^d$ that:
|
||||
\begin{align}
|
||||
&U^\theta_n \lp t,x\rp = \frac{\mathbbm{1}_\N\lp n \rp}{M^n} \lb \sum^{M^n}_{k=1} g \lp x + \mathcal{Y}^{(\theta,0,-k)}_{t,T}\rp\rb \nonumber\\
|
||||
U^\theta_n \lp t,x\rp &= \frac{\mathbbm{1}_\N\lp n \rp}{M^n} \lb \sum^{M^n}_{k=1} g \lp x + \mathcal{Y}^{(\theta,0,-k)}_{t,T}\rp\rb \nonumber\\
|
||||
&+ \sum^{n-1}_{i=0} \frac{T-t}{M^{n-i}} \lb \sum^{M^{n-i}}_{k=1} \lp \lp f \circ U^{(\theta,i,k)}_i\rp - \mathbbm{1}_\N \lp i \rp \lp f \circ U^{(\theta,-i,k)}_{\max \{ i-1,0\}} \rp \rp \lp \mathcal{U}^{(\theta,i,k)}_t,x+ \mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}}\rp\rb
|
||||
\end{align}
|
||||
it is then the case that:
|
||||
|
|
@ -46,7 +46,7 @@ We now look at neural networks in the context of multi-level Picard iterations.
|
|||
% \boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{T-t}{M^{n-i}} \rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}} \lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,i,k}_{i, \mathcal{U}_t^{(\theta,i,k)} \rp \rp \rp
|
||||
% \end{align}
|
||||
\begin{align}
|
||||
&\mathsf{U}^\theta_{n,t} \\&= \lb \bigoplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathsf{G}\bullet \aff_{\mathbb{I}_d, \mathcal{Y}^{(\theta,0,-k}_{t,T}} \rp \rp \rb \nonumber \\
|
||||
\mathsf{U}^\theta_{n,t} &= \lb \bigoplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathsf{G}\bullet \aff_{\mathbb{I}_d, \mathcal{Y}^{(\theta,0,-k}_{t,T}} \rp \rp \rb \nonumber \\
|
||||
&\boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{T-t}{M^{n-i}}\rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}}\lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,i,k)}_{i,\mathcal{U}_t^{(\theta,i,k)}} \rp \bullet \aff_{\mathbb{I}_d}, \mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}} \rp\rp \rb\rb \nonumber\\
|
||||
&\boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{(t-T)\mathbbm{1}_\N}{M^{n-i}}\rp \circledast\lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}} \lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,-i,k)}_{\max \{i-1,0\}, \mathcal{U}_t^{(\theta,i,k)}}\rp \bullet \aff_{\mathbb{I}_d,\mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}}} \rp \rp\rb \rb
|
||||
\end{align}
|
||||
|
|
|
|||
|
|
@ -1,7 +1,5 @@
|
|||
\chapter{Introduction and Basic Notions About Neural Networks}
|
||||
We seek here to introduce a unified framework for artificial neural networks. This framework borrows from the work presented in \cite{grohsetal}, which was in turn inspired by work done in \cite{petersen_optimal_2018}. The most recent exposition of this framework can be found in \cite{bigbook}, and it is this exposition that our work will be based on and extended upon.
|
||||
|
||||
With this framework in place, we wish to study ANNs from the perspective of trying to see the number of parameters required to define a neural network to solve certain PDEs. The \textit{curse of dimensionality} here refers to the number of parameters and depths of neural networks necessary to approximate functions to a certain accuracy. Specifically a scheme is said to have beat the curse of dimensionality if the number of parameters and depths necessary to approximate an underlying function to an accuracy (specifically the upper bound on the the 1-norm difference between the approximant and the function over the entire domain), only grows polynomially or at-least sub-exponentially on $\frac{1}{\ve}$.
|
||||
We seek here to introduce a unified framework for artificial neural networks. This framework borrows from the work presented in \cite{grohsetal} and work done by Joshua Padgett, Benno Kuckuk, and Arnulf Jentzen (unpublished). With this framework in place, we wish to study ANNs from the perspective of trying to see the number of parameters required to define a neural network to solve certain PDEs. The \textit{curse of dimensionality} here refers to the number of parameters necessary to model PDEs and their growth (exponential or otherwise) as dimensions $d$ increase.
|
||||
\section{The Basic Definition of ANNs and instantiations of ANNs}
|
||||
|
||||
\begin{definition}[Rectifier Function]
|
||||
|
|
@ -10,19 +8,13 @@ With this framework in place, we wish to study ANNs from the perspective of tryi
|
|||
\rect(x) = \max \left\{ 0,x\right\}
|
||||
\end{align}
|
||||
\end{definition}
|
||||
\begin{remark}
|
||||
By analogy the multidimensional rectifier function, defined for $x = \lb x_1 \: x_2 \: \cdots \right.\\ \left. \: x_n\rb^\intercal \in \R^n$ is:
|
||||
\begin{align}
|
||||
\rect ([x]_*) = \left[ \max\{ 0,x_1\} \: \max \{ 0,x_2\}\: \cdots \max\{ 0,x_n\}\right]^\intercal
|
||||
\end{align}
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Artificial Neural Networks]\label{5.1.2}\label{def:nn_def}
|
||||
Denote by $\neu$ the set given by:
|
||||
\begin{align}
|
||||
\neu = \bigcup_{L\in \N} \bigcup_{l_0,l_1,...,l_L \in \N} \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp
|
||||
\end{align}
|
||||
An artificial neural network is a tuple $\lp \nu, \param, \dep, \inn, \out, \hid, \lay, \wid \rp $ where $\nu \in \neu$ and is equipped with the following functions (referred to as auxiliary functions) satisfying for all \\$\nu \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$ that:
|
||||
An artificial neural network is a tuple $\lp \nu, \param, \dep, \inn, \out, \hid, \lay, \wid \rp $ where $\nu \in \neu$ and is equipped with the following functions (referred to as auxiliary functions) satisfying for all $\nu \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\param: \neu \rightarrow \N$ denoting the number of parameters of $\nu$, given by:
|
||||
\begin{align}\label{paramdef}
|
||||
|
|
@ -58,7 +50,7 @@ With this framework in place, we wish to study ANNs from the perspective of tryi
|
|||
\end{align}
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
Note that this implies that $\nu = ((W_1,b_1),(W_2,b_2),...(W_L,b_L)) \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$. Note that we denote by $\we_{(\cdot ), \nu}: (\we_{n,\nu})_{n\in \{1,2,...,L\}}: \{1,2,...,L\} \rightarrow \lp \bigcup_{m,k \in \N}\R^{m \times k} \rp $ and also $\bi_{(\cdot),\nu}: \lp \bi_{n,\nu} \rp_{\{1,2,...,L\}}: \{1,2,...,L\} \rightarrow \lp \bigcup_{m \in \N}\R^m \rp$ the functions that satisfy for all $n \in \{1,2,...,L\}$ that $\we_{i,\nu} = W_i$ i.e. the weights matrix for neural network $\nu$ at layer $i$ and $\bi_{i,\nu} = b_i$, i.e. the bias vector for neural network $\nu$ at layer $i$.
|
||||
Note that this implies that that $\nu = ((W_1,b_1),(W_2,b_2),...(W_L,b_L)) \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$. Note that we also denote by $\we_{(\cdot ), \nu}: (\we_{n,\nu})_{n\in \{1,2,...,L\}}: \{1,2,...,L\} \rightarrow \lp \bigcup_{m,k \in \N}\R^{m \times k} \rp $ and also $\bi_{(\cdot),\nu}: \lp \bi_{n,\nu} \rp_{\{1,2,...,L\}}: \{1,2,...,L\} \rightarrow \lp \bigcup_{m \in \N}\R^m \rp$ the functions that satisfy for all $n \in \{1,2,...,L\}$ that $\we_{i,\nu} = W_i$ i.e. the weights matrix for neural network $\nu$ at layer $i$ and $\bi_{i,\nu} = b_i$, i.e. the bias vector for neural network $\nu$ at layer $i$.
|
||||
|
||||
We will call $l_0$ the \textit{starting width} and $l_L$ the \textit{finishing width}. Together, they will be referred to as \textit{end-widths}.
|
||||
\begin{remark}
|
||||
|
|
@ -66,24 +58,33 @@ With this framework in place, we wish to study ANNs from the perspective of tryi
|
|||
|
||||
Note that we develop this definition to closely align to popular deep-learning frameworks such as \texttt{PyTorch}, \texttt{TensorFlow}, and \texttt{Flux}, where, in principle, it is always possible to know the parameter count, depth, number of layers, and other auxiliary information.
|
||||
|
||||
We will often say let $\nu\in \neu$, and it is implied that the tuple $\nu$ with the auxiliary functions is what is being referred to. This is analogous to when we say that $X$ is a topological but we mean the pair $\lp X,\tau\rp$, i.e. $X$ endowed with topology $\tau$, or when we say that $Y$ is a measurable space when we mean the triple $\lp X,\Omega, \mu\rp$, i.e. $X$, endowed with $\sigma-$algebra $\Omega$, and measure $\mu$.
|
||||
We will often say let $\nu\in \neu$, and it is implied that the tuple $\nu$ with the auxiliary functions is what is being referred to.
|
||||
\end{remark}
|
||||
|
||||
|
||||
\begin{definition}[Instantiations of Artificial Neural Networks with Activation Functions]\label{def:rlz}\label{def:inst}
|
||||
\begin{definition}[Instantiations of Artificial Neural Networks with Activation Functions]\label{def:rlz}
|
||||
Let $\act \in C \lp \R, \R \rp$, we denote by $\real_{\act}: \neu \rightarrow \lp \bigcup_{k,l \in \N} C \lp \R^k, \R^l \rp \rp$ the function satisfying for all $L \in \N$, $l_0,l_1,...,l_L \in \N$, $\nu = \lp \lp W_1, b_1 \rp , \lp W_2, b_2\rp ,...,\lp W_L, b_L \rp \rp \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$, $x_0 \in \R^{l_0}, x_1 \in \R^{l_1},...,x_{L-1} \in \R^{l_L-1}$ and with $\forall k \in \N \cap (0,L):x_k = \act \lp \lb W_kx_k+b_k \rb_{*,*} \rp$such that:
|
||||
\begin{align}\label{5.1.11}
|
||||
\real_{\act}\lp \nu \rp \in C \lp \R^{l_0}, \R^{l_L} \rp & \text{ and } & \lp \real_{\act}\lp \nu\rp \rp \lp x_0 \rp = W_Lx_{L-1}+b_L
|
||||
\end{align}
|
||||
\end{definition}
|
||||
\begin{figure}
|
||||
We will often denote the instantiated neural network $\nu^{l_0,l_L}$ taking $\R^{l_0}$ to $\R^{l_L}$ as $\nu^{l_0,l_L}: \R^{l_0} \rightarrowtail \R^{l_L}$ or simply as $\R^{l_0} \overset{\nu}{\rightarrowtail} \R^{l_L}$ where $l_0$ and $l_L$ are obvious.
