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

@@ -111,6 +111,8 @@
 \item 对于向量$\textbf{x} \in \mathbb{R}^m$，一层神经网络首先把他经过\textbf{\alert{线性变换}}映射到$\mathbb{R}^m$，之后经过\textbf{{\color{blue}激活函数}}变换成$\textbf{y} \in \mathbb{R}^n$
 \end{itemize}
 
+\vspace{1em}
+
 \begin{center}
 \begin{tikzpicture}
 
@@ -136,10 +138,19 @@
 \end{pgfonlayer}
 
 \end{tikzpicture}
-\end{center}
 
+\begin{figure}[htp!]
+
+\includegraphics[scale=0.24]{./Figures/wf.png}
+% \begin{tikzpicture}
+%     \node [rectangle,inner sep=0.2em,fill=red!20] [fit = (x) (w) (b)] (linear) {};
+%     \node [anchor=north] (linearlabel) at ([yshift=-1.1em]linear.south) {\alert{线性变换}}
+\end{figure}
+\tikz {\node () at (0,0) {}; \node () at (0,10) {};}
+\end{center}
 \end{frame}
 
+
 %%%------------------------------------------------------------------------------------------------------------
 %%% 线性变换
 \begin{frame}{线性变换}
@@ -152,7 +163,7 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
 \item<2-> 线性变换的一种几何解释：
 \end{itemize}
 
-\vspace{-1em}
+\vspace{-0.5em}
 \visible<2->{
 \begin{center}
 \begin{tikzpicture}
@@ -163,32 +174,133 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
 \node [anchor=west] (plus) at (w.east) {\Large{$+$}};
 \node [anchor=west] (b) at (plus.east) {\Large{$\textbf{b}$}};
 
+\tikzstyle{neuron} = [rectangle,draw,thick,fill=red!30,red!35,minimum height=2em,minimum width=2em,font=\small]
+\node[neuron,anchor=north] (a1) at ([xshift=-6em,yshift=-4em]x.south) {};
+\draw[->,thick] ([xshift=-2em,yshift=0em]a1.south) to ([xshift=3em,yshift=0em]a1.south);
+\draw[->,thick] ([xshift=0em,yshift=-4em]a1.west) to ([xshift=0em,yshift=2em]a1.west);
+\node[below] at ([xshift=0.5em,yshift=-1em]a1.west){0};
+\node[below] at ([xshift=2em,yshift=-1em]a1.west){1};
+\node[below] at ([xshift=-0.5em,yshift=2em]a1.west){1};
+\node [anchor=west] (x) at ([xshift=-0.7em,yshift=1em]a1.south) {\Large{$\textbf{F}$}};
+
 \visible<3->{
 \node [anchor=center,fill=green!20] (w2) at (w) {\Large{$\textbf{w}$}};
 \node [anchor=north,inner sep=1pt] (wlabel) at ([yshift=-0.7em]w.south) {\small{旋转(rotation)}};
 \draw [<-] ([yshift=-0.2em]w2.south) -- (wlabel.north);
+
+\tikzstyle{neuron} = [rectangle,draw,thick,fill=red!30,red!35,minimum height=2em,minimum width=2em,font=\small]
+\node[neuron,anchor=north] (a2) at ([xshift=10em,yshift=0em]a1.south) {};
+\draw[->,thick] ([xshift=-2em,yshift=0em]a2.north) to ([xshift=3em,yshift=0em]a2.north);
+\draw[->,thick] ([xshift=0em,yshift=-2em]a2.west) to ([xshift=0em,yshift=4em]a2.west);
+\node[above] at ([xshift=0.5em,yshift=1em]a2.west){0};
+\node[above] at ([xshift=2em,yshift=1em]a2.west){1};
+\node[below] at ([xshift=-0.5em,yshift=0em]a2.west){-1};
+\node [anchor=west] (x) at ([xshift=-3.5cm,yshift=2em]a2.north) {\scriptsize{
+    $w=\begin{bmatrix}
+    1&0&0\\
+    0&-1&0\\
+    0&0&1
+    \end{bmatrix}$}
+    };
+
+\node [anchor=west,rotate = 180] (x) at ([xshift=0.7em,yshift=1em]a2.south) {\Large{$\textbf{F}$}};
+
+
+\draw[-stealth, line width=2pt,dashed] ([xshift=4em,yshift=0em]a1.south) to ([xshift=-3em,yshift=0em]a2.north);
 }
 
