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Toy-MT-Introduction
Commit f6e47ea1 by Lee
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Add softmax figure

parent f7c5d4e6
正在显示 包含 228 行增加 和 48 行删除
Section06-Neural-Machine-Translation/section06-test.tex
......@@ -160,67 +160,122 @@
% RNN translation model
\begin{scope}[local bounding box=RNNMT]
% RNN Encoder
\node[rnnnode,minimum height=0.5\base,fill=green!30!white,anchor=west] (eemb1) at (0,0) {\tiny{$e_x()$}};
\node[rnnnode,fill=blue!30!white,anchor=south] (enc1) at ([yshift=0.3\base]eemb1.north) {};
\node[rnnnode,fill=purple!30!white] (enclabel1) at (enc1) {\tiny{$\textbf{h}_{m}$}};
\node[wordnode,left=0.4\base of enc1] (init1) {$\cdots$};
\node[wordnode,left=0.4\base of eemb1] (init2) {$\cdots$};
\node[wordnode,below=0pt of eemb1] () {$\langle$eos$\rangle$};
% RNN Decoder
\node[rnnnode,minimum height=0.5\base,fill=green!30!white,anchor=south] (demb1) at ([yshift=\base]enc1.north) {\tiny{$e_y()$}};
\node[rnnnode,fill=blue!30!white,anchor=south] (dec1) at ([yshift=0.3\base]demb1.north) {{\tiny{$\textbf{s}_1$}}};
\node[rnnnode,minimum height=0.5\base,fill=red!30!white,anchor=south] (softmax1) at ([yshift=0.3\base]dec1.north) {\tiny{Softmax}};
\node[wordnode,right=0.4\base of demb1] (end1) {$\cdots$};
\node[wordnode,right=0.4\base of dec1] (end2) {$\cdots$};
\node[wordnode,right=0.4\base of softmax1] (end3) {$\cdots$};
% Decoder input words
\node[wordnode,below=0pt of demb1] (decwordin) {$\langle$sos$\rangle$};
% Decoder output words
\node[wordnode,above=0pt of softmax1] (decwordout) {Do};
% Connections
\draw[-latex'] (init1.east) to (enc1.west);
\draw[-latex'] (dec1.east) to (end2.west);
\draw[-latex'] (eemb1) to (enc1);
\draw[-latex'] (demb1) to (dec1);
\draw[-latex'] (dec1.north) to (softmax1.south);
\coordinate (bridge) at ([yshift=0.4\base]enc1.north west);
\draw[-latex'] (enc1.north) to [out=90,in=0] (bridge) to [out=180,in=180] (dec1.west);
\coordinate (eemb0) at (0,0);
\foreach \x [count=\y from 0] in {1,2,...,3}
\node[rnnnode,minimum height=0.5\base,fill=green!30!white,anchor=west] (eemb\x) at ([xshift=0.4\base]eemb\y.east) {\tiny{$e_x()$}};
\foreach \x in {1,2,...,3}
\node[rnnnode,fill=blue!30!white,anchor=south] (enc\x) at ([yshift=0.3\base]eemb\x.north) {};
\node[] (enclabel1) at (enc1) {\tiny{$\textbf{h}_{m-2}$}};
\node[] (enclabel2) at (enc2) {\tiny{$\textbf{h}_{m-1}$}};
\node[rnnnode,fill=purple!30!white] (enclabel3) at (enc3) {\tiny{$\textbf{h}_{m}$}};
\node[wordnode,left=0.4\base of enc1] (init1) {$\cdots$};
\node[wordnode,left=0.4\base of eemb1] (init2) {$\cdots$};
\node[wordnode,below=0pt of eemb1] () {走};
\node[wordnode,below=0pt of eemb2] () {吗};
\node[wordnode,below=0pt of eemb3] () {$\langle$eos$\rangle$};
% RNN Decoder
\foreach \x in {1,2,...,3}
\node[rnnnode,minimum height=0.5\base,fill=green!30!white,anchor=south] (demb\x) at ([yshift=\base]enc\x.north) {\tiny{$e_y()$}};
\foreach \x in {1,2,...,3}
\node[rnnnode,fill=blue!30!white,anchor=south] (dec\x) at ([yshift=0.3\base]demb\x.north) {{\tiny{$\textbf{s}_\x$}}};
\foreach \x in {1,2,...,3}
\node[rnnnode,minimum height=0.5\base,fill=red!30!white,anchor=south] (softmax\x) at ([yshift=0.3\base]dec\x.north) {\tiny{Softmax}};
\node[wordnode,right=0.4\base of demb3] (end1) {$\cdots$};
\node[wordnode,right=0.4\base of dec3] (end2) {$\cdots$};
\node[wordnode,right=0.4\base of softmax3] (end3) {$\cdots$};
% Decoder input words
\node[wordnode,below=0pt of demb1] (decwordin) {$\langle$sos$\rangle$};
\ExtractX{$(demb2.south)$}
\ExtractY{$(decwordin.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {Do};