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.5]{nn-example.png}
|
||||
\begin{neuralnetwork}[height=4, title = {A neural network $\nu$ with $\lay(\nu) = \lp 4,5,4,2\rp$}, nodesize = 10pt, maintitleheight=1em]
|
||||
\newcommand{\x}[2]{$x$}
|
||||
\newcommand{\y}[2]{$x$}
|
||||
\newcommand{\hfirst}[2]{\small $h$}
|
||||
\newcommand{\hsecond}[2]{\small $h$}
|
||||
\inputlayer[count=3, bias=true, title=, text=\x]
|
||||
\hiddenlayer[count=4, bias=true, title=, text=\hfirst] \linklayers
|
||||
\hiddenlayer[count=3, bias=true, title=, text=\hsecond] \linklayers
|
||||
\outputlayer[count=2, title=, text=\y] \linklayers
|
||||
\end{neuralnetwork}
|
||||
|
||||
\end{center}
|
||||
\caption{A neural network $\nu$ with $\lay \lp \nu \rp = \lp 6,8,6,3\rp$}
|
||||
\end{figure}
|
||||
\begin{remark}
|
||||
For an R implementation see Listings \ref{nn_creator}, \ref{aux_fun}, \ref{activations}, and \ref{instantiation}.
|
||||
For an R implementation see Listings \ref{nn_creator}, \ref{aux_fun}, \ref{activations}, and \ref{instantiation}
|
||||
\end{remark}
|
||||
|
||||
\begin{lemma}\label{5.1.8}
|
||||
|
|
@ -100,17 +101,17 @@ With this framework in place, we wish to study ANNs from the perspective of tryi
|
|||
\end{align}
|
||||
This ensures that there exist $l_0,l_1,...,l_L,L \in \N$ such that:
|
||||
\begin{align}
|
||||
\nu \in \lp \bigtimes^L_{j=1} \lb \R^{l_j \times l_{j-1}} \times \R^{l_j} \rb \rp
|
||||
\nu \in \lp \bigtimes^L_{j=1} \lb \R^{l_j \times l_{j-1}} \times \R^{\l_j} \rb \rp
|
||||
\end{align}
|
||||
This also ensures that $\lay(\nu) = \lp l_0,l_1,...,l_L \rp \in \N^{L+1} = \N^{\dep(\nu)+1}$ and further that $\inn(\nu) = l_0$, $\out(\nu) = l_L$, and that $\dep(\nu) = L$. Together with ($\ref{5.1.11}$), this proves the lemma.
|
||||
\end{proof}
|
||||
\section{Compositions of ANNs}
|
||||
The first operation we want to be able to do is to compose neural networks. Note that the composition is not done in an obvious way; for instance, note that the last layer of the first component of the composition is superimposed with the first layer of the second component of the composition.
|
||||
|
||||
\subsection{Composition}
|
||||
\begin{definition}[Compositions of ANNs]\label{5.2.1}\label{def:comp}
|
||||
We denote by $\lp \cdot \rp \bullet \lp \cdot \rp: \{ \lp \nu_1,\nu_2 \rp \in \neu \times \neu: \inn(\nu_1) = \out (\nu_1) \} \rightarrow \neu$ the function satisfying for all $L,M \in \N, l_0,l_1,...,l_L, m_0, m_1,...,m_M \in \N$, $\nu_1 = \lp \lp W_1, b_1 \rp, \lp W_2, b_2 \rp,...,\lp W_L,b_L \rp \rp \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$, and $\nu_2 = \\ \lp \lp W'_1, b'_1 \rp, \lp W'_2, b'_2 \rp,... \lp W'_M, b'_M \rp \rp \in \lp \bigtimes^M_{k=1} \lb \R^{m_k \times m_{k-1}} \times \R^{m_k}\rb \rp$ with $l_0 = \inn(\nu_1)= \out(\nu_2) = m_M$ and :
|
||||
We denote by $\lp \cdot \rp \bullet \lp \cdot \rp: \{ \lp \nu_1,\nu_2 \rp \in \neu \times \neu: \inn(\nu_1) = \out (\nu_1) \} \rightarrow \neu$ the function satisfying for all $L,M \in \N, l_0,l_1,...,l_L, m_0, m_1,...,m_M \in \N$, $\nu_1 = \lp \lp W_1, b_1 \rp, \lp W_2, b_2 \rp,...,\lp W_L,b_L \rp \rp \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$, and $\nu_2 = \lp \lp W'_1, b'_1 \rp, \lp W'_2, b'_2 \rp,... \lp W'_M, b'_M \rp \rp \in \lp \bigtimes^M_{k=1} \lb \R^{m_k \times m_{k-1}} \times \R^{m_k}\rb \rp$ with $l_0 = \inn(\nu_1)= \out(\nu_2) = m_M$ and :
|
||||
\begin{align}\label{5.2.1}
|
||||
&\nu_1 \bullet \nu_2 =\\ &\begin{cases}
|
||||
\nu_1 \bullet \nu_2 = \begin{cases}
|
||||
(( W'_1,b'_1 ), ( W'_2,b'_2 ), ...( W'_{M-1}, b'_{M-1}), ( W_1W'_M, W_1b'_{M} + b_1), (W_2, b_2 ),\\..., ( W_L,b_L )) & :( L> 1 ) \land ( M > 1 ) \\
|
||||
((W_1W'_1,W_1b'_1+b_1),(W_2,b_2), (W_3,b_3),...,(W_Lb_L)) & :(L>1) \land (M=1) \\
|
||||
((W'_1, b'_1),(W'_2,b'_2), ..., (W'_{M-1}, b'_{M-1})(W_1, b'_M + b_1)) &:(L=1) \land (M>1) \\
|
||||
|
|
@ -141,7 +142,7 @@ The first operation we want to be able to do is to compose neural networks. Note
|
|||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
This is a consequence of (\ref{5.2.1}), which implies (i)\textemdash (ii).
|
||||
This is a consequence of (\ref{5.2.1}), which implies both (i) and (ii).
|
||||
\end{proof}
|
||||
\begin{lemma} \label{5.2.3}
|
||||
Let $\nu_1,\nu_2,\nu_3 \in \neu$ satisfy that $\inn(\nu_1) = \out(\nu_2)$ and $\inn(\nu_2) = \out(\nu_3)$, it is then the case\\ that:
|
||||
|
|
@ -167,7 +168,7 @@ The following Lemma will be important later on, referenced numerous times, and f
|
|||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Note that Items (i)---(iii) are a simple consequence of Definition \ref{5.2.1}. Specifically, given neural networks $\nu_1,\nu_2 \in \neu$, and $\dep\lp \nu_1\rp = n$ and $\dep \lp \nu_2\rp = m$, note that for all four cases, we have that the depth of the composed neural network $\nu_1 \bullet \nu_2$ is given by $n-1+m-1+1=n+m-1$ proving Item (i). Note that the outer neural network loses its last layer, yielding Item (ii) in all four cases. Finally since, for all $\nu \in \neu$ it is the case that $\hid \lp \nu\rp =\dep \lp \nu\rp-1$, Item (i) yields Item (iii).
|
||||
Note that Items (i)---(iii) are a simple consequence of Definition \ref{5.2.1}. Specifically, given neural networks $\nu_1,\nu_2 \in \neu$, and $\dep\lp \nu_1\rp = n$ and $\dep \lp \nu_2\rp = m$, note that for all four cases, we have that the depth of the composed neural network $\nu_1 \bullet \nu_2$ is given by $n-1+m-1=n+m-1$ proving Item (i). Note that the outer neural network loses its last layer, yielding Item (ii) in all four cases. Finally since, for all $\nu \in \neu$ it is the case that $\hid \lp \nu\rp =\dep \lp \nu\rp-1$, Item (i) yields Item (iii).
|
||||
|
||||
Now, suppose it is the case that $\nu_3 = \nu_1\bullet \nu_2$ and that:
|
||||
\begin{align}
|
||||
|
|
@ -252,9 +253,7 @@ The following Lemma will be important later on, referenced numerous times, and f
|
|||
\end{align}
|
||||
This and (\ref{comp_cont}) then prove Item (v), hence proving the lemma.
|
||||
\end{proof}
|
||||
\section{Stacking of ANNs}
|
||||
We will introduce here the important concept of stacking of ANNs. Given an input vector $x\in \R^d$, it is sometimes very helpful to imagine two neural networks working on them simultaneously, whence we have stacking. Because vectors are ordered tuples, stacking $\nu_1$ and $\nu_2$ is not necessarily the same as stacking $\nu_2$ and $\nu_1$. We will thus forego the phrase "parallelization" used in e.g. \cite{grohs2019spacetime} and \cite{bigbook}, and opt to use the term "stacking". This because parallelization implies commutativity, but it is clearly not the case that $\nu_1 \boxminus \nu_2$ is the same as $\nu_2 \boxminus \nu_1$.
|
||||
\subsection{Stacking of ANNs of Equal Depth}
|
||||
\section{Stacking of ANNs of Equal Depth}
|
||||
\begin{definition}[Stacking of ANNs of same depth]\label{5.2.5}\label{def:stacking}
|
||||
Let $L,n\in \N$, and let $\nu_1,\nu_2,\hdots, \nu_n \in \neu$, such that $\dep\lp \nu_1\rp= \dep \lp \nu_2\rp= \cdots = \dep\lp \nu_n\rp = L$. As such, for all $i \in \{1,\hdots,n\}$, let it also be the case that $\lay\lp \nu_i\rp = \lp \lp W_1^i,b^i_1\rp, \lp W^i_2,b^i_2\rp,\hdots, \lp W_L^i,b_L^i\rp \rp$. We then denote by $\boxminus^n_{i=1}\nu_i$, the neural network whose layer architecture is given by:
|
||||
\begin{align*}
|
||||
|
|
@ -264,13 +263,13 @@ We will introduce here the important concept of stacking of ANNs. Given an input
|
|||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
For an \texttt{R} implementation see Listing \ref{stk}
|
||||
For an \texttt{R} implementation see Listing \ref{par}
|
||||
\end{remark}
|
||||
\begin{lemma}\label{inst_of_stk}
|
||||
Let $\nu_1,\nu_2\in \neu$, with $\dep\lp \nu_1\rp = \dep\lp \nu_2\rp$, $x_1 \in \R^{m_1}$, $x_2 \in \R^{m_2}$, and $\mathfrak{x} \in \R^{m_1+m_2}$. Let $\inst_{\rect}\lp \nu_1\rp: \R^{m_1} \rightarrow \R^{n_1}$, and $\inst_{\rect}:\R^{m_2} \rightarrow \R^{n_2}$. It is then the case that $\real_{\rect}\lp \nu_1\boxminus\nu_2\rp\lp \mathfrak{x}\rp = \inst_{\rect}\lp \nu_1\rp\lp x_1\rp \frown \inst_{\rect}\lp \nu_2\rp\lp x_2\rp$.