 \visible<4->{
 \node [anchor=center,fill=purple!20] (b2) at (b) {\Large{$\textbf{b}$}};
 \node [anchor=west] (blabel) at ([xshift=1.5em]b2.east) {平移(shift)};
 \draw [<-] ([xshift=0.2em]b2.east) -- (blabel.west);
+
+\tikzstyle{neuron} = [rectangle,draw,thick,fill=red!30,red!35,minimum height=2em,minimum width=2em,font=\small]
+\node[neuron,anchor=north] (a3) at ([xshift=11em,yshift=2.05em]a2.south) {};
+\draw[->,thick] ([xshift=-3em,yshift=0em]a3.north) to ([xshift=2em,yshift=0em]a3.north);
+\draw[->,thick] ([xshift=-1em,yshift=-2em]a3.west) to ([xshift=-1em,yshift=4em]a3.west);
+\node[above] at ([xshift=-0.5em,yshift=1em]a3.west){0};
+\node[above] at ([xshift=1em,yshift=1em]a3.west){1};
+\node[left] at ([xshift=-0.75em,yshift=-0.5em]a3.west){-1};
+\node [anchor=west,rotate = 180] (x) at ([xshift=0.7em,yshift=1em]a3.south) {\Large{$\textbf{F}$}};
+
+
+\node [anchor=west] (x) at ([xshift=-4cm,yshift=2em]a3.north) {\scriptsize{
+    $b=\begin{bmatrix}
+    0.5&0&0\\
+    0&0&0\\
+    0&0&0
+    \end{bmatrix}$}
+    };
+\draw[-stealth, line width=2pt,dashed] ([xshift=3em,yshift=1em]a2.east) to ([xshift=-3em,yshift=1em]a3.west);
 }
 
 \end{tikzpicture}
 \end{center}
 }
 
+
 \end{frame}
 
+
 %%%------------------------------------------------------------------------------------------------------------
 %%% 线性变换：更复杂的实例
-\begin{frame}{线性变换（续）}
+\begin{frame}[fragile]{线性变换（续）}
 \begin{itemize}
 \item 线性变换也适用于更加复杂的情况，这也给神经网络提供了拟合不同数据分布的能力
+    \begin{itemize}
+    \item 比如，我们可以把三维图形投影到二维平面上
+    \item 再比如，我们也可以把二维平面上的图形映射到三维平面上
+    \end{itemize}
 \end{itemize}
+
+\vspace{1em}
+
+\newcommand{\plane}[1]{
+(-1.95, #1, 1.35) --
+++(3.6, 0.6, 0.0) --
+++(0.3, -1.8, -2.7) --
+++(-3.6, -0.6, -0.0) --
+cycle}
+\newcommand{\nullspacepicture}{
+% bottom part of the row space line
+\draw (0,0,0) -- (0.3,-1.8,1.233);
+% five planes
+\draw[fill=gray!20]\plane{-0.2};
+\draw[fill=gray!20]\plane{0.2};
+\draw[fill=blue!70!gray]\plane{0.6};
+\draw[fill=gray!20]\plane{1};
+\draw[fill=gray!20]\plane{1.4};
+% top part of the row space line
+\draw (-.094,.562,-.385) -- (-0.3,1.8,-1.233);
+}
+\newcommand{\rangepicture}[1]{
+% axes
+\draw[help lines,->] (-2,0) -- (2,0);
+\draw[help lines,->] (0,-2) -- (0,2);
+% the line and circles
+\draw (1,-2) -- (-1,2);
+\draw[fill=#1] (0,0) circle (2.5pt);
+\draw[fill=gray!50] (0.2,-0.4) circle (2.5pt);
+\draw[fill=gray!50] (0.4,-0.8) circle (2.5pt);
+\draw[fill=gray!50] (-0.2,0.4) circle (2.5pt);
+\draw[fill=gray!50] (-0.4,0.8) circle (2.5pt);
+}
+
+\begin{tikzpicture}[scale=0.95]
+\centering
+\nullspacepicture
+% the label
+\node at (-2,1.8) {$\mathbb{R}^3$};
+% arrow between diagrams
+\path[->] (3,0) edge[bend left] node[above] {线性变换} (4.5,0);
+\begin{scope}[xshift=7cm]
+\rangepicture{blue!70!gray}
+\node at (1.8,1.8) {$\mathbb{R}^2$};
+\end{scope}
+\end{tikzpicture}
 \end{frame}
 
+
 %%%------------------------------------------------------------------------------------------------------------
 %%% 激活函数
 \begin{frame}{激活函数}
@@ -237,10 +349,10 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
 \begin{scope}[xshift=9.5em]
 \draw [->,thick] (-1.5,0) -- (1.5,0);
 \draw [->,thick] (0,-0.1) -- (0,1.5);
-\draw [-,very thick,ublue,domain=-1.2:1.2,samples=100] plot (\x, {0.5*2^\x});
+\draw [-,very thick,ublue,domain=-1.2:1.2,samples=100] plot (\x, {0.5 * exp(\x)});
 \node [anchor=west] (ylabel) at (0,1.3) {$y$};
 \node [anchor=north] (xlabel) at (1.3,0) {$x$};
-\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = 0.5 \cdot  2^x$}};
+\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = 0.5 \cdot  \exp(x)$}};
 \node [anchor=south west] (flabel) at ([xshift=-1.8em]func.north west) {\footnotesize{Exponential:}};
 \end{scope}
 
@@ -271,6 +383,117 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
 \subsection{多层神经网络}
 
