More stuff
This commit is contained in:
parent
3d09719471
commit
c0593180fc
|
|
@ -483,3 +483,23 @@ Note that $\mathfrak{C}_{d,\mathfrak{N}_{d,\epsilon},\mathfrak{N}_{d,\epsilon}}$
|
|||
% \end{align}
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -8,6 +8,7 @@ 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}
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ We will now take the modified and simplified version of Multi-level Picard intro
|
|||
\begin{lemma}[R\textemdash,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 \Gamma \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 \mathsf{G}_d \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}
|
||||
|
|
@ -79,6 +79,7 @@ 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}
|
||||
\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:
|
||||
|
|
@ -332,7 +333,7 @@ This proves Item (v) and hence the whole lemma.
|
|||
% 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{Cpy}_{n}{}_{,}{}_{1}$};
|
||||
\draw (41,250.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}_{n}{}_{,}{}_{1}$};
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
|
|
@ -791,7 +792,7 @@ 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 $\real_{\rect} \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \rp \in C \lp \R^{\mathfrak{n}\lp N+1 \rp}\times \R^{\mathfrak{n} d}, \R \rp$
|
||||
\item $\real_{\rect} \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \rp \lp \fX \rp\in C \lp \R^{\mathfrak{n}\lp N+1 \rp}\times \R^{\mathfrak{n} d}, \R \rp$
|
||||
\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\\
|
||||
|
|
@ -802,7 +803,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
\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\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}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \lp \fX \rp\right| \nonumber\\
|
||||
&\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:
|
||||
|
|
@ -828,8 +829,7 @@ 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\\
|
||||
&\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}
|
||||
\ \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\\
|
||||
|
|
@ -867,7 +867,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
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_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
|
||||
&\les \frac{\fk_2 }{\fn^{\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{corollary}
|
||||
|
||||
|
|
@ -881,13 +881,13 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
|
|||
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_p }{n^{\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}}
|
||||
&\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}
|
||||
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 have 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:
|
||||
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 \\
|
||||
|
|
|
|||
|
|
@ -813,6 +813,23 @@ year = {2021}
|
|||
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},
|
||||
}
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
Binary file not shown.
|
|
@ -631,6 +631,65 @@ 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}
|
||||
|
|
@ -1084,6 +1143,72 @@ Affine neural networks present an important class of neural networks. By virtue
|
|||
This is a consequence of a finite number of applications of Lemma \ref{lem:diamondplus}. This proves the Lemma.
|
||||
\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 $\lambda \triangleright \nu = \aff_{\lambda \mathbb{I}_{\out(\nu)},0} \bullet \nu$.
|
||||
|
|
|
|||
Binary file not shown.
Binary file not shown.
Binary file not shown.
Loading…
Reference in New Issue