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Let $\lay\lp \nu_1\rp = \lp \lp W_1,b_1 \rp,\lp W_2,b_2\rp,\hdots, \lp W_L,b_L\rp\rp$ and \\ $\lay \lp \nu_2\rp = \lp \lp \fW_1, \fb_1\rp, \lp \fW_2,\fb_2\rp,\hdots,\lp \fW_L,\fb_L\rp\rp$, and as such it is the case according to Definition \ref{def:stacking} that:
|
||||
Let $\lay\lp \nu_1\rp = \lp \lp W_1,b_1 \rp,\lp W_2,b_2\rp,\hdots, \lp W_L,b_L\rp\rp$ and $\lay \lp \nu_2\rp = \lp \lp \fW_1, \fb_1\rp, \lp \fW_2,\fb_2\rp,\hdots, \lp \fW_L,\fb_L\rp\rp$, and as such it is the case according to Definition \ref{def:stacking} that:
|
||||
\begin{align*}
|
||||
\lay \lp \nu_1 \boxminus\nu_2\rp = \lp \lp \diag\lp W_1,\fW_1\rp , b_1 \frown \fb_1\rp,\right.\\ \left.\lp \diag\lp W_2,\fW_2\rp , b_2 \frown \fb_2\rp, \right.\\ \left. \vdots \hspace{2.5cm}\right.\\ \left. \lp \diag\lp W_L,\fW_L\rp , b_L^1 \frown \fb_L\rp\rp
|
||||
\end{align*}
|
||||
|
|
@ -352,7 +351,6 @@ We will introduce here the important concept of stacking of ANNs. Given an input
|
|||
0 \les \param\lp \nu_3\rp - \param\lp \nu_2\rp &\les \param\lp \nu_1 \boxminus\nu_3\rp - \param\lp \nu_1\boxminus\nu_2\rp \nonumber\\
|
||||
\param\lp \nu_1 \boxminus\nu_2\rp &\les \param\lp \nu_1 \boxminus \nu_2\rp \nonumber
|
||||
\end{align}
|
||||
This completes the proof of the Corollary.
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
|
@ -371,11 +369,11 @@ We will introduce here the important concept of stacking of ANNs. Given an input
|
|||
Let $\act \in C \lp \R, \R \rp$, $n \in \N$, and $\nu = \boxminus_{i=1}^n \nu_i$ satisfy the condition that $\dep(\nu_1) = \dep(\nu_2) =...=\dep(\nu_n)$. It is then the case that $\real_{\act} \lp \nu \rp \in C \lp \R^{\sum_{i=1}^n \inn(\nu_i)}, \R^{\sum^n_{i=1}\out(\nu_i)} \rp $
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Let $L = \dep(\nu_1)$, and let $l_{i,0},l_{i,1}...l_{i,L} \in \N$ satisfy for all $i \in \{ 1,2,...,n\}$ that $\lay(\nu_i) = \lp l_{i,0}, l_{i,1},...,l_{i,L} \rp $. Furthermore let $\lp \lp W_{i,1},b_{i,1}\rp, \lp W_{i,2},b_{i,2} \rp , ..., \lp W_{i,L},b_{i,L} \rp \rp \in \\ \lp \bigtimes^L_{j=1} \lb \R^{l_{i,j} \times l_{i,j-1}} \times \R^{l_{i,j}} \rb \rp $ satisfy for all $i \in \{ 1,2,...,n\}$ that:
|
||||
Let $L = \dep(\nu_1)$, and let $l_{i,0},l_{i,1}...l_{i,L} \in \N$ satisfy for all $i \in \{ 1,2,...,n\}$ that $\lay(\nu_i) = \lp l_{i,0}, l_{i,1},...,l_{i,L} \rp $. Furthermore let $\lp \lp W_{i,1},b_{i,1}\rp, \lp W_{i,2},b_{i,2} \rp , ..., \lp W_{i,L},b_{i,L} \rp \rp \in \lp \bigtimes^L_{j=1} \lb \R^{l_{i,j} \times l_{i,j-1}} \times \R^{l_{i,j}} \rb \rp $ satisfy for all $i \in \{ 1,2,...,n\}$ that:
|
||||
\begin{align}
|
||||
\nu_i = \lp \lp W_{i,1},b_{i,1} \rp , \lp W_{i,2}, b_{i,2}\rp ,...,\lp W_{i,L},b_{i,L} \rp \rp
|
||||
\end{align}
|
||||
Let $\alpha_j \in \N$ with $j \in \{0,1,...,L\}$ satisfy that $\alpha_j = \sum^n_{i=1} l_{i,j}$ and let \\ $\lp \lp A_1,b_1 \rp, \lp A_2,b_2 \rp,...,\lp A_L,b_L \rp \rp \in \lp \bigtimes^L_{j=1} \lb \R^{\alpha_{j} \times \alpha_{j-1}} \times \R^{\alpha_{j}} \rb \rp $ satisfy that:
|
||||
Let $\alpha_j \in \N$ with $j \in \{0,1,...,L\}$ satisfy that $\alpha_j = \sum^n_{i=1} l_{i,j}$ and let $\lp \lp A_1,b_1 \rp, \lp A_2,b_2 \rp,...,\lp A_L,b_L \rp \rp \in \lp \bigtimes^L_{j=1} \lb \R^{\alpha_{j} \times \alpha_{j-1}} \times \R^{\alpha_{j}} \rb \rp $ satisfy that:
|
||||
\begin{align}\label{5.3.5}
|
||||
\boxminus_{i=1}^n \nu_i = \lp \lp A_1,b_1 \rp, \lp A_2,b_2 \rp,...,\lp A_L,b_L \rp \rp
|
||||
\end{align}
|
||||
|
|
@ -391,10 +389,10 @@ We will introduce here the important concept of stacking of ANNs. Given an input
|
|||
This proves the lemma.
|
||||
\end{proof}
|
||||
|
||||
\subsection{Stacking of ANNs of Unequal Depth}
|
||||
We will often encounter neural networks that we want to stack but have unequal depth. Definition \ref{5.2.5} only deals with neural networks of the same depth. We will facilitate this situation by introducing a form of padding for our shorter neural network. Hence, they come out to the same length before stacking them. This padding will be via the tunneling neural network, as shown below.
|
||||
\section{Stacking of ANNs of Unequal Depth}
|
||||
We will often encounter neural networks that we want to stack but have unequal depth. Definition \ref{5.2.5} only deals with neural networks of the same depth. We will facilitate this situation by introducing a form of ``padding" for our neural network. Hence, they come out to the same length before stacking them. This padding will be via the "tunneling" neural network, as shown below.
|
||||
\begin{definition}[Identity Neural Network]\label{7.2.1}
|
||||
Let $d\in \N$. We will denote by $\id_d \in \neu$ the neural network satisfying for all $d \in \N$ that:
|
||||
We will denote by $\id_d \in \neu$ the neural network satisfying for all $d \in \N$ that:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item \begin{align}
|
||||
\id_1 = \lp \lp \begin{bmatrix}
|
||||
|
|
@ -410,10 +408,10 @@ We will often encounter neural networks that we want to stack but have unequal d
|
|||
\item \begin{align}\label{7.2.2}
|
||||
\id_d = \boxminus^d_{i=1} \id_1
|
||||
\end{align}
|
||||
For $d \in \N \cap \lb 2,\infty\rp$.
|
||||
For $d>1$.
|
||||
\end{enumerate}
|
||||
\begin{remark}
|
||||
We will discuss some properties of $\id_d$ in Section \ref{sec_tun}.
|
||||
We will discuss some properties of $\id$ in Section \ref{sec_tun}.
|
||||
\end{remark}
|
||||
\end{definition}
|
||||
\begin{definition}[The Tunneling Neural Network]
|
||||
|
|
@ -428,16 +426,16 @@ We will often encounter neural networks that we want to stack but have unequal d
|
|||
We will drop the requirement for $d$ and $\tun_n$ by itself will be used to denote $\tun_n^1$.
|
||||
\end{definition}
|
||||
\begin{remark}
|
||||
We will discuss some properties of the $\tun^d_n$ network in Section \ref{sec_tun}. We will also discuss properties of wider tunneling neural network in Lemma \ref{tun_mult}.
|
||||
We will discuss some properties of the $\tun^d_n$ network in Section \ref{sec_tun}.
|
||||
\end{remark}
|
||||
\begin{definition}
|
||||
Let $n \in \N$, and $\nu_1,\nu_2,...,\nu_n \in \neu$. We will define the stacking of unequal length neural networks, denoted $\DDiamond^n_{i=1}\nu_i$ as the neural network given by:
|
||||
\begin{align}
|
||||
\DDiamond^n_{i=1}\nu_i =
|
||||
\boxminus^n_{i=1} \lb \tun_{\max_i \left\{\dep \lp \nu_i \rp\right\} +1 - \dep \lp \nu_i\rp}^{\out \lp \nu_i\rp} \bullet \nu_i \rb
|
||||
\boxminus^n_{i=1} \lb \tun_{\max_i \left\{\dep \lp \nu_i \rp\right\} +1 - \dep \lp \nu_i\rp} \bullet \nu_i \rb
|
||||
\end{align}
|
||||
\end{definition}
|
||||
Diagrammatically, this can be thought of as shown below.
|
||||
Diagrammatically, this can be thought of as:
|
||||
\begin{figure}
|
||||
\begin{center}
|
||||
|
||||
|
|
@ -492,9 +490,8 @@ Diagrammatically, this can be thought of as shown below.
|
|||
\end{proof}
|
||||
|
||||
\section{Affine Linear Transformations as ANNs and Their Properties.}
|
||||
Affine neural networks present an important class of neural networks. By virtue of them being only one layer deep, they may be instantiated with any activation function whatsoever and still retain their affine transformative properties, see Definition \ref{def:inst}. In addition, when composing, they are subsumed into the function being somposed to, i.e. they do not change the depth of a neural network once composed into it, see Lemma \ref{comp_prop}.
|
||||
\begin{definition}\label{5.3.1}\label{def:aff}
|
||||
Let $m,n \in \N$, $W \in \R^{m \times n}$, $b \in \R^m$.We denote by $\aff_{W,b} \in \lp \R^{m\times n} \times \R^m \rp \subsetneq \neu$ the neural network given by $\aff_{W,b} = ((W,b))$.
