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Section05-Neural-Networks-and-Language-Modeling/section05-test.tex
...@@ -111,6 +111,8 @@ ...@@ -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$\textbf{y} \in \mathbb{R}^n$ \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} \end{itemize}
\vspace{1em}
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
...@@ -136,10 +138,19 @@ ...@@ -136,10 +138,19 @@
\end{pgfonlayer} \end{pgfonlayer}
\end{tikzpicture} \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} \end{frame}
%%%------------------------------------------------------------------------------------------------------------ %%%------------------------------------------------------------------------------------------------------------
%%% 线性变换 %%% 线性变换
\begin{frame}{线性变换} \begin{frame}{线性变换}
...@@ -152,7 +163,7 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber ...@@ -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： \item<2-> 线性变换的一种几何解释：
\end{itemize} \end{itemize}
\vspace{-1em} \vspace{-0.5em}
\visible<2->{ \visible<2->{
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
...@@ -163,32 +174,133 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber ...@@ -163,32 +174,133 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
\node [anchor=west] (plus) at (w.east) {\Large{$+$}}; \node [anchor=west] (plus) at (w.east) {\Large{$+$}};
\node [anchor=west] (b) at (plus.east) {\Large{$\textbf{b}$}}; \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->{ \visible<3->{
\node [anchor=center,fill=green!20] (w2) at (w) {\Large{$\textbf{w}$}}; \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)}}; \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); \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->{ \visible<4->{
\node [anchor=center,fill=purple!20] (b2) at (b) {\Large{$\textbf{b}$}}; \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)}; \node [anchor=west] (blabel) at ([xshift=1.5em]b2.east) {平移(shift)};
\draw [<-] ([xshift=0.2em]b2.east) -- (blabel.west); \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{tikzpicture}
\end{center} \end{center}
} }
\end{frame} \end{frame}
%%%------------------------------------------------------------------------------------------------------------ %%%------------------------------------------------------------------------------------------------------------
%%% 线性变换：更复杂的实例 %%% 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} \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 \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} \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} \end{frame}
%%%------------------------------------------------------------------------------------------------------------ %%%------------------------------------------------------------------------------------------------------------
%%% 激活函数 %%% 激活函数
\begin{frame}{激活函数} \begin{frame}{激活函数}
...@@ -237,10 +349,10 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber ...@@ -237,10 +349,10 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
\begin{scope}[xshift=9.5em] \begin{scope}[xshift=9.5em]
\draw [->,thick] (-1.5,0) -- (1.5,0); \draw [->,thick] (-1.5,0) -- (1.5,0);
\draw [->,thick] (0,-0.1) -- (0,1.5); \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=west] (ylabel) at (0,1.3) {$y$};
\node [anchor=north] (xlabel) at (1.3,0) {$x$}; \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:}}; \node [anchor=south west] (flabel) at ([xshift=-1.8em]func.north west) {\footnotesize{Exponential:}};
\end{scope} \end{scope}
...@@ -271,6 +383,117 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber ...@@ -271,6 +383,117 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
\subsection{多层神经网络} \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} \subsection{参数学习 - 反向传播}
\end{CJK} \end{CJK}
......
Section05-Neural-Networks-and-Language-Modeling/section05.tex
...@@ -991,10 +991,10 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber ...@@ -991,10 +991,10 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
\begin{scope}[xshift=9.5em] \begin{scope}[xshift=9.5em]
\draw [->,thick] (-1.5,0) -- (1.5,0); \draw [->,thick] (-1.5,0) -- (1.5,0);
\draw [->,thick] (0,-0.1) -- (0,1.5); \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=west] (ylabel) at (0,1.3) {$y$};
\node [anchor=north] (xlabel) at (1.3,0) {$x$}; \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:}}; \node [anchor=south west] (flabel) at ([xshift=-1.8em]func.north west) {\footnotesize{Exponential:}};
\end{scope} \end{scope}
......
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