page of MLP for function approximation

Section05-Neural-Networks-and-Language-Modeling/section05-test.tex

@@ -390,18 +390,19 @@ cycle}
 \item 我们可以重复上面的过程，构建\textbf{多层神经网络}
 \end{itemize}
 
-\vspace{-0.5em}
+\vspace{-1.0em}
 \begin{center}
 \begin{tikzpicture}
 \begin{scope}[]
 
 \def\neuronsep{1.6}
 
-\tikzstyle{neuronnode} = [minimum size=1.5em,circle,draw,ublue,very thick,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}]
+\tikzstyle{neuronnode} = [minimum size=1.7em,circle,draw,ublue,very thick,inner sep=1pt, fill=white,align=center,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}]
 
 %%% layer 1
 \foreach \n in {1,...,5}{
-    \node [neuronnode] (neuron0\n) at (\n * \neuronsep,0) {};
+    \node [neuronnode] (neuron0\n) at (\n * \neuronsep,0) {\tiny{$f_1$}\\[-1ex] \tiny{$\sum$}};
+    \draw [-,ublue] (neuron0\n.east) -- (neuron0\n.west);
 }
 
 \foreach \n in {1,...,5}{
@@ -421,19 +422,20 @@ cycle}
 \node [rectangle,inner sep=0.2em,fill=red!20] [fit = (neuron01) (neuron05)] (layer01) {};
 \end{pgfonlayer}
 
-\node [anchor=west] (layer00label) at ([xshift=1.2em]x5.east) {输入层};
+\node [anchor=west] (layer00label) at ([xshift=1.25em]x5.east) {\alert{输入层}};
 
 \visible<2->{
-\node [anchor=west] (layer01label) at ([xshift=1em]layer01.east) {第一层};
+\node [anchor=west] (layer01label) at ([xshift=1em]layer01.east) {第二层};
 }
 \visible<4->{
-\node [anchor=west] (layer01label2) at (layer01label.east) {(\alert{输入层})};
+\node [anchor=west] (layer01label2) at (layer01label.east) {(\alert{隐层})};
 }
 
 %%% layer 2
 \visible<2->{
 \foreach \n in {2,...,4}{
-    \node [neuronnode] (neuron1\n) at (\n * \neuronsep,4em) {};
+    \node [neuronnode] (neuron1\n) at (\n * \neuronsep,4em) {\tiny{$f_2$}\\[-1ex] \tiny{$\sum$}};
+    \draw [-,ublue] (neuron1\n.east) -- (neuron1\n.west);
 }
 
 \foreach \n in {2,...,4}{
@@ -454,7 +456,7 @@ cycle}
 }
 \end{pgfonlayer}
 
-\node [anchor=west] (layer02label) at ([xshift=4.9em]layer02.east) {第二层};
+\node [anchor=west] (layer02label) at ([xshift=4.9em]layer02.east) {第三层};
 \visible<4->{
 \node [anchor=west] (layer02label2) at (layer02label.east) {(\alert{隐层})};
 }
@@ -463,7 +465,8 @@ cycle}
 %%% layer 3
 \visible<3->{
 \foreach \n in {1,...,5}{
-    \node [neuronnode] (neuron2\n) at (\n * \neuronsep,8em) {};
+    \node [neuronnode] (neuron2\n) at (\n * \neuronsep,8em) {\tiny{$f_3$}\\[-1ex] \tiny{$\sum$}};
+    \draw [-,ublue] (neuron2\n.east) -- (neuron2\n.west);
 }
 
 \foreach \n in {1,...,5}{
@@ -483,7 +486,7 @@ cycle}
 }
 \end{pgfonlayer}
 
-\node [anchor=west] (layer03label) at ([xshift=1em]layer03.east) {第三层};
+\node [anchor=west] (layer03label) at ([xshift=1em]layer03.east) {第四层};
 \visible<4->{
 \node [anchor=west] (layer03label2) at (layer03label.east) {(\alert{输出层})};
 }
@@ -497,11 +500,77 @@ cycle}
 