|
||||
Let $m,n \in \N$, $W \in \R^{m \times n}$, $b \in \R^m$.We denote by $\aff_{W,b} \in \lp \R^{m\times n} \times \R^m \rp \subseteq \neu$ the neural network given by $\aff_{W,b} = ((W,b))$.
|
||||
\end{definition}
|
||||
\begin{lemma}\label{5.3.2}\label{aff_prop}
|
||||
Let $m,n \in \N$, $W \in \R^{m\times n}$, $b \in \R^m$. It is then the case that:
|
||||
|
|
@ -505,7 +502,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Note that $(i)$ is a consequence of Definition \ref{5.1.2} and \ref{5.3.1}. Note next that $\aff_{W,b} = (W,b) \in (\R^{m\times n} \times \R^m) \subsetneq \neu$. Note that ($\ref{5.1.11}$) then tells us that $\real_{\act} (\aff_{W,b}) = Wx+b$ which in turn proves $(ii)$ and $(iii)$
|
||||
Note that $(i)$ is a consequence of Definition \ref{5.1.2} and \ref{5.3.1}. Note next that $\aff_{W,b} = (W,b) \in (\R^{m\times n} \times \R^m) \subseteq \neu$. Note that ($\ref{5.1.11}$) then tells us that $\real_{\act} (\aff_{W,b}) = Wx+b$ which in turn proves $(ii)$ and $(iii)$
|
||||
\end{proof}
|
||||
\begin{remark}\label{remark:5.4.3}\label{param_of_aff}
|
||||
Given $W\in \R^{m\times n}$, and $b \in \R^{m \times 1}$, it is the case that according to Definition (\ref{paramdef}) we have: $\param(\aff_{W,b})= m\times n + m$
|
||||
|
|
@ -555,14 +552,14 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\begin{proof}
|
||||
Let it be the case that $\lay \lp \nu\rp = \lp l_0,l_1,...,l_L\rp$ for $l_0,l_1,...,l_L,L \in \N$. Lemma \ref{5.3.3}, Item (i), and Lemma \ref{comp_prop} then tells us that:
|
||||
\begin{align}
|
||||
&\param \lp \aff_{W,b} \bullet \nu \rp\\ &= \lb \sum^{L-1}_{m=1} l_m \lp l_{m-1}+1\rp\rb + \out \lp \aff_{W,b}\rp \lp l_{L-1}+1\rp \nonumber \\
|
||||
\param \lp \aff_{W,b} \bullet \nu \rp &= \lb \sum^{L-1}_{m=1} l_m \lp l_{m-1}+1\rp\rb + \out \lp \aff_{W,b}\rp \lp l_{L-1}+1\rp \nonumber \\
|
||||
&= \lb \sum^{L-1}_{m=1} l_m \lp l_{m-1}+1 \rp\rb+ \lb \frac{\out\lp \aff_{W,b}\rp}{l_L}\rb l_L\lp l_{L-1}+1 \rp \nonumber \\
|
||||
&\les \lb \max \left\{ 1, \frac{\out(\aff_{W,b})}{l_L}\right\}\rb \lb \sum^{L-1}_{m=1} l_m \lp l_{m-1}+1\rp\rb + \lb \max\left\{ 1,\frac{\out\lp \aff_{W,b}\rp}{l_L}\right\}\rb l_L \lp l_{L-1}+1\rp \nonumber\\
|
||||
&= \lb \max\left\{ 1, \frac{\out \lp \aff_{W,b}\rp}{l_L}\right\}\rb \lb \sum^L_{m=1}l_m \lp l_{m-1} +1\rp\rb = \lb \max\left\{ 1, \frac{\out \lp \aff_{W,b}\rp}{l_L}\right\}\rb \param \lp \nu\rp \nonumber
|
||||
\end{align}
|
||||
and further that:
|
||||
\begin{align}
|
||||
&\param \lp \nu \bullet\aff_{W,b} \rp \\ &= \lb \sum^{L}_{m=2} l_m \lp l_{m-1}+1\rp\rb + l_{1}\lp \inn \lp \aff_{W,b}\rp+1\rp \nonumber \\
|
||||
\param \lp \nu \bullet\aff_{W,b} \rp &= \lb \sum^{L}_{m=2} l_m \lp l_{m-1}+1\rp\rb + l_{1}\lp \inn \lp \aff_{W,b}\rp+1\rp \nonumber \\
|
||||
&= \lb \sum^{L}_{m=2} l_m \lp l_{m-1}+1 \rp\rb+ \lb \frac{\inn \lp \aff_{W,b}\rp+1}{l_0+1}\rb l_1\lp l_{0}+1 \rp \nonumber \\
|
||||
&\les \lb \max \left\{ 1, \frac{\inn(\aff_{W,b})+1}{l_0+1}\right\}\rb \lb \sum^{L}_{m=2} l_m \lp l_{m-1}+1\rp\rb + \lb \max\left\{ 1,\frac{\inn\lp \aff_{W,b}\rp+1}{l_0+1}\right\}\rb l_1 \lp l_{0}+1\rp \nonumber\\
|
||||
&= \lb \max\left\{ 1, \frac{\inn \lp \aff_{W,b}\rp+1}{l_0+1}\right\}\rb \lb \sum^L_{m=1}l_m \lp l_{m-1} +1\rp\rb = \lb \max\left\{ 1, \frac{\inn \lp \aff_{W,b}\rp+1}{\inn\lp \nu\rp+1}\right\}\rb \param \lp \nu\rp \nonumber
|
||||
|
|
@ -578,7 +575,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
|
||||
\section{Sums of ANNs of Same End-widths}
|
||||
|
||||
\begin{definition}[The $\cpy_{n,k}$ Network]\label{def:cpy}
|
||||
\begin{definition}[The $\cpy$ Network]\label{def:cpy}
|
||||
We define the neural network, $\cpy_{n,k} \in \neu$ for $n,k\in \N$ as the neural network given by:
|
||||
\begin{align}
|
||||
\cpy_{n,k} = \aff_{\underbrace{\lb \mathbb{I}_{k} \: \mathbb{I}_k \: \cdots \: \mathbb{I}_k \rb^T}_{n-\text{many}},\mymathbb{0}_{nk}}
|
||||
|
|
@ -587,7 +584,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
See Listing \ref{affn}.
|
||||
See Listing \ref{affn}
|
||||
\end{remark}
|
||||
\begin{lemma}\label{dep_cpy}\label{lem:param_cpy}
|
||||
Let $n,k \in \N$ and let $\cpy_{n,k} \in \neu$, it is then the case for all $n,k \in \N$ that:
|
||||
|
|
@ -596,11 +593,11 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\item $\param\lp \cpy_{n,k} \rp = nk^2+nk$
|
||||
\end{enumerate}
|
||||
\begin{proof}
|
||||
Note that $(i)$ is a consequence of Definition \ref{5.3.1}, and (ii) follows from the structure of $\cpy_{n,k}$.
|
||||
Note that $(i)$ is a consequence of Definition \ref{5.3.1} and (ii) follows from the structure of $\cpy_{n,k}$.
|
||||
\end{proof}
|
||||
|
||||
\end{lemma}
|
||||
\begin{definition}[The $\sm_{n,k}$ Network]\label{def:sm}
|
||||
\begin{definition}[The $\sm$ Network]\label{def:sm}
|
||||
We define the neural network $\sm_{n,k}$ for $n,k \in \N$ as the neural network given by:
|
||||
\begin{align}
|
||||
\sm_{n,k} = \aff_{\underbrace{\lb \mathbb{I}_k \: \mathbb{I}_k \: \cdots \: \mathbb{I}_k\rb}_{n-\text{many}}, \mymathbb{0}_{k}}
|
||||
|
|
@ -631,65 +628,6 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\begin{remark}
|
||||
For an \texttt{R} implementation, see Listing \ref{nn_sum}.