\ExtractX{$(demb3.south)$}
\ExtractY{$(decwordin.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {you};
% Decoder output words
\node[wordnode,above=0pt of softmax1] (decwordout) {Do};
\ExtractX{$(softmax2.north)$}
\ExtractY{$(decwordout.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {you};
\ExtractX{$(softmax3.north)$}
\ExtractY{$(decwordout.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {know};
% Connections
\draw[-latex'] (init1.east) to (enc1.west);
\draw[-latex'] (dec3.east) to (end2.west);
\foreach \x in {1,2,...,3}
\draw[-latex'] (eemb\x) to (enc\x);
\foreach \x in {1,2,...,3}
\draw[-latex'] (demb\x) to (dec\x);
\foreach \x in {1,2,...,3}
\draw[-latex'] (dec\x.north) to (softmax\x.south);
\foreach \x [count=\y from 2] in {1,2}
{
\draw[-latex'] (enc\x.east) to (enc\y.west);
\draw[-latex'] (dec\x.east) to (dec\y.west);
}
\coordinate (bridge) at ([yshift=0.4\base]enc2.north west);
\draw[-latex'] (enc3.north) .. controls +(north:0.3\base) and +(east:\base) .. (bridge) .. controls +(west:2.7\base) and +(west:0.3\base) .. (dec1.west);
\end{scope}
\begin{scope}
\coordinate (start) at (7\base,0);
\coordinate (start) at (8.5\base,-0.3\base);
\node [anchor=center,minimum width=6.7em,minimum height=1.8em,draw,rounded corners=0.3em] (hidden) at (start) {};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!20] (cell01) at ([xshift=0.2em]hidden.west) {\footnotesize{.2}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!10] (cell02) at (cell01.east) {\footnotesize{-1}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=white] (cell03) at (cell02.east) {\footnotesize{$\cdots$}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!50] (cell04) at (cell03.east) {\footnotesize{5}};
\node [anchor=center,minimum width=5.7em,minimum height=1.3em,draw,rounded corners=0.3em] (hidden) at (start) {};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!20] (cell01) at ([xshift=0.2em]hidden.west) {\scriptsize{.2}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell02) at (cell01.east) {\scriptsize{-1}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=white] (cell03) at (cell02.east) {\scriptsize{$\cdots$}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!50] (cell04) at (cell03.east) {\scriptsize{5}};
\node [anchor=south,minimum width=12.9em,minimum height=1.8em,draw,rounded corners=0.3em] (target) at ([yshift=3em]hidden.north) {};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!10] (cell11) at ([xshift=0.2em]target.west) {\footnotesize{-2}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!10] (cell12) at (cell11.east) {\footnotesize{-1}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!70] (cell13) at (cell12.east) {\footnotesize{6}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=white] (cell14) at (cell13.east) {\footnotesize{$\cdots$}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!30] (cell15) at (cell14.east) {\footnotesize{.7}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!10] (cell16) at (cell15.east) {\footnotesize{-3}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!10] (cell17) at (cell16.east) {\footnotesize{-1}};
\node [anchor=west,minimum width=1.5em,minimum size=1.5em,fill=ugreen!20] (cell18) at (cell17.east) {\footnotesize{.2}};
