\relax \providecommand\hyper@newdestlabel[2]{} \citation{grohsetal} \@writefile{toc}{\contentsline {chapter}{\numberline {5}Introduction and Basic Notions About Neural Networks}{67}{chapter.5}\protected@file@percent } \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} \@writefile{toc}{\contentsline {section}{\numberline {5.1}The Basic Definition of ANNs and realizations of ANNs}{67}{section.5.1}\protected@file@percent } \newlabel{5.1.2}{{5.1.2}{67}{Artificial Neural Networks}{theorem.5.1.2}{}} \newlabel{5.1.2@cref}{{[definition][2][5,1]5.1.2}{[1][67][]67}} \newlabel{paramdef}{{5.1.3}{68}{Artificial Neural Networks}{equation.5.1.3}{}} \newlabel{paramdef@cref}{{[equation][3][5,1]5.1.3}{[1][68][]68}} \citation{petersen_optimal_2018} \citation{grohs2019spacetime} \newlabel{widthdef}{{5.1.9}{69}{Artificial Neural Networks}{equation.5.1.9}{}} \newlabel{widthdef@cref}{{[equation][9][5,1]5.1.9}{[1][68][]69}} \newlabel{5.1.11}{{5.1.10}{69}{Realizations of Artificial Neural Networks with Activation Functions}{equation.5.1.10}{}} \newlabel{5.1.11@cref}{{[equation][10][5,1]5.1.10}{[1][69][]69}} \newlabel{5.1.8}{{5.1.6}{70}{}{theorem.5.1.6}{}} \newlabel{5.1.8@cref}{{[lemma][6][5,1]5.1.6}{[1][70][]70}} \@writefile{toc}{\contentsline {section}{\numberline {5.2}Compositions of ANNs}{70}{section.5.2}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {5.2.1}Composition}{71}{subsection.5.2.1}\protected@file@percent } \newlabel{5.2.1}{{5.2.1}{71}{Compositions of ANNs}{theorem.5.2.1}{}} \newlabel{5.2.1@cref}{{[definition][1][5,2]5.2.1}{[1][70][]71}} \newlabel{5.2.1}{{5.2.1}{71}{Compositions of ANNs}{equation.5.2.1}{}} \newlabel{5.2.1@cref}{{[equation][1][5,2]5.2.1}{[1][71][]71}} \newlabel{depthofcomposition}{{5.2.3}{71}{}{theorem.5.2.3}{}} \newlabel{depthofcomposition@cref}{{[lemma][3][5,2]5.2.3}{[1][71][]71}} \newlabel{5.2.3}{{5.2.4}{71}{}{theorem.5.2.4}{}} \newlabel{5.2.3@cref}{{[lemma][4][5,2]5.2.4}{[1][71][]71}} \citation{grohs2019spacetime} \citation{grohs2019spacetime} \newlabel{comp_prop}{{5.2.5}{72}{}{theorem.5.2.5}{}} \newlabel{comp_prop@cref}{{[lemma][5][5,2]5.2.5}{[1][72][]72}} \newlabel{comp_x}{{5.2.6}{73}{Composition}{equation.5.2.6}{}} \newlabel{comp_x@cref}{{[equation][6][5,2]5.2.6}{[1][73][]73}} \newlabel{comp_y}{{5.2.7}{73}{Composition}{equation.5.2.7}{}} \newlabel{comp_y@cref}{{[equation][7][5,2]5.2.7}{[1][73][]73}} \newlabel{comp_z}{{5.2.8}{73}{Composition}{equation.5.2.8}{}} \newlabel{comp_z@cref}{{[equation][8][5,2]5.2.8}{[1][73][]73}} \newlabel{comp_cont}{{5.2.10}{74}{Composition}{equation.5.2.10}{}} \newlabel{comp_cont@cref}{{[equation][10][5,2]5.2.10}{[1][74][]74}} \newlabel{(5.2.12)}{{5.2.13}{75}{Composition}{equation.5.2.13}{}} \newlabel{(5.2.12)@cref}{{[equation][13][5,2]5.2.13}{[1][74][]75}} \newlabel{(5.2.13)}{{5.2.14}{75}{Composition}{equation.5.2.14}{}} \newlabel{(5.2.13)@cref}{{[equation][14][5,2]5.2.14}{[1][75][]75}} \@writefile{toc}{\contentsline {section}{\numberline {5.3}Parallelization of ANNs of Equal