|
||||
\end{remark}
|
||||
\begin{remark}
|
||||
We may diagrammatically refer to this network as:
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{center}
|
||||
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
%uncomment if require: \path (0,433); %set diagram left start at 0, and has height of 433
|
||||
|
||||
%Shape: Rectangle [id:dp9509582141653736]
|
||||
\draw (470,170) -- (540,170) -- (540,210) -- (470,210) -- cycle ;
|
||||
%Shape: Rectangle [id:dp042468147108538634]
|
||||
\draw (330,100) -- (400,100) -- (400,140) -- (330,140) -- cycle ;
|
||||
%Shape: Rectangle [id:dp46427980442406214]
|
||||
\draw (330,240) -- (400,240) -- (400,280) -- (330,280) -- cycle ;
|
||||
%Straight Lines [id:da8763809527154822]
|
||||
\draw (470,170) -- (401.63,121.16) ;
|
||||
\draw [shift={(400,120)}, rotate = 35.54] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da9909123473315302]
|
||||
\draw (470,210) -- (401.63,258.84) ;
|
||||
\draw [shift={(400,260)}, rotate = 324.46] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da8497218496635237]
|
||||
\draw (570,190) -- (542,190) ;
|
||||
\draw [shift={(540,190)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Shape: Rectangle [id:dp11197066111784415]
|
||||
\draw (210,170) -- (280,170) -- (280,210) -- (210,210) -- cycle ;
|
||||
%Straight Lines [id:da5201326815013356]
|
||||
\draw (330,120) -- (281.41,168.59) ;
|
||||
\draw [shift={(280,170)}, rotate = 315] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da4370325799656589]
|
||||
\draw (330,260) -- (281.41,211.41) ;
|
||||
\draw [shift={(280,210)}, rotate = 45] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da012890543438617508]
|
||||
\draw (210,190) -- (182,190) ;
|
||||
\draw [shift={(180,190)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
|
||||
% Text Node
|
||||
\draw (481,182.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{n}{}_{,}{}_{k}$};
|
||||
% Text Node
|
||||
\draw (351,110.4) node [anchor=north west][inner sep=0.75pt] {$\nu _{1}$};
|
||||
% Text Node
|
||||
\draw (351,252.4) node [anchor=north west][inner sep=0.75pt] {$\nu _{2}$};
|
||||
% Text Node
|
||||
\draw (574,180.4) node [anchor=north west][inner sep=0.75pt] {$x$};
|
||||
% Text Node
|
||||
\draw (441,132.4) node [anchor=north west][inner sep=0.75pt] {$x$};
|
||||
% Text Node
|
||||
\draw (437,232.4) node [anchor=north west][inner sep=0.75pt] {$x$};
|
||||
% Text Node
|
||||
\draw (221,180.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}_{n}{}_{,}{}_{k}$};
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\caption{Neural Network diagram of a neural network sum.}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
|
||||
\subsection{Neural Network Sum Properties}
|
||||
\begin{lemma}\label{paramsum}
|
||||
|
|
@ -719,7 +657,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\begin{corollary}\label{corsum}
|
||||
Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$ satisfy that $\lay \lp \nu_1\rp = \lay \lp \nu_2\rp= \cdots =\lay \lp \nu_n\rp$. It is then the case that:
|
||||
\begin{align}
|
||||
\param \lp \bigoplus_{i=1}^n \nu_i\rp \les n^2 \cdot \param \lp \nu_1\rp
|
||||
\param \lp \bigoplus_{i=1}^n \nu_i\rp \les n^2\param \lp \nu_1\rp
|
||||
\end{align}
|
||||
\end{corollary}
|
||||
\begin{proof}
|
||||
|
|
@ -846,8 +784,8 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\end{align}
|
||||
Applying Claim \ref{5.4.5} and especially the third case of Definition \ref{5.2.1} to to the above then gives us:
|
||||
\begin{align}
|
||||
&\aff_{\lb \mathbb{I}_{\out(\nu_1)} \: \mathbb{I}_{\out(\nu_1)} \rb,0}\bullet \lb \nu_1 \boxminus \nu_2 \rb \bullet \aff_{\lb \mathbb{I}_{\inn \lp \nu_2 \rp } \: \mathbb{I}_{\inn \lp \nu_2 \rp} \rb^T,0} = \nonumber\\
|
||||
&\lp \lp \begin{bmatrix}
|
||||
&\aff_{\lb \mathbb{I}_{\out(\nu_1)} \: \mathbb{I}_{\out(\nu_1)} \rb,0}\bullet \lb \nu_1 \boxminus \nu_2 \rb \bullet \aff_{\lb \mathbb{I}_{\inn \lp \nu_2 \rp } \: \mathbb{I}_{\inn \lp \nu_2 \rp} \rb^T,0} \nonumber\\
|
||||
&= \lp \lp \begin{bmatrix}
|
||||
W_1 \\
|
||||
W'_1
|
||||
\end{bmatrix} ,\begin{bmatrix}
|
||||
|
|
@ -858,11 +796,11 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\lp \begin{bmatrix}
|
||||
W_2 & 0\\
|
||||
0 & W'_2
|
||||
\end{bmatrix} ,\begin{bmatrix}
|
||||
\end{bmatrix} \begin{bmatrix}
|
||||
b_2 \\
|
||||
b_2'
|
||||
\end{bmatrix} \rp \right.,..., \nonumber \\
|
||||
&\left. \lp \begin{bmatrix}
|
||||
\end{bmatrix} \rp \right.,..., \nonumber
|
||||
\left. \lp \begin{bmatrix}
|
||||
\mathbb{I}_{\out(\nu_2)} \: \mathbb{I}_{\out(\nu_2)}
|
||||
\end{bmatrix}\begin{bmatrix}
|
||||
W_L & 0 \\
|
||||
|
|
@ -955,29 +893,27 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
W'_1x+b_1'
|
||||
\end{bmatrix} \nonumber
|
||||
\end{align}
|
||||
The full instantiation of (\ref{5.4.10}) with activation function $\fa \in C \lp \R, \R\rp$ is then given by:
|
||||
The full instantiation of (\ref{5.4.10}) is then given by:
|
||||
\begin{align}
|
||||
\begin{bmatrix}
|
||||
\real \lp \begin{bmatrix}
|
||||
W_L \quad W'_L
|
||||
\end{bmatrix}\begin{bmatrix}
|
||||
\act\lp W_{L-1}(...\act(W_2\lp \act\lp W_1x+b_1 \rp\rp + b_2) + ... )+ b_{L-1} \rp\\
|
||||
\act\lp W'_{L-1}(...\act(W'_2\lp\act\lp W'_1x + b'_1 \rp \rp + b'_2)+...)+b'_{L-1}\rp
|
||||
\end{bmatrix} + b_L+b'_L \label{5.4.12}
|
||||
W_{L-1}(...(W_2\lp W_1x+b_1 \rp + b_2) + ... )+ b_{L-1} \\
|
||||
W'_{L-1}(...(W'_2 \lp W'_1x + b'_1 \rp + b'_2)+...)+b'_{L-1}
|
||||
\end{bmatrix} + b_L+b'_L \rp \label{5.4.12}
|
||||
\end{align}
|
||||
The full instantiation of (\ref{5.4.11}) is then given by:
|
||||
\begin{align}
|
||||
\begin{bmatrix}
|
||||
\real \lp \begin{bmatrix}
|
||||
W_L' \quad W_L
|
||||
\end{bmatrix}\begin{bmatrix}
|
||||
\act\lp W'_{L-1}(...\act(W'_2\lp \act \lp W'_1x+b'_1 \rp\rp + b'_2) + ... )+ b'_{L-1}\rp \\
|
||||
\act\lp W_{L-1}(...\act(W_2 \lp\act\lp W_1x + b_1 \rp\rp + b_2)+...)+b_{L-1} \rp
|
||||
\end{bmatrix} + b_L+b'_L \label{5.4.13}
|
||||
W'_{L-1}(...(W'_2\lp W'_1x+b'_1 \rp + b'_2) + ... )+ b'_{L-1} \\
|
||||
W_{L-1}(...(W_2 \lp W_1x + b_1 \rp + b_2)+...)+b_{L-1}
|
||||
\end{bmatrix} + b_L+b'_L \rp \label{5.4.13}
|
||||
\end{align}
|
||||
Since (\ref{5.4.12}) and (\ref{5.4.13}) are the same this proves that $\nu_1 \oplus \nu_2 = \nu_2 \oplus \nu_1$.
|
||||
\end{proof}
|
||||
\begin{remark}
|
||||
This is a special case of \cite[Lemma~3.28]{Grohs_2022}.
|
||||
\end{remark}
|
||||
\begin{lemma}\label{5.4.7}
|
||||
Let $ l_0,l_1,...,l_L \in \N$. Let $\nu \in \neu$ with $\lay(\nu) = \lp l_0,l_1,...,l_L \rp$. There then exists a neural network $\zero_{l_0,l_1,...,l_L} \in \neu$ such that $\real(\nu \oplus \zero_{l_0,l_1,...,l_L}) = \real(\zero_{l_0,l_1,...,l_L} \oplus \nu) = \nu $.
|
||||
\end{lemma}
|
||||
|
|
@ -1104,7 +1040,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
This is the consequence of a finite number of applications of Lemma \ref{5.5.11}. This proves the Lemma.
|
||||
This is the consequence of a finite number of applications of Lemma \ref{5.5.11}.
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
|
@ -1140,75 +1076,9 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\end{align}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
This is a consequence of a finite number of applications of Lemma \ref{lem:diamondplus}. This proves the Lemma.
|
||||
This is a consequence of a finite number of applications of Lemma \ref{lem:diamondplus}.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
We may represent this kind of sum as the neural network diagram shown below:
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{center}
|
||||
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
%uncomment if require: \path (0,433); %set diagram left start at 0, and has height of 433
|
||||
|
||||
%Shape: Rectangle [id:dp9509582141653736]
|
||||
\draw (470,170) -- (540,170) -- (540,210) -- (470,210) -- cycle ;
|
||||
%Shape: Rectangle [id:dp042468147108538634]
|
||||
\draw (200,100) -- (400,100) -- (400,140) -- (200,140) -- cycle ;
|
||||
%Shape: Rectangle [id:dp46427980442406214]
|
||||
\draw (330,240) -- (400,240) -- (400,280) -- (330,280) -- cycle ;
|
||||
%Straight Lines [id:da8763809527154822]
|
||||
\draw (470,170) -- (401.63,121.16) ;
|
||||
\draw [shift={(400,120)}, rotate = 35.54] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da9909123473315302]
|
||||
\draw (470,210) -- (401.63,258.84) ;
|
||||
\draw [shift={(400,260)}, rotate = 324.46] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da8497218496635237]
|
||||
\draw (570,190) -- (542,190) ;
|
||||
\draw [shift={(540,190)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Shape: Rectangle [id:dp11197066111784415]
|
||||
\draw (80,170) -- (150,170) -- (150,210) -- (80,210) -- cycle ;
|
||||
%Straight Lines [id:da5201326815013356]
|
||||
\draw (200,130) -- (151.56,168.75) ;
|
||||
\draw [shift={(150,170)}, rotate = 321.34] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da4370325799656589]
|
||||
\draw (330,260) -- (312,260) ;
|
||||
\draw [shift={(310,260)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Straight Lines [id:da012890543438617508]
|
||||
\draw (80,190) -- (52,190) ;
|
||||
\draw [shift={(50,190)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
%Shape: Rectangle [id:dp2321426611089945]
|
||||
\draw (200,240) -- (310,240) -- (310,280) -- (200,280) -- cycle ;
|
||||
%Straight Lines [id:da03278204116412775]
|
||||
\draw (200,260) -- (151.41,211.41) ;
|
||||
\draw [shift={(150,210)}, rotate = 45] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
|
||||
|
||||
% Text Node
|
||||
\draw (481,182.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{n}{}_{,}{}_{k}$};
|
||||
% Text Node
|
||||
\draw (301,110.4) node [anchor=north west][inner sep=0.75pt] {$\nu _{1}$};
|
||||
% Text Node
|
||||
\draw (351,252.4) node [anchor=north west][inner sep=0.75pt] {$\nu _{2}$};
|
||||
% Text Node
|
||||
\draw (574,180.4) node [anchor=north west][inner sep=0.75pt] {$x$};
|
||||
% Text Node
|
||||
\draw (441,132.4) node [anchor=north west][inner sep=0.75pt] {$x$};
|
||||
% Text Node
|
||||
\draw (437,232.4) node [anchor=north west][inner sep=0.75pt] {$x$};
|
||||
% Text Node
|
||||
\draw (91,180.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}_{n}{}_{,}{}_{k}$};
|
||||
% Text Node
|
||||
\draw (238,252.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$};
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
\caption{Neural network diagram of a neural network sum of unequal depth networks.}
|
||||
\end{center}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
\section{Linear Combinations of ANNs and Their Properties}
|
||||
\begin{definition}[Scalar left-multiplication with an ANN]\label{slm}
|
||||
Let $\lambda \in \R$. We will denote by $(\cdot ) \triangleright (\cdot ): \R \times \neu \rightarrow \neu$ the function that satisfy for all $\lambda \in \R$ and $\nu \in \neu$ that $\lambda \triangleright \nu = \aff_{\lambda \mathbb{I}_{\out(\nu)},0} \bullet \nu$.