\node [anchor=south,minimum width=10.9em,minimum height=1.3em,draw,rounded corners=0.3em] (target) at ([yshift=2.4em]hidden.north) {};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell11) at ([xshift=0.2em]target.west) {\scriptsize{-2}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell12) at (cell11.east) {\scriptsize{-1}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!70] (cell13) at (cell12.east) {\scriptsize{6}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=white] (cell14) at (cell13.east) {\scriptsize{$\cdots$}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!30] (cell15) at (cell14.east) {\scriptsize{.7}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell16) at (cell15.east) {\scriptsize{-3}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell17) at (cell16.east) {\scriptsize{-1}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!20] (cell18) at (cell17.east) {\scriptsize{.2}};
\node [anchor=south,minimum width=1em,minimum height=0.2em,fill=ublue!80,inner sep=0pt] (label1) at ([yshift=3em]cell11.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w1) at (label1.north) {$\langle$eos$\rangle$};
\node [anchor=south,minimum width=1em,minimum height=0.3em,fill=ublue!80,inner sep=0pt] (label2) at ([yshift=3em]cell12.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w2) at (label2.north) {$\langle$sos$\rangle$};
\node [anchor=south,minimum width=1em,minimum height=1.5em,fill=ublue!80,inner sep=0pt] (label3) at ([yshift=3em]cell13.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w3) at (label3.north) {Do};
\node [anchor=south,font=\scriptsize] (w4) at ([yshift=3em]cell14.north) {$\cdots$};
\node [anchor=south,minimum width=1em,minimum height=0.5em,fill=ublue!80,inner sep=0pt] (label5) at ([yshift=3em]cell15.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w5) at (label5.north) {I};
\node [anchor=south,minimum width=1em,minimum height=0.1em,fill=ublue!80,inner sep=0pt] (label6) at ([yshift=3em]cell16.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w6) at (label6.north) {you};
\node [anchor=south,minimum width=1em,minimum height=0.3em,fill=ublue!80,inner sep=0pt] (label7) at ([yshift=3em]cell17.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w7) at (label7.north) {?};
\node [anchor=south,minimum width=1em,minimum height=0.4em,fill=ublue!80,inner sep=0pt] (label8) at ([yshift=3em]cell18.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w8) at (label8.north) {have};
\filldraw [fill=red!20,draw=white] (target.south west) -- (target.south east) -- ([xshift=-0.2em,yshift=0.1em]hidden.north east) -- ([xshift=0.2em,yshift=0.1em]hidden.north west);
\draw [->,thick] ([xshift=0.2em,yshift=0.1em]hidden.north west) -- (target.south west);
\draw [->,thick] ([xshift=-0.2em,yshift=0.1em]hidden.north east) -- (target.south east);
\node [anchor=south,rounded corners=0.3em] (softmax) at ([yshift=0.5em]target.north) {\scriptsize{$p(\hat{s})_i=\frac{e^{\hat{s}_i}}{\sum_j e^{\hat{s}_j}}$}};
\begin{pgfonlayer}{background}
\filldraw [fill=blue!20,draw=white] ([yshift=0.1em]cell11.north west) {[rounded corners=0.3em] -- (softmax.west)} -- (label1.south west) -- (label8.south east) {[rounded corners=0.3em] -- (softmax.east)} -- ([yshift=0.1em]cell18.north east) -- ([yshift=0.1em]cell11.north west);
\end{pgfonlayer}
\node [anchor=south] () at ([yshift=0.9em]hidden.north) {$y=Wx$};
\node [anchor=south] () at ([yshift=0.6em]hidden.north) {\scriptsize{$\hat{s}=Ws$}};
\node [draw=red,thick,densely dashed,rounded corners=3pt,fit=(cell01) (cell11) (label1) (label8) (target) (hidden) (w3)] (output) {};
\end{scope}
\draw [->,thick,densely dashed,red] ([yshift=-0.2em]softmax3.east) .. controls +(east:2\base) and +(west:\base) .. (output.west);
\end{tikzpicture}
\end{center}
\end{frame}
\end{frame}
%%%------------------------------------------------------------------------------------------------------------
\section{Transformer}
......