Depth}{76}{section.5.3}\protected@file@percent } \newlabel{5.2.5}{{5.3.1}{76}{Parallelization of ANNs of same depth}{theorem.5.3.1}{}} \newlabel{5.2.5@cref}{{[definition][1][5,3]5.3.1}{[1][76][]76}} \newlabel{5.4.2}{{5.3.2}{77}{Parallelization of ANNs of same depth}{equation.5.3.2}{}} \newlabel{5.4.2@cref}{{[equation][2][5,3]5.3.2}{[1][76][]77}} \newlabel{5.3.2}{{5.3.3}{77}{}{theorem.5.3.3}{}} \newlabel{5.3.2@cref}{{[remark][3][5,3]5.3.3}{[1][77][]77}} \newlabel{(5.3.3)}{{5.3.3}{77}{}{equation.5.3.3}{}} \newlabel{(5.3.3)@cref}{{[equation][3][5,3]5.3.3}{[1][77][]77}} \newlabel{paramofparallel}{{5.3.4}{77}{}{theorem.5.3.4}{}} \newlabel{paramofparallel@cref}{{[lemma][4][5,3]5.3.4}{[1][77][]77}} \newlabel{cor:sameparal}{{5.3.4.1}{78}{}{corollary.5.3.4.1}{}} \newlabel{cor:sameparal@cref}{{[corollary][1][]5.3.4.1}{[1][78][]78}} \newlabel{5.4.3}{{5.3.5}{79}{}{theorem.5.3.5}{}} \newlabel{5.4.3@cref}{{[lemma][5][5,3]5.3.5}{[1][79][]79}} \newlabel{5.3.4}{{5.3.6}{79}{}{theorem.5.3.6}{}} \newlabel{5.3.4@cref}{{[lemma][6][5,3]5.3.6}{[1][79][]79}} \newlabel{5.3.5}{{5.3.10}{79}{Parallelization of ANNs of Equal Depth}{equation.5.3.10}{}} \newlabel{5.3.5@cref}{{[equation][10][5,3]5.3.10}{[1][79][]79}} \@writefile{toc}{\contentsline {section}{\numberline {5.4}Parallelization of ANNs of Unequal Depth}{80}{section.5.4}\protected@file@percent } \newlabel{7.2.1}{{5.4.1}{80}{Identity Neural Network}{theorem.5.4.1}{}} \newlabel{7.2.1@cref}{{[definition][1][5,4]5.4.1}{[1][80][]80}} \newlabel{7.2.2}{{5.4.2}{80}{Identity Neural Network}{equation.5.4.2}{}} \newlabel{7.2.2@cref}{{[equation][2][5,4]5.4.2}{[1][80][]80}} \@writefile{toc}{\contentsline {section}{\numberline {5.5}Affine Linear Transformations as ANNs and Their Properties.}{82}{section.5.5}\protected@file@percent } \newlabel{5.3.1}{{5.5.1}{82}{}{theorem.5.5.1}{}} \newlabel{5.3.1@cref}{{[definition][1][5,5]5.5.1}{[1][81][]82}} \newlabel{5.3.2}{{5.5.2}{82}{}{theorem.5.5.2}{}} \newlabel{5.3.2@cref}{{[lemma][2][5,5]5.5.2}{[1][82][]82}} \newlabel{remark:5.4.3}{{5.5.3}{82}{}{theorem.5.5.3}{}} \newlabel{remark:5.4.3@cref}{{[remark][3][5,5]5.5.3}{[1][82][]82}} \newlabel{5.3.3}{{5.5.5}{82}{}{theorem.5.5.5}{}} \newlabel{5.3.3@cref}{{[lemma][5][5,5]5.5.5}{[1][82][]82}} \newlabel{affcor}{{5.5.5.1}{83}{}{corollary.5.5.5.1}{}} \newlabel{affcor@cref}{{[corollary][1][]5.5.5.1}{[1][83][]83}} \@writefile{toc}{\contentsline {section}{\numberline {5.6}Sums of ANNs of Same End-widths}{84}{section.5.6}\protected@file@percent } \newlabel{dep_cpy}{{5.6.2}{84}{}{theorem.5.6.2}{}} \newlabel{dep_cpy@cref}{{[lemma][2][5,6]5.6.2}{[1][84][]84}} \newlabel{lem:5.5.4}{{5.6.6}{85}{}{theorem.5.6.6}{}} \newlabel{lem:5.5.4@cref}{{[lemma][6][5,6]5.6.6}{[1][85][]85}} \newlabel{5.4.3}{{5.6.3}{85}{Sum of ANNs of the same depth and same end widths}{equation.5.6.3}{}} \newlabel{5.4.3@cref}{{[equation][3][5,6]5.6.3}{[1][85][]85}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.6.1}Neural Network Sum Properties}{85}{subsection.5.6.1}\protected@file@percent } \newlabel{paramsum}{{5.6.9}{85}{}{theorem.5.6.9}{}} \newlabel{paramsum@cref}{{[lemma][9][5,6]5.6.9}{[1][85][]85}} \newlabel{(5.5.6)}{{5.6.6}{86}{Neural Network Sum Properties}{equation.5.6.6}{}} \newlabel{(5.5.6)@cref}{{[equation][6][5,6]5.6.6}{[1][86][]86}} \newlabel{corsum}{{5.6.9.1}{86}{}{corollary.5.6.9.1}{}} \newlabel{corsum@cref}{{[corollary][1][]5.6.9.1}{[1][86][]86}} \newlabel{depth_prop}{{5.6.10}{87}{}{theorem.5.6.10}{}} \newlabel{depth_prop@cref}{{[lemma][10][5,6]5.6.10}{[1][87][]87}} \newlabel{5.4.6}{{5.6.11}{87}{}{theorem.5.6.11}{}} \newlabel{5.4.6@cref}{{[lemma][11][5,6]5.6.11}{[1][87][]87}} \newlabel{5.4.10}{{5.6.15}{88}{Neural Network Sum Properties}{equation.5.6.15}{}} \newlabel{5.4.10@cref}{{[equation][15][5,6]5.6.15}{[1][88][]88}} \citation{Grohs_2022} \newlabel{5.4.11}{{5.6.16}{89}{Neural Network Sum Properties}{equation.5.6.16}{}} \newlabel{5.4.11@cref}{{[equation][16][5,6]5.6.16}{[1][88][]89}} \newlabel{5.4.12}{{5.6.17}{89}{Neural Network Sum Properties}{equation.5.6.17}{}} \newlabel{5.4.12@cref}{{[equation][17][5,6]5.6.17}{[1][89][]89}} \newlabel{5.4.13}{{5.6.18}{89}{Neural Network Sum Properties}{equation.5.6.18}{}} \newlabel{5.4.13@cref}{{[equation][18][5,6]5.6.18}{[1][89][]89}} \newlabel{5.4.7}{{5.6.12}{89}{}{theorem.5.6.12}{}} \newlabel{5.4.7@cref}{{[lemma][12][5,6]5.6.12}{[1][89][]89}} \newlabel{5.4.17}{{5.6.22}{90}{Neural Network Sum Properties}{equation.5.6.22}{}} \newlabel{5.4.17@cref}{{[equation][22][5,6]5.6.22}{[1][90][]90}} \newlabel{5.4.8}{{5.6.13}{90}{}{theorem.5.6.13}{}} \newlabel{5.4.8@cref}{{[lemma][13][5,6]5.6.13}{[1][90][]90}} \newlabel{5.4.18}{{5.6.23}{91}{Neural Network Sum Properties}{equation.5.6.23}{}} \newlabel{5.4.18@cref}{{[equation][23][5,6]5.6.23}{[1][90][]91}} \newlabel{5.4.19}{{5.6.24}{91}{Neural Network Sum Properties}{equation.5.6.24}{}} \newlabel{5.4.19@cref}{{[equation][24][5,6]5.6.24}{[1][91][]91}} \newlabel{5.5.11}{{5.6.16}{91}{}{theorem.5.6.16}{}} \newlabel{5.5.11@cref}{{[lemma][16][5,6]5.6.16}{[1][91][]91}} \newlabel{5.5.20}{{5.6.25}{92}{Neural Network Sum Properties}{equation.5.6.25}{}} \newlabel{5.5.20@cref}{{[equation][25][5,6]5.6.25}{[1][91][]92}} \newlabel{5.4.12}{{5.6.27}{92}{Neural Network Sum Properties}{equation.5.6.27}{}} \newlabel{5.4.12@cref}{{[equation][27][5,6]5.6.27}{[1][92][]92}} \newlabel{nn_sum_cont}{{5.6.17}{92}{}{theorem.5.6.17}{}} \newlabel{nn_sum_cont@cref}{{[lemma][17][5,6]5.6.17}{[1][92][]92}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.6.2}Sum of ANNs of Unequal Depth But Same End-widths}{92}{subsection.5.6.2}\protected@file@percent } \newlabel{lem:diamondplus}{{5.6.19}{93}{}{theorem.5.6.19}{}} \newlabel{lem:diamondplus@cref}{{[lemma][19][5,6]5.6.19}{[1][92][]93}} \@writefile{toc}{\contentsline {section}{\numberline {5.7}Linear Combinations of ANNs and Their Properties}{93}{section.5.7}\protected@file@percent } \newlabel{slm}{{5.7.1}{93}{Scalar left-multiplication with an