|
||||
|
|
@ -1242,7 +1112,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\begin{align}
|
||||
\lay \lp \lambda \triangleright \nu \rp = \lay \lp \aff_{\lambda \mathbb{I}_{\out(\nu)},0} \bullet \nu \rp = \lp l_0, l_1,...,l_{L-1}, \out(\nu) \rp = \lay(\nu)
|
||||
\end{align}
|
||||
Which proves (i). Item (ii)\textemdash(iii) of Lemma $\ref{5.3.2}$ then prove that for all $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu)}$, that $\real_{\act} \lp \lambda \triangleright \nu \rp \in C \lp \R^{\inn(\nu),\out(\nu)} \rp$ given by:
|
||||
Which proves $(i)$. Item $(ii)-(iii)$ of Lemma $\ref{5.3.2}$ then prove that for all $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu)}$, that $\real_{\act} \lp \lambda \triangleright \nu \rp \in C \lp \R^{\inn(\nu),\out(\nu)} \rp$ given by:
|
||||
\begin{align}
|
||||
\lp \real_{\act} \lp \lambda \triangleright \nu \rp \rp \lp x \rp &= \lp \real_{\act} \lp \aff_{\lambda \mathbb{I}_{\out(\nu),0}} \bullet \nu \rp \rp \lp x \rp \nonumber\\
|
||||
&= \lambda \mathbb{I}_{\out(\nu)} \lp \lp \real_{\act} \lp \nu \rp \rp \lp x \rp \rp = \lambda \lp \lp \real_{\act} \lp \nu \rp \rp \lp x \rp \rp
|
||||
|
|
@ -1270,7 +1140,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\begin{align}
|
||||
\lay(\nu \triangleleft\lambda) = \lay \lp \nu \bullet \aff_{\lambda \mathbb{I}_{\inn(\nu)}}\rp = \lp \inn(\nu), l_1,l_2,...,l_L \rp = \lay(\nu)
|
||||
\end{align}
|
||||
Which proves $(i)$. Item (v)\textemdash(vi) of Lemma \ref{5.3.3} then prove that for all $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu)}$ that $\real_{\act} \lp \nu \triangleleft \lambda \rp \in C\lp \R^{\inn(\nu),\out(\nu)} \rp$ given by:
|
||||
Which proves $(i)$. Item (v)--(vi) of Lemma \ref{5.3.3} then prove that for all $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu)}$ that $\real_{\act} \lp \nu \triangleleft \lambda \rp \in C\lp \R^{\inn(\nu),\out(\nu)} \rp$ given by:
|
||||
\begin{align}
|
||||
\lp \real_{\act} \lp \nu \triangleleft \lambda \rp \rp \lp x \rp &= \lp \real_{\act} \lp \nu \bullet \aff_{\lambda \mathbb{I}_{\inn(\nu),0}} \rp \rp \lp x \rp \nonumber\\
|
||||
&= \lp \real_{\act} \lp \nu \rp \rp \lp \aff_{\lambda \mathbb{I}_{\inn(\nu)}} \rp \lp x \rp \nonumber\\
|
||||
|
|
@ -1293,17 +1163,17 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
&= \begin{bmatrix}
|
||||
W_L \quad W'_L
|
||||
\end{bmatrix}\begin{bmatrix}
|
||||
\act \lp W_{L-1}(...(\act \lp W_2\lp \act \lp W_1\lambda x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp \\
|
||||
\act \lp W'_{L-1}(...(\act \lp W'_2\lp \act \lp W'_1\lambda x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp \\
|
||||
\inst_{\rect} \lp W_{L-1}(...(\inst_{\rect} \lp W_2\lp \inst_{\rect} \lp W_1\lambda x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp \\
|
||||
\inst_{\rect} \lp W'_{L-1}(...(\inst_{\rect} \lp W'_2\lp \inst_{\rect} \lp W'_1\lambda x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp \\
|
||||
\end{bmatrix} + b_L+b'_L \nonumber
|
||||
\end{align}
|
||||
Note that:
|
||||
\begin{align}
|
||||
\lp \real_{\act} \lp \nu \rp \rp \lp \lambda x \rp = W_L \cdot \act \lp W_{L-1}(...(\act \lp W_2\lp \act \lp W_1\lambda x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp + b_L
|
||||
\lp \real_{\act} \lp \nu \rp \rp \lp \lambda x \rp = W_L \cdot \inst_{\rect} \lp W_{L-1}(...(\inst_{\rect} \lp W_2\lp \inst_{\rect} \lp W_1\lambda x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp + b_L
|
||||
\end{align}
|
||||
and that:
|
||||
\begin{align}
|
||||
\lp \real_{\act} \lp \mu \rp \rp \lp \lambda x \rp = W'_L\cdot\act \lp W'_{L-1}(...(\act \lp W'_2\lp \act \lp W'_1\lambda x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp + b'_L
|
||||
\lp \real_{\act} \lp \mu \rp \rp \lp \lambda x \rp = W'_L\cdot\inst_{\rect} \lp W'_{L-1}(...(\inst_{\rect} \lp W'_2\lp \inst_{\rect} \lp W'_1\lambda x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp + b'_L
|
||||
\end{align}
|
||||
This, together with Lemma \ref{5.5.11}, completes the proof.
|
||||
\end{proof}
|
||||
|
|
@ -1450,16 +1320,19 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
&= \sum^v_{i=u} \lp \real_{\act} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i \rp \rp \lp x \rp\\
|
||||
&=\sum^v_{i=u} \lp \real_{\act} \lp \nu_i \rp \rp \lp c_i x+b_i \rp \nonumber
|
||||
\end{align}
|
||||
This establishes items (ii)\textemdash(iii); thus, the proof is complete.
|
||||
This establishes items (ii)--(iii); thus, the proof is complete.
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma}\label{5.6.9}
|
||||
Let $L \in \N$, $u,v \in \Z$ with $u\leqslant v$. Let $c_u, c_{u+1},...,c_v \in \R$. $\nu_u, \nu_{u+1},...,\nu_v, \mu \in \neu$, $B_u, B_{u+1},...,B_v \in \R^{\inn(\nu_u)}$, $\act \in C\lp \R, \R \rp$, satisfy for all $j \in \N \cap [u,v]$ that $L = \max_{i\in \N \cap \lb u,v \rb}\\ \dep(\nu_i)$, $\inn(\nu_j) = \inn(\nu_u)$, $\out(\nu_j) = \inn(\mathfrak{I})= \out(\mathfrak{I})$, $\hid(\mathfrak{I}) = 1$, $\real_{\act} (\mathfrak{I}) = \mathbb{I}_\R$, and that:
|
||||
Let $L \in \N$, $u,v \in \Z$ with $u\leqslant v$. Let $c_u, c_{u+1},...,c_v \in \R$. $\nu_u, \nu_{u+1},...,\nu_v, \mu, \mathfrak{I} \in \neu$, $B_u, B_{u+1},...,B_v \in \R^{\inn(\nu_u)}$, $\act \in C\lp \R, \R \rp$, satisfy for all $j \in \N \cap [u,v]$ that $L = \max_{i\in \N \cap \lb u,v \rb} \dep(\nu_i)$, $\inn(\nu_j) = \inn(\nu_u)$, $\out(\nu_j) = \inn(\mathfrak{I})= \out(\mathfrak{I})$, $\hid(\mathfrak{I}) = 1$, $\real_{\act} (\mathfrak{I}) = \mathbb{I}_\R$, and that:
|
||||
\begin{align}
|
||||
\mu = \dplus^v_{i = u, \mathfrak{I}} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i), },b_i} \rp \rp
|
||||
\mu = \boxplus^v_{i = u, \mathfrak{I}} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i), },b_i} \rp \rp
|
||||
\end{align}
|
||||
We then have that:
|
||||
We then have:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item it holds that:
|
||||
\begin{align}
|
||||
\lay(\mu) = \lp \inn(\nu_u ), \sum^v_{i=u}\wid_1 \lp \ex_{L,\mathfrak{I}} \lp \nu_i \rp \rp ,\sum^v_{i=u}\wid_2 \lp \ex_{L,\mathfrak{I}} \lp \nu_i\rp\rp,...,\sum^v_{i=u} \wid_{L-1} \lp \ex_{I,\mathfrak{I}} \lp \nu_i \rp , \out \lp \nu_u \rp \rp \rp
|
||||
\end{align}
|
||||
\item it holds that $\real_{\act}(\mu) \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $, and that,
|
||||
\item it holds for all $ x \in \R^{\inn(\nu_u)}$ that:
|
||||
\begin{align}
|
||||
|
|
@ -1468,8 +1341,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Note that Item(i) from Lemma \ref{5.6.5} establish Item(i) and (\ref{5.5.20}); in addition, items (v) \textemdash (vi) from Lemma \ref{5.3.3} tell us that for all $i \in \N \cap [u,v]$, $x \in \R^{\inn(\nu_u}$, it holds that \\
|
||||
$\real_{\act} \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)}\rp \rp $ and further that:
|
||||
Note that Item(i) from Lemma \ref{5.6.5} establish Item(i) and (\ref{5.5.20}); in addition, items (v) and (vi) from Lemma \ref{5.3.3} tell us that for all $i \in \N \cap [u,v]$, $x \in \R^{\inn(\nu_u}$, it holds that $\real_{\act} \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)}\rp \rp $ and further that:
|
||||
\begin{align}
|
||||
\lp \real_{\act} \lp \nu_i\bullet \aff_{\mathbb{I}_{\inn(\nu_i)},B_i} \rp \rp \lp x \rp = \lp \real_{\act} \lp \nu_i \rp \rp \lp x + b_k \rp
|
||||
\end{align}
|
||||
|
|
@ -1491,7 +1363,7 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
This establishes Items(ii)--(iii), thus proving the lemma.