Section06-Neural-Machine-Translation/section06.tex
......@@ -1145,6 +1145,131 @@ NLP问题的隐含结构假设 & 无隐含结构假设，端到端学习 \\
\item Softmax
\end{itemize}
%%% 图
\begin{center}
\begin{tikzpicture}
\setlength{\base}{0.9cm}
\tikzstyle{rnnnode} = [rounded corners=1pt,minimum height=0.5\base,minimum width=1\base,draw,inner sep=0pt,outer sep=0pt]
\tikzstyle{wordnode} = [font=\tiny]
% RNN translation model
\begin{scope}[local bounding box=RNNMT]
% RNN Encoder
\coordinate (eemb0) at (0,0);
\foreach \x [count=\y from 0] in {1,2,...,3}
\node[rnnnode,minimum height=0.5\base,fill=green!30!white,anchor=west] (eemb\x) at ([xshift=0.4\base]eemb\y.east) {\tiny{$e_x()$}};
\foreach \x in {1,2,...,3}
\node[rnnnode,fill=blue!30!white,anchor=south] (enc\x) at ([yshift=0.3\base]eemb\x.north) {};
\node[] (enclabel1) at (enc1) {\tiny{$\textbf{h}_{m-2}$}};
\node[] (enclabel2) at (enc2) {\tiny{$\textbf{h}_{m-1}$}};
\node[rnnnode,fill=purple!30!white] (enclabel3) at (enc3) {\tiny{$\textbf{h}_{m}$}};
\node[wordnode,left=0.4\base of enc1] (init1) {$\cdots$};
\node[wordnode,left=0.4\base of eemb1] (init2) {$\cdots$};
\node[wordnode,below=0pt of eemb1] () {走};
\node[wordnode,below=0pt of eemb2] () {吗};
\node[wordnode,below=0pt of eemb3] () {$\langle$eos$\rangle$};
% RNN Decoder
\foreach \x in {1,2,...,3}
\node[rnnnode,minimum height=0.5\base,fill=green!30!white,anchor=south] (demb\x) at ([yshift=\base]enc\x.north) {\tiny{$e_y()$}};
\foreach \x in {1,2,...,3}
\node[rnnnode,fill=blue!30!white,anchor=south] (dec\x) at ([yshift=0.3\base]demb\x.north) {{\tiny{$\textbf{s}_\x$}}};
\foreach \x in {1,2,...,3}
\node[rnnnode,minimum height=0.5\base,fill=red!30!white,anchor=south] (softmax\x) at ([yshift=0.3\base]dec\x.north) {\tiny{Softmax}};
\node[wordnode,right=0.4\base of demb3] (end1) {$\cdots$};
\node[wordnode,right=0.4\base of dec3] (end2) {$\cdots$};
\node[wordnode,right=0.4\base of softmax3] (end3) {$\cdots$};
% Decoder input words
\node[wordnode,below=0pt of demb1] (decwordin) {$\langle$sos$\rangle$};
\ExtractX{$(demb2.south)$}
\ExtractY{$(decwordin.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {Do};
\ExtractX{$(demb3.south)$}
\ExtractY{$(decwordin.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {you};
% Decoder output words
\node[wordnode,above=0pt of softmax1] (decwordout) {Do};
\ExtractX{$(softmax2.north)$}
\ExtractY{$(decwordout.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {you};
\ExtractX{$(softmax3.north)$}
\ExtractY{$(decwordout.base)$}
\node[wordnode,anchor=base] () at (\XCoord,\YCoord) {know};
% Connections
\draw[-latex'] (init1.east) to (enc1.west);
\draw[-latex'] (dec3.east) to (end2.west);
\foreach \x in {1,2,...,3}
\draw[-latex'] (eemb\x) to (enc\x);
\foreach \x in {1,2,...,3}
\draw[-latex'] (demb\x) to (dec\x);
\foreach \x in {1,2,...,3}
\draw[-latex'] (dec\x.north) to (softmax\x.south);
\foreach \x [count=\y from 2] in {1,2}
{
\draw[-latex'] (enc\x.east) to (enc\y.west);
\draw[-latex'] (dec\x.east) to (dec\y.west);
}
\coordinate (bridge) at ([yshift=0.4\base]enc2.north west);
\draw[-latex'] (enc3.north) .. controls +(north:0.3\base) and +(east:\base) .. (bridge) .. controls +(west:2.7\base) and +(west:0.3\base) .. (dec1.west);
\end{scope}
\begin{scope}
\coordinate (start) at (8.5\base,-0.3\base);
\node [anchor=center,minimum width=5.7em,minimum height=1.3em,draw,rounded corners=0.3em] (hidden) at (start) {};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!20] (cell01) at ([xshift=0.2em]hidden.west) {\scriptsize{.2}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell02) at (cell01.east) {\scriptsize{-1}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=white] (cell03) at (cell02.east) {\scriptsize{$\cdots$}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!50] (cell04) at (cell03.east) {\scriptsize{5}};