ANN}{theorem.5.7.1}{}} \newlabel{slm@cref}{{[definition][1][5,7]5.7.1}{[1][93][]93}} \newlabel{5.6.3}{{5.7.4}{94}{}{theorem.5.7.4}{}} \newlabel{5.6.3@cref}{{[lemma][4][5,7]5.7.4}{[1][93][]94}} \newlabel{5.6.4}{{5.7.5}{94}{}{theorem.5.7.5}{}} \newlabel{5.6.4@cref}{{[lemma][5][5,7]5.7.5}{[1][94][]94}} \newlabel{5.6.5}{{5.7.8}{96}{}{theorem.5.7.8}{}} \newlabel{5.6.5@cref}{{[lemma][8][5,7]5.7.8}{[1][96][]96}} \citation{Grohs_2022} \citation{Grohs_2022} \newlabel{5.3.15}{{5.7.20}{97}{Linear Combinations of ANNs and Their Properties}{equation.5.7.20}{}} \newlabel{5.3.15@cref}{{[equation][20][5,7]5.7.20}{[1][97][]97}} \newlabel{5.6.13}{{5.7.21}{97}{Linear Combinations of ANNs and Their Properties}{equation.5.7.21}{}} \newlabel{5.6.13@cref}{{[equation][21][5,7]5.7.21}{[1][97][]97}} \citation{Grohs_2022} \citation{Grohs_2022} \newlabel{5.6.6}{{5.7.9}{98}{}{theorem.5.7.9}{}} \newlabel{5.6.6@cref}{{[lemma][9][5,7]5.7.9}{[1][98][]98}} \citation{Grohs_2022} \newlabel{5.6.22}{{5.7.30}{99}{Linear Combinations of ANNs and Their Properties}{equation.5.7.30}{}} \newlabel{5.6.22@cref}{{[equation][30][5,7]5.7.30}{[1][99][]99}} \newlabel{5.6.23}{{5.7.31}{99}{Linear Combinations of ANNs and Their Properties}{equation.5.7.31}{}} \newlabel{5.6.23@cref}{{[equation][31][5,7]5.7.31}{[1][99][]99}} \citation{Grohs_2022} \newlabel{5.6.9}{{5.7.10}{100}{}{theorem.5.7.10}{}} \newlabel{5.6.9@cref}{{[lemma][10][5,7]5.7.10}{[1][100][]100}} \citation{grohs2019spacetime} \citation{Grohs_2022} \citation{grohs2019spacetime} \citation{Grohs_2022} \citation{vaswani_attention_2017} \citation{arik_tabnet_2021} \citation{8099678} \@writefile{toc}{\contentsline {section}{\numberline {5.8}Neural Network Diagrams}{103}{section.5.8}\protected@file@percent } \@setckpt{neural_network_introduction}{ \setcounter{page}{106} \setcounter{equation}{0} \setcounter{enumi}{3} \setcounter{enumii}{0} \setcounter{enumiii}{0} \setcounter{enumiv}{0} \setcounter{footnote}{0} \setcounter{mpfootnote}{0} \setcounter{part}{2} \setcounter{chapter}{5} \setcounter{section}{8} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} \setcounter{figure}{0} \setcounter{table}{0} \setcounter{@pps}{0} \setcounter{@ppsavesec}{0} \setcounter{@ppsaveapp}{0} \setcounter{AM@survey}{0} \setcounter{parentequation}{0} \setcounter{section@level}{1} \setcounter{Item}{111} \setcounter{Hfootnote}{0} \setcounter{Hy@AnnotLevel}{0} \setcounter{bookmark@seq@number}{33} \setcounter{NAT@ctr}{0} \setcounter{ALG@line}{0} \setcounter{ALG@rem}{0} \setcounter{ALG@nested}{0} \setcounter{ALG@Lnr}{1} \setcounter{ALG@blocknr}{1} \setcounter{ALG@storecount}{0} \setcounter{ALG@tmpcounter}{0} \setcounter{@stackindex}{0} \setcounter{ROWcellindex@}{0} \setcounter{lstnumber}{1} \setcounter{theorem}{0} \setcounter{corollary}{0} \setcounter{lstlisting}{0} \setcounter{nn@layerindex@counter}{4} \setcounter{nn@lastlayerstart@counter}{0} \setcounter{nn@thislayerstart@counter}{1} \setcounter{nn@lastlayercount@counter}{3} \setcounter{nn@thislayercount@counter}{2} \setcounter{nn@lastlayerindex@counter}{2} \setcounter{nn@thislayerindex@counter}{3} }