|
||||
\end{proof}
|
||||
\begin{lemma}
|
||||
Let $L \in \N$, $u,v \in \Z$ with $u\leqslant v$. Let $c_u, c_{u+1},...,c_v \in \R$. $\nu_u, \nu_{u+1},...,\nu_v, \mu, \mathfrak{I} \in \neu$, $B_u, B_{u+1},...,B_v \in \R^{\inn(\nu_u)}$, $\act \in C\lp \R, \R \rp$, satisfy for all $j \in \N \cap [u,v]$ that $L =\\ \max_{i\in \N \cap \lb u,v \rb} \dep(\nu_i)$, $\inn(\nu_j) = \inn(\nu_u)$, $\out(\nu_j) = \inn(\mathfrak{I})= \out(\mathfrak{I})$, $\hid(\mathfrak{I}) = 1$, $\real_{\act} (\mathfrak{I}) = \mathbb{I}_\R$, and that:
|
||||
Let $L \in \N$, $u,v \in \Z$ with $u\leqslant v$. Let $c_u, c_{u+1},...,c_v \in \R$. $\nu_u, \nu_{u+1},...,\nu_v, \mu, \mathfrak{I} \in \neu$, $B_u, B_{u+1},...,B_v \in \R^{\inn(\nu_u)}$, $\act \in C\lp \R, \R \rp$, satisfy for all $j \in \N \cap [u,v]$ that $L = \max_{i\in \N \cap \lb u,v \rb} \dep(\nu_i)$, $\inn(\nu_j) = \inn(\nu_u)$, $\out(\nu_j) = \inn(\mathfrak{I})= \out(\mathfrak{I})$, $\hid(\mathfrak{I}) = 1$, $\real_{\act} (\mathfrak{I}) = \mathbb{I}_\R$, and that:
|
||||
\begin{align}
|
||||
\mu = \boxplus^v_{i = u, \mathfrak{I}} \lp \lp \aff_{\mathbb{I} _{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i \rp
|
||||
\end{align}
|
||||
|
|
|
|||
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@ -1,26 +1,7 @@
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% Set chapter and section font sizes to 12pt
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\titleformat{\part}[display]
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{\normalfont\fontsize{12}{12}\selectfont\bfseries}{\partname\ \thepart}{0pt}{\fontsize{12}{12}\selectfont}[\vspace{1in}]
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\titleformat{\chapter}[display]{\normalfont\fontsize{12}{12} \bfseries }{\chaptertitlename\ \thechapter.}{1ex}{}{}
|
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\titlespacing{\chapter}{0pt}{-20pt}{0pt}
|
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|
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\documentclass[11pt]{report}
|
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|
||||
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|
||||
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|
||||
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|
||||
% Set chapter heading font size to 12pt
|
||||
|
||||
|
||||
|
||||
\usepackage{pdfpages}
|
||||
\usepackage[]{amsmath}
|
||||
|
|
@ -28,21 +9,7 @@
|
|||
\usepackage{mathtools}
|
||||
\numberwithin{equation}{section}
|
||||
\usepackage[]{amssymb}
|
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|
||||
\usepackage{geometry}
|
||||
|
||||
\geometry{
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||||
left=1in, % Left margin
|
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right=1in, % Right margin
|
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top=1in, % Top margin
|
||||
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|
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|
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||||
\usepackage{url}
|
||||
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
|
||||
\usepackage[margin=1in]{geometry}
|
||||
\usepackage[]{soul}
|
||||
\usepackage[]{bbm}
|
||||
\usepackage[]{cancel}
|
||||
|
|
@ -60,7 +27,7 @@
|
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pdfauthor={Shakil Rafi},
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filecolor = magenta,
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urlcolor = cyan
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@ -97,7 +64,7 @@
|
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numbersep = 5pt,
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breakatwhitespace=false,
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tabsize=4,
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morekeywords={install.packages, library, ggplot, aes, geom_bar, source, }
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morekeywords={install.packages, library, ggplot, aes, geom_bar}
|
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}
|
||||
|
||||
\usepackage{fontspec}
|
||||
|
|
@ -121,7 +88,7 @@
|
|||
\DeclareMathOperator{\supp}{supp}
|
||||
\DeclareMathOperator{\domain}{Domain}
|
||||
\DeclareMathOperator{\lip}{Lip}
|
||||
\DeclareMathOperator{\var}{\mathbb{V}}
|
||||
\DeclareMathOperator{\var}{Var}
|
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|
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\DeclareMathOperator{\unif}{Unif}
|
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\DeclareMathOperator{\dropout}{Dropout}
|
||||
|
|
@ -140,7 +107,7 @@
|
|||
\DeclareMathOperator{\lay}{\mathsf{L}}
|
||||
\DeclareMathOperator{\dep}{\mathsf{D}}
|
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\DeclareMathOperator{\we}{Weight}
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|
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|
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\DeclareMathOperator{\aff}{\mathsf{Aff}}
|
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|
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\DeclareMathOperator{\real}{\mathfrak{I}}
|
||||
|
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@ -179,7 +146,6 @@
|
|||
\DeclareMathOperator{\inst}{\mathfrak{I}}
|
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@ -1,5 +1,4 @@
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@ -7,66 +6,194 @@
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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@ -1,6 +1,6 @@
|
|||
\chapter{That $u$ is a Viscosity Solution}
|
||||
|
||||
Our goal this chapter is to use Feynman-Kac to see that the solutions to certain versions of the heat equations can be expressed as the expectation of a stocahstic integral. Parts of this work is heavily inspired from \cite{crandall_lions} and esp. \cite{Beck_2021}.
|
||||
We can extend the work for the heat equation to generic parabolic partial differential equations. We do this by first introducing viscosity solutions to Kolmogorov PDEs as given in Crandall \& Lions \cite{crandall_lions} and further extended, esp. in \cite{Beck_2021}.
|
||||
%\subsection{The case without $f$}
|
||||
%\subsection{Linear Algebra Preliminaries}
|
||||
%\begin{lemma}
|
||||
|
|
@ -44,7 +44,7 @@ Our goal this chapter is to use Feynman-Kac to see that the solutions to certain
|
|||
% Leading to a contradiction. Thus there are no generalized eigenvectors of order 2 or higher, and so $A$ must be diagonalizable.
|
||||
%\end{proof}
|
||||
\section{Some Preliminaries}
|
||||
We take work previously pioneered by \cite{Ito1942a} and \cite{Ito1946}, and then seek to re-apply concepts applied in \cite{Beck_2021} and \cite{BHJ21}.
|
||||
We take work previously pioneered by \cite{Ito1942a} and \cite{Ito1946}, and then seek to re-apply concepts first applied in \cite{Beck_2021} and \cite{BHJ21}.
|
||||
\begin{lemma}\label{lemma:2.7}
|
||||
Let $d,m \in \N$, $T \in (0,\infty)$. Let $\mu \in C^{1,2}([0,T] \times \R^d, \R^d)$ and $\sigma \in C^{1,2}([0,T] \times \R^d, \R^{d\times m})$ satisfying that they have non-empty compact supports and let $\mathfrak{S}= \supp(\mu)\cup \supp(\sigma) \subseteq [0,T] \times \R^d$. Let $( \Omega, \mathcal{F}, \mathbb{P}, ( \mathbb{F}_t )_{t \in [0,T]})$ be a filtered probability space satisfying usual conditions. Let $W:[0,T ]\times \Omega \rightarrow \R^m$ be a standard $(\mathbb{F}_t)_{t\in [0,T]}$ -Brownian motion, and let $\mathcal{X}:[0,T] \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_t)_{t\in [0,T]}$-adapted stochastic process with continuous sample paths satisfying for all $t \in [0,T]$ with $\mathbb{P}$-a.s. that:
|
||||
\begin{align}
|
||||
|
|
@ -394,8 +394,7 @@ Note that (\ref{2.13}) and (\ref{2.14}) together prove that $u(T,x) = g(x)$. Thi
|
|||
%For each row $j$ we therefore have $x_k + \sqrt{2} \mathfrak{W}^{i,j,d}_{t-s}$
|
||||
%
|
||||
\begin{lemma} \label{maxviscosity}
|
||||
Let $d\in \N$, $T \in \lp 0,\infty \rp$, $\mathfrak{t} \in \lp 0,T \rp$, let $\mathcal{O} \subseteq \R^d$ be an open set, let $\mathfrak{r} \in \mathcal{O}$, $\phi \in C^{1,2}\lp \lp 0,T \rp \times \mathcal{O},\R \rp$, let $G: \lp 0,T \rp \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ be degenerate elliptic and let $u_d: (0,T) \times \mathcal{O} \rightarrow \R$ be a viscosity solution of \\
|
||||
$\lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + G \lp t,x,u(t,x), \lp \nabla_x u_D \rp \lp t,x \rp, \lp \Hess_x u_d \rp \lp t,x \rp \rp \geqslant 0$ for $(t,x) \in (0,T) \times \mathcal{O}$, and assume that $u-\phi$ has a local maximum at $(\mathfrak{t}, \mathfrak{r}) \in (0,T) \times \mathcal{O}$, then:
|
||||
Let $d\in \N$, $T \in \lp 0,\infty \rp$, $\mathfrak{t} \in \lp 0,T \rp$, let $\mathcal{O} \subseteq \R^d$ be an open set, let $\mathfrak{r} \in \mathcal{O}$, $\phi \in C^{1,2}\lp \lp 0,T \rp \times \mathcal{O},\R \rp$, let $G: \lp 0,T \rp \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ be degenerate elliptic and let $u_d (0,T) \times \mathcal{O} \rightarrow \R$ be a viscosity solution of $\lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + G \lp t,x,u(t,x), \lp \nabla_x u_D \rp \lp t,x \rp, \lp \Hess_x u_d \rp \lp t,x \rp \rp \geqslant 0$ for $(t,x) \in (0,T) \times \mathcal{O}$, and assume that $u-\phi$ has a local maximum at $(\mathfrak{t}, \mathfrak{r}) \in (0,T) \times \mathcal{O}$, then:
|
||||
\begin{align}
|
||||
\lp \frac{\partial}{\partial t} \phi \rp \lp \mathfrak{t},\mathfrak{r}\rp + G \lp \mathfrak{t}, \mathfrak{r}, u(\mathfrak{t}, \mathfrak{r}), \lp \nabla _x \phi \rp \lp \mathfrak{t}, \mathfrak{r} \rp, \lp \Hess_x \phi\rp\lp \mathfrak{t}, \mathfrak{r} \rp \rp \geqslant 0
|
||||
\end{align}
|
||||
|
|
@ -427,7 +426,7 @@ Note that (\ref{2.13}) and (\ref{2.14}) together prove that $u(T,x) = g(x)$. Thi
|
|||
\end{align}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Let $(t_0, x_0) \in (0,T) \times \mathcal{O}$. Let $\phi_\epsilon \in C^{1,2}((0,T) \times \mathcal{O}, \R)$ satisfy for all $\epsilon \in (0, \infty)$, $s \in (0,T)$, $y \in \mathcal{O}$ that $\phi_0(t_0,x_0) = u_0(t_0,x_0)$, $\phi_0(t_0,x_0) \geqslant u_0(t_0,x_0)$, and:
|
||||
Let $(t_o, x_o) \in (0,T) \times \mathcal{O}$. Let $\phi_\epsilon \in C^{1,2}((0,T) \times \mathcal{O}, \R)$ satisfy for all $\epsilon \in (0, \infty)$, $s \in (0,T)$, $y \in \mathcal{O}$ that $\phi_0(t_0,x_0) = u_0(t_0,x_0)$, $\phi_0(t_0,x_0) \geqslant u_0(t_0,x_0)$, and:
|
||||
\begin{align}\label{phieps}
|
||||
\phi_\varepsilon(s,y) = \phi_o(s,y) + \varepsilon\lp \lv s - t_0 \rv + \| y - x_0 \|_E \rp
|
||||
\end{align}
|
||||
|
|
@ -576,7 +575,7 @@ Taken together these prove the corollary.