\node [anchor=south,minimum width=10.9em,minimum height=1.3em,draw,rounded corners=0.3em] (target) at ([yshift=2.4em]hidden.north) {};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell11) at ([xshift=0.2em]target.west) {\scriptsize{-2}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell12) at (cell11.east) {\scriptsize{-1}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!70] (cell13) at (cell12.east) {\scriptsize{6}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=white] (cell14) at (cell13.east) {\scriptsize{$\cdots$}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!30] (cell15) at (cell14.east) {\scriptsize{.7}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell16) at (cell15.east) {\scriptsize{-3}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!10] (cell17) at (cell16.east) {\scriptsize{-1}};
\node [anchor=west,minimum width=1em,minimum size=1em,fill=ugreen!20] (cell18) at (cell17.east) {\scriptsize{.2}};
\node [anchor=south,minimum width=1em,minimum height=0.2em,fill=ublue!80,inner sep=0pt] (label1) at ([yshift=3em]cell11.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w1) at (label1.north) {$\langle$eos$\rangle$};
\node [anchor=south,minimum width=1em,minimum height=0.3em,fill=ublue!80,inner sep=0pt] (label2) at ([yshift=3em]cell12.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w2) at (label2.north) {$\langle$sos$\rangle$};
\node [anchor=south,minimum width=1em,minimum height=1.5em,fill=ublue!80,inner sep=0pt] (label3) at ([yshift=3em]cell13.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w3) at (label3.north) {Do};
\node [anchor=south,font=\scriptsize] (w4) at ([yshift=3em]cell14.north) {$\cdots$};
\node [anchor=south,minimum width=1em,minimum height=0.5em,fill=ublue!80,inner sep=0pt] (label5) at ([yshift=3em]cell15.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w5) at (label5.north) {I};
\node [anchor=south,minimum width=1em,minimum height=0.1em,fill=ublue!80,inner sep=0pt] (label6) at ([yshift=3em]cell16.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w6) at (label6.north) {you};
\node [anchor=south,minimum width=1em,minimum height=0.3em,fill=ublue!80,inner sep=0pt] (label7) at ([yshift=3em]cell17.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w7) at (label7.north) {?};
\node [anchor=south,minimum width=1em,minimum height=0.4em,fill=ublue!80,inner sep=0pt] (label8) at ([yshift=3em]cell18.north) {};
\node [anchor=west,rotate=90,font=\scriptsize] (w8) at (label8.north) {have};
\filldraw [fill=red!20,draw=white] (target.south west) -- (target.south east) -- ([xshift=-0.2em,yshift=0.1em]hidden.north east) -- ([xshift=0.2em,yshift=0.1em]hidden.north west);
\draw [->,thick] ([xshift=0.2em,yshift=0.1em]hidden.north west) -- (target.south west);
\draw [->,thick] ([xshift=-0.2em,yshift=0.1em]hidden.north east) -- (target.south east);
\node [anchor=south,rounded corners=0.3em] (softmax) at ([yshift=0.5em]target.north) {\scriptsize{$p(\hat{s})_i=\frac{e^{\hat{s}_i}}{\sum_j e^{\hat{s}_j}}$}};
\begin{pgfonlayer}{background}
\filldraw [fill=blue!20,draw=white] ([yshift=0.1em]cell11.north west) {[rounded corners=0.3em] -- (softmax.west)} -- (label1.south west) -- (label8.south east) {[rounded corners=0.3em] -- (softmax.east)} -- ([yshift=0.1em]cell18.north east) -- ([yshift=0.1em]cell11.north west);
\end{pgfonlayer}
\node [anchor=south] () at ([yshift=0.6em]hidden.north) {\scriptsize{$\hat{s}=Ws$}};
\node [draw=red,thick,densely dashed,rounded corners=3pt,fit=(cell01) (cell11) (label1) (label8) (target) (hidden) (w3)] (output) {};
\end{scope}
\draw [->,thick,densely dashed,red] ([yshift=-0.2em]softmax3.east) .. controls +(east:2\base) and +(west:\base) .. (output.west);
\end{tikzpicture}
\end{center}
\end{frame}
%%%------------------------------------------------------------------------------------------------------------
......
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