 %%%------------------------------------------------------------------------------------------------------------
+%%% 一层 -> 多层
+\begin{frame}{更多的层}
+\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}
+\end{itemize}
+
+\vspace{-0.5em}
+\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}]
+
+%%% layer 1
+\foreach \n in {1,...,5}{
+    \node [neuronnode] (neuron0\n) at (\n * \neuronsep,0) {};
+}
+
+\foreach \n in {1,...,5}{
+    \foreach \m in {1,...,5}{
+        \draw [<-] (neuron0\m.south) -- ([yshift=-2em]neuron0\n.south);
+    }
+    \node [anchor=north] (x\n) at ([yshift=-2em]neuron0\n.south) {$x_\n$};
+    \visible<1>{
+    \draw [<-,thick] ([yshift=1.5em]neuron0\n.north) -- (neuron0\n.north);
+    \node [anchor=south] (y\n) at ([yshift=1.5em]neuron0\n.north) {$y_\n$};
+    }
+}
+
+\node [anchor=west] (w1label) at ([xshift=-0.5em,yshift=0.5em]x5.north east) {$\textbf{w}_1$};
+
+\begin{pgfonlayer}{background}
+\node [rectangle,inner sep=0.2em,fill=red!20] [fit = (neuron01) (neuron05)] (layer01) {};
+\end{pgfonlayer}
+
+\visible<2->{
+\node [anchor=west] (layer01label) at ([xshift=1em]layer01.east) {第一层};
+}
+\visible<4->{
+\node [anchor=west] (layer01label2) at (layer01label.east) {(\alert{输入层})};
+}
+
+%%% layer 2
+\visible<2->{
+\foreach \n in {2,...,4}{
+    \node [neuronnode] (neuron1\n) at (\n * \neuronsep,4em) {};
+}
+
+\foreach \n in {2,...,4}{
+    \foreach \m in {1,...,5}{
+        \draw [<-] (neuron1\n.south) -- (neuron0\m.north);
+    }
+    \visible<2>{
+    \draw [<-,thick] ([yshift=1.5em]neuron1\n.north) -- (neuron1\n.north);
+    \node [anchor=south] (y\n) at ([yshift=1.5em]neuron1\n.north) {$y_\n$};
+    }
+}
+
+\node [anchor=west] (w2label) at ([xshift=-2.5em,yshift=5.0em]x5.north east) {$\textbf{w}_2$};
+
+\begin{pgfonlayer}{background}
+\visible<2->{
+\node [rectangle,inner sep=0.2em,fill=ugreen!20] [fit = (neuron12) (neuron14)] (layer02) {};
+}
+\end{pgfonlayer}
+
+\node [anchor=west] (layer02label) at ([xshift=4.9em]layer02.east) {第二层};
+\visible<4->{
+\node [anchor=west] (layer02label2) at (layer02label.east) {(\alert{隐层})};
+}
+}
+
+%%% layer 3
+\visible<3->{
+\foreach \n in {1,...,5}{
+    \node [neuronnode] (neuron2\n) at (\n * \neuronsep,8em) {};
+}
+
+\foreach \n in {1,...,5}{
+    \foreach \m in {2,...,4}{
+        \draw [<-] (neuron2\n.south) -- (neuron1\m.north);
+    }
+    
+    \node [anchor=south] (y\n) at ([yshift=1.5em]neuron2\n.north) {$y_\n$};
+    \draw [<-,thick] ([yshift=1.5em]neuron2\n.north) -- (neuron2\n.north);
+}
+
+\node [anchor=west] (w3label) at ([xshift=-2.5em,yshift=8.5em]x5.north east) {$\textbf{w}_3$};
+
+\begin{pgfonlayer}{background}
+\visible<3->{
+\node [rectangle,inner sep=0.2em,fill=blue!20] [fit = (neuron21) (neuron25)] (layer03) {};
+}
+\end{pgfonlayer}
+
+\node [anchor=west] (layer03label) at ([xshift=1em]layer03.east) {第三层};
+\visible<4->{
+\node [anchor=west] (layer03label2) at (layer03label.east) {(\alert{输出层})};
+}
+}
+
+\end{scope}
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+
+%%%------------------------------------------------------------------------------------------------------------
 \subsection{参数学习 - 反向传播}
 
 \end{CJK}

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

@@ -991,10 +991,10 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
 \begin{scope}[xshift=9.5em]
 \draw [->,thick] (-1.5,0) -- (1.5,0);
 \draw [->,thick] (0,-0.1) -- (0,1.5);
-\draw [-,very thick,ublue,domain=-1.2:1.2,samples=100] plot (\x, {0.5*2^\x});
+\draw [-,very thick,ublue,domain=-1.2:1.2,samples=100] plot (\x, {0.5 * exp(\x)});
 \node [anchor=west] (ylabel) at (0,1.3) {$y$};
 \node [anchor=north] (xlabel) at (1.3,0) {$x$};
-\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = 0.5 \cdot  2^x$}};
+\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = 0.5 \cdot  \exp(x)$}};
 \node [anchor=south west] (flabel) at ([xshift=-1.8em]func.north west) {\footnotesize{Exponential:}};
 \end{scope}