|
|||
\begin{align}\label{limsupis0}
|
||||
\limsup_{n \rightarrow \infty} \lb \sup_{t\in[0,T]} \sup_{x\in \mathcal{O}} \lp \right\|\mu_n(t,x) - \mu_0(t,x)\left\|_E + \left\|\sigma_n(t,x)-\sigma_0(t,x)\right\|_F \rp \rb = 0
|
||||
\end{align}
|
||||
Let $\lp \Omega, \mathcal{F}, \mathbb{R} \rp$ be a stochastic basis and let $W: [0,T] \times \Omega \rightarrow \R^m$ be a standard\\ $(\mathbb{F}_t)_{t\in [0,T]}$-Brownian motion for every $t\in [0,T]$, $x \in \mathcal{O}$, let $\mathcal{X}^{t,x} = (\mathcal{X}^{t,x}_s)_{s\in [t,T]}: [t,T] \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_s)_{s\in [t,T]}$ adapted stochastic process with continuous sample paths, satisfying for all $s \in [t, T]$ we have $\mathbb{P}$-a.s.
|
||||
Let $\lp \Omega, \mathcal{F}, \mathbb{R} \rp$ be a stochastic basis and let $W: [0,T] \times \Omega \rightarrow \R^m$ be a standard $(\mathbb{F}_t)_{t\in [0,T]}$-Brownian motion for every $t\in [0,T]$, $x \in \mathcal{O}$, let $\mathcal{X}^{t,x} = (\mathcal{X}^{t,x}_s)_{s\in [t,T]}: [t,T] \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_s)_{s\in [t,T]}$ adapted stochastic process with continuous sample paths, satisfying for all $s \in [t, T]$ we have $\mathbb{P}$-a.s.
|
||||
\begin{align}\label{xnasintuvxn}
|
||||
\mathcal{X}^{n,t,x}_s = x + \int^s_t \mu_n(r,\mathcal{X}^{n,t,x}_s) dr + \int^s_t \sigma_n(r,\mathcal{X}^{n,t,x}_r) dW_r
|
||||
\end{align}
|
||||
|
|
@ -626,7 +625,7 @@ Taken together these prove the corollary.
|
|||
\end{align}
|
||||
However assumption (\ref{lipformun}) then gives us that for all $n\in \N$, $t \in [0,T]$, $s \in [t,T]$, and $x \in \mathcal{O}$ that:
|
||||
\begin{align}
|
||||
&\E \lb \left\|\mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E^2 \rb \leqslant 4L^2(T+1) \int^s_t\E\lb \left\|\mathcal{X}^{n,t,x}_r-\mathcal{X}^{0,t,x}_r \right\|_E^2 \rb dr \nonumber \\
|
||||
\E \lb \left\|\mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E^2 \rb &\leqslant 4L^2(T+1) \int^s_t\E\lb \left\|\mathcal{X}^{n,t,x}_r-\mathcal{X}^{0,t,x}_r \right\|_E^2 \rb dr \nonumber \\
|
||||
&+4T(T+1) \lb \sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y) - \mu_0(r,y) \right\|_E^2 + \left\| \sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb \nonumber
|
||||
\end{align}
|
||||
|
||||
|
|
@ -636,12 +635,12 @@ Taken together these prove the corollary.
|
|||
&\leqslant 4T(T+1) \lb \sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \| \mu_n(r,y)-\mu_0(r,y) \|_E^2 + \|\sigma_n(r,y) - \sigma_)(r,y) \|_F^2 \rp \rb e^{4L^2T(T+1)}
|
||||
\end{align}
|
||||
Applying $\limsup_{n\rightarrow \infty}$ to both sides and applying (\ref{limsupis0}) gives us for all $n \in \N$, $t \in [0,T]$, $s\in [t,T]$, $x \in \mathcal{O}$ that:
|
||||
\begin{align*}
|
||||
&\limsup_{n\rightarrow \infty} \E \lb \left\| \mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E^2 \rb \nonumber
|
||||
\leqslant \\ &\limsup_{n\rightarrow \infty} \lb 4T(T+1) \lb \sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y)-\mu_0(r,y) \right\|_E^2 + \left\|\sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb e^{4L^2T(T+1)} \rb \nonumber
|
||||
\nonumber\\ &\leqslant \\ &4T(T+1) \lb \limsup_{n\rightarrow \infty} \lb\sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y)-\mu_0(r,y) \right\|_E^2 + \left\|\sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb\rb e^{4L^2T(T+1)} \nonumber \\
|
||||
\begin{align}
|
||||
&\limsup_{n\rightarrow \infty} \E \lb \left\| \mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E^2 \rb \nonumber\\
|
||||
&\leqslant \limsup_{n\rightarrow \infty} \lb 4T(T+1) \lb \sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y)-\mu_0(r,y) \right\|_E^2 + \left\|\sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb e^{4L^2T(T+1)} \rb \nonumber \\
|
||||
&\leqslant 4T(T+1) \lb \limsup_{n\rightarrow \infty} \lb\sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y)-\mu_0(r,y) \right\|_E^2 + \left\|\sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb\rb e^{4L^2T(T+1)} \nonumber \\
|
||||
&\leqslant 0 \nonumber
|
||||
\end{align*}
|
||||
\end{align}
|
||||
This completes the proof.
|
||||
\end{proof}
|
||||
|
||||
|
|
@ -798,7 +797,7 @@ Since for all $n\in \N$, it is the case that $\mathcal{S} = \lp \supp(\mathfrak{
|
|||
\end{align}
|
||||
and finally let, for every $n\in \N$, $t \in [0,T]$, $x \in \mathcal{O}$, there be $\mathfrak{t}^{t,x}_n: \Omega \rightarrow [t,T]$ which satisfy $\mathfrak{t}^{t,x}_n = \inf \lp \{ s \in [t,T], \max \{V(s,\mathfrak{X}^{t,x}_s),V(s,\mathcal{X}^{t,x}_s)\} \geqslant n \} \cup \{T\} \rp$. We may apply Lemma \ref{2.19} with $\mu \curvearrowleft \mathfrak{m}_n$, $\sigma \curvearrowleft \mathfrak{s}_n$, $g \curvearrowleft \mathfrak{g}_k$ to show that for all $n,k \in \N$ we have that $\mathfrak{u}^{n,k}$ is a viscosity solution to:
|
||||
\begin{align}\label{2.89}
|
||||
&\lp \frac{\partial}{\partial t} \mathfrak{u}^{n,k} \rp (t,x) + \frac{1}{2} \Trace \lp \mathfrak{s}_n(t,x) \lb \mathfrak{s}_n(t,x) \rb^* \lp \Hess_x \mathfrak{u}^{n,k} \rp (t,x) \rp + \la \mathfrak{m}_n(t,x), \lp \nabla_x(\mathfrak{u}^{n,k} \rp(t,x) \ra \nonumber\\& = 0 \nonumber
|
||||
\lp \frac{\partial}{\partial t} \mathfrak{u}^{n,k} \rp (t,x) + \frac{1}{2} \Trace \lp \mathfrak{s}_n(t,x) \lb \mathfrak{s}_n(t,x) \rb^* \lp \Hess_x \mathfrak{u}^{n,k} \rp (t,x) \rp + \la \mathfrak{m}_n(t,x), \lp \nabla_x(\mathfrak{u}^{n,k} \rp(t,x) \ra = 0
|
||||
\end{align}
|
||||
for $(t,x) \in (0,T) \times \R^d$. But note that items (i)-(iii) and \ref{2.86} give us that, in line with \cite[Lemma~3.5]{Beck_2021}:
|
||||
\begin{align}
|
||||
|
|
@ -842,7 +841,7 @@ Since for all $n\in \N$, it is the case that $\mathcal{S} = \lp \supp(\mathfrak{
|
|||
\end{proof}
|
||||
|
||||
|
||||
\section{Solutions, Characterization, and Computational \\ Bounds}
|
||||
\section{Solutions, Characterization, and Computational Bounds to the Kolmogorov Backward Equations}
|
||||
% \begin{proof}
|
||||
% From Feynman-Kac, especially from \cite[(1.5)]{hutzenthaler_strong_2021} and setting $f=0$ in the notation of \cite[(1.5)]{hutzenthaler_strong_2021} we have that:
|
||||
% \begin{align}
|
||||
|
|
@ -891,7 +890,8 @@ Let $T \in (0,\infty)$. Let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ be a proba
|
|||
\begin{proof}
|
||||
This is a consequence of Lemma \ref{lem:3.4} and \ref{2.19}.
|
||||
\end{proof}
|
||||
\begin{corollary}\label{lem:3.19} Let $T \in (0,\infty)$,\\ let $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, let $u_d \in C^{1,2} \left( \left[ 0,T \right] \times \R^d, \R \right)$, $d \in \N$ satisfy for all $d \in \N$, $t \in [0,T]$, $x \in \R^d$ that:
|
||||
\newpage
|
||||
\begin{corollary}\label{lem:3.19} Let $T \in (0,\infty)$, let $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, let $u_d \in C^{1,2} \left( \left[ 0,T \right] \times \R^d, \R \right)$, $d \in \N$ satisfy for all $d \in \N$, $t \in [0,T]$, $x \in \R^d$ that:
|
||||
\begin{align}
|
||||
\left( \frac{\partial}{\partial t} u_d \right) \left(t,x\right) + \frac{1}{2}\left(\nabla^2_x u_d\right) \left(t,x\right) = 0
|
||||
\end{align}
|
||||
|
|
@ -905,7 +905,7 @@ Then for all $d\in \N$, $t \in [0,T]$, $x \in \R^d$ it holds that:
|
|||
\end{align}
|
||||
\end{corollary}
|
||||
\begin{proof}
|
||||
This is a special case of Theorem \ref{thm:3.21}. It is the case where $\sigma_d(x) = \mathbb{I}_d$, the uniform identity function where $\mathbb{I}_d$ is the identity matrix in dimension $d$ for $d \in \N$, and $\mu_d(x) = \mymathbb{0}_{d}$ where $\mymathbb{0}_d$ is the zero vector in dimension $d$ for $d \in \N$.
|
||||
This is a special case of Theorem \ref{thm:3.21}. It is the case where $\sigma_d(x) = \mathbb{I}_d$, the uniform identity function where $\mathbb{I}_d$ is the identity matrix in dimension $d$ for $d \in \N$, and $\mu_d(x) = \mymathbb{0}_{d,1}$ where $\mymathbb{0}_d$ is the zero vector in dimension $d$ for $d \in \N$.
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
|
|
|||
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Reference in New Issue