 %%%------------------------------------------------------------------------------------------------------------
 %%% 两层神经网络可以逼近任何函数
-\begin{frame}{多层神经网络的函数逼近能力}
+\begin{frame}{多层神经网络可以逼近任意函数}
 \begin{itemize}
-\item \textbf{单层神经网络}：线性变换 + 激活函数（非线性）
-\item 我们可以重复上面的过程，构建\textbf{多层神经网络}
+\item 以一个简单的两层网络为例（隐层激活函数：sigmoid）
 \end{itemize}
+
+\begin{center}
+\begin{tikzpicture}
+
+%% a two-layer neural network
+\begin{scope}
+\tikzstyle{neuronnode} = [minimum size=1.7em,circle,draw,ublue,very thick,inner sep=1pt, fill=white,align=center,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}]
+
+%% input and hidden layers
+\node [neuronnode] (n10) at (0,0) {\tiny{$f$}\\[-1ex] \tiny{$\sum$}};
+\node [neuronnode] (n11) at (1.5,0) {\tiny{$f$}\\[-1ex] \tiny{$\sum$}};
+\draw [-,ublue] (n10.west) -- (n10.east);
+\draw [-,ublue] (n11.west) -- (n11.east);
+\node [anchor=north] (x1) at ([yshift=-6em]n11.south) {$x_1$};
+\node [anchor=north] (b) at ([yshift=-6em]n10.south) {$b$};
+\draw [->,thick,red] (b.north) -- ([yshift=-0.1em]n10.south);
+\draw [->,thick,blue] (b.north) -- ([yshift=-0.1em]n11.250);
+\draw [->,thick,ugreen] (x1.north) -- ([yshift=-0.1em]n10.290);
+\draw [->,thick,purple] (x1.north) -- ([yshift=-0.1em]n11.south);
+
+%% output layers
+\node [neuronnode] (n20) at (0.75,5em) {\scriptsize{$\sum$}};
+\draw [->,thick] ([yshift=0.1em]n10.north) -- ([yshift=-0.1em]n20.250);
+\draw [->,thick] ([yshift=0.1em]n11.north) -- ([yshift=-0.1em]n20.290);
+
+\node [] (y) at ([yshift=3em]n20.north) {$y$};
+\draw [->,thick] ([yshift=0.1em]n20.north) -- (y.south);
+
+%% weight and bias
+\node [anchor=center,rotate=90] (b0) at ([yshift=3em,xshift=-0.5em]b.north) {\tiny{$b=0$}};
+\node [anchor=center,rotate=59,fill=white,inner sep=1pt] (b1) at ([yshift=4.9em,xshift=2.2em]b.north) {\tiny{$b=0$}};
+\node [anchor=center,rotate=90] (w1) at ([yshift=3em,xshift=0.5em]x1.north) {\tiny{$w=1$}};
+\node [anchor=center,rotate=-59,fill=white,inner sep=1pt] (w2) at ([yshift=1.2em,xshift=-1.2em]x1.north) {\tiny{$w=0$}};
+
+\node [anchor=center,rotate=62,fill=white,inner sep=1pt] (w21) at ([yshift=2em,xshift=0.5em]n10.north) {\tiny{$w=0$}};
+\node [anchor=center,rotate=-62,fill=white,inner sep=1pt] (w22) at ([yshift=2em,xshift=-0.5em]n11.north) {\tiny{$w=0$}};
+
+\begin{scope}
+\node [anchor=west] (flabel) at ([xshift=1.2in]y.east) {\footnotesize{sigmoid:}};
+\node [anchor=west,inner sep=2pt] (flabel2) at (flabel.east) {\footnotesize{$f(x)=1/(1+e^{-x})$}};
+\node [anchor=north west,inner sep=2pt] (flabel3) at (flabel2.south west) {\footnotesize{$s=x_1 \cdot w + b$}};
+\draw [->,thick,dotted] ([yshift=-0.3em,xshift=-0.1em]n11.60)  .. controls +(east:1) and +(west:2) ..  ([xshift=-0.2em]flabel.west) ;
+
+\begin{pgfonlayer}{background}
+\node [rectangle,inner sep=0.2em,fill=blue!20,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] [fit = (flabel) (flabel2) (flabel3)] (funcbox) {};
+\end{pgfonlayer}
+\end{scope}
+
+\begin{scope}[xshift=2.8in,yshift=0.2in]
+\draw [->,thick] (-2,0) -- (2,0);
+\draw [->,thick] (0,0) -- (0,2);
+\draw [-] (-0.05,1) -- (0.05,1);
+\node [anchor=east,inner sep=1pt] (label1) at (0,1) {\tiny{1}};
+\node [anchor=south east,inner sep=1pt] (label2) at (0,0) {\tiny{0}};
+\draw [-,very thick,ublue,domain=-2:2,samples=100] plot (\x,{1/(1+exp(-2*\x))});
+\draw [-,very thick,ublue,domain=-2:2,samples=100] plot (\x,{1/(1+exp(-4*\x))});
+\draw [-,very thick,ublue,rounded corners=0.2em] (-2,0) -- (0,0) -- (0,1) -- (2,1);
+\draw [-,very thick,ublue,rounded corners=0.2em] (-2,0) -- (0.25,0) -- (0.25,1) -- (2,1);
+\draw [-,very thick,ublue,rounded corners=0.2em] (-2,0) -- (0.5,0) -- (0.5,1) -- (2,1);
+
+\node [anchor=north west,align=left] (wblabel) at (-2,2) {\scriptsize{$w_1=1$}\\[-0ex] \scriptsize{\ $b_1=0$}};
+\end{scope}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
 \end{frame}
 
 %%%------------------------------------------------------------------------------------------------------------