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linear transformation and activation function

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正在显示 包含 95 行增加 和 69 行删除
Section05-Neural-Networks-and-Language-Modeling/section05-test.tex
...@@ -105,90 +105,108 @@ ...@@ -105,90 +105,108 @@
\subsection{数学基础：张量计算} \subsection{数学基础：张量计算}
%%%------------------------------------------------------------------------------------------------------------ %%%------------------------------------------------------------------------------------------------------------
%%% 层的概念 %%% 神经网络的作用
\begin{frame}{``层"的概念} \begin{frame}{神经网络：线性变换 + 激活函数}
\begin{itemize} \begin{itemize}
\item 对于一个问题（相同输入），可能会有多个输出，这时可以把\alert{多个相同的神经元并列起来}，构成一\alert{``层"} \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$\textbf{y} \in \mathbb{R}^n$
\begin{itemize}
\item 比如，天气预报需要同时预测湿度和温度
\end{itemize}
\end{itemize} \end{itemize}
\vspace{-2em}
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
\begin{scope}
\tikzstyle{neuronnode} = [minimum size=1.5em,circle,draw,ublue,very thick,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] \node [anchor=center] (y) at (0,0) {\Large{$\textbf{y}$}};
\node [anchor=west] (eq) at (y.east) {\Large{$=$}};
\node [anchor=west] (func) at (eq.east) {\Large{$f$}};
\node [anchor=west] (brace01) at (func.east) {\Large{$($}};
\node [anchor=west] (x) at (brace01.east) {\Large{$\textbf{x}$}};
\node [anchor=west] (dot) at (x.east) {\Large{$\cdot$}};
\node [anchor=west] (w) at (dot.east) {\Large{$\textbf{w}$}};
\node [anchor=west] (plus) at (w.east) {\Large{$+$}};
\node [anchor=west] (b) at (plus.east) {\Large{$\textbf{b}$}};
\node [anchor=west] (brace02) at (b.east) {\Large{$)$}};
\node [anchor=center,fill=blue!20] (func2) at (func) {\LARGE{$f$}};
\node [anchor=north] (funclabel) at ([yshift=-1.1em]func.south) {\blue{激活函数}};
\draw [<-] ([yshift=-0.2em]func2.south) -- (funclabel.north);
\node [anchor=center,neuronnode] (neuron00) at (0,0) {}; \begin{pgfonlayer}{background}
\visible<2->{ \node [rectangle,inner sep=0.2em,fill=red!20] [fit = (x) (w) (b)] (linear) {};
\node [anchor=center,neuronnode] (neuron01) at ([yshift=-3em]neuron00) {}; \node [anchor=north] (linearlabel) at ([yshift=-1.1em]linear.south) {\alert{线性变换}};
} \draw [<-] ([yshift=-0.2em]linear.south) -- (linearlabel.north);
\visible<3->{ \end{pgfonlayer}
\node [anchor=center,neuronnode] (neuron02) at ([yshift=-3em]neuron01) {};
}
\node [anchor=east] (x0) at ([xshift=-6em]neuron00.west) {$x_0$}; \end{tikzpicture}
\node [anchor=east] (x1) at ([xshift=-6em]neuron01.west) {$x_1$}; \end{center}
\node [anchor=east] (x2) at ([xshift=-6em]neuron02.west) {$b$};
\node [anchor=west] (y0) at ([xshift=4em]neuron00.east) {$y_0$}; \end{frame}
\draw [->] (x0.east) -- (neuron00.180) node [pos=0.1,above] {\tiny{$w_{00}$}}; %%%------------------------------------------------------------------------------------------------------------
\draw [->] (x1.east) -- (neuron00.200) node [pos=0.1,above] {\tiny{$w_{10}$}}; %%% 线性变换
\draw [->] (x2.east) -- (neuron00.220) node [pos=0.05,above,yshift=0.3em] {\tiny{$b_{0}$}}; \begin{frame}{线性变换}
\draw [->] (neuron00.east) -- (y0.west); \begin{itemize}
\item 对于线性空间$V$，任意$\textbf{a}$，$\textbf{b} \in V$和数域中的任意$\alpha$，线性变换$T(\cdot)$需满足
\begin{eqnarray}
T(\textbf{a} + \textbf{b}) & = & T(\textbf{a}) + T(\textbf{b}) \nonumber \\
T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
\end{eqnarray}
\item<2-> 线性变换的一种几何解释：
\end{itemize}
\vspace{-1em}
\visible<2->{ \visible<2->{
\node [anchor=west] (y1) at ([xshift=4em]neuron01.east) {$y_1$}; \begin{center}
\draw [->] (x0.east) -- (neuron01.160) node [pos=0.4,above] {\tiny{$w_{01}$}}; \begin{tikzpicture}
\draw [->] (x1.east) -- (neuron01.180) node [pos=0.35,above,yshift=-0.2em] {\tiny{$w_{11}$}};
\draw [->] (x2.east) -- (neuron01.200) node [pos=0.4,below] {\tiny{$b_{1}$}}; \node [anchor=west] (x) at (0,0) {\Large{$\textbf{x}$}};
\draw [->] (neuron01.east) -- (y1.west); \node [anchor=west] (dot) at (x.east) {\Large{$\cdot$}};
} \node [anchor=west] (w) at (dot.east) {\Large{$\textbf{w}$}};
\node [anchor=west] (plus) at (w.east) {\Large{$+$}};
\node [anchor=west] (b) at (plus.east) {\Large{$\textbf{b}$}};
\visible<3->{ \visible<3->{
\node [anchor=west] (y2) at ([xshift=4em]neuron02.east) {$y_2$}; \node [anchor=center,fill=green!20] (w2) at (w) {\Large{$\textbf{w}$}};
\draw [->] (x0.east) -- (neuron02.140) node [pos=0.1,below,yshift=-0.2em] {\tiny{$w_{02}$}}; \node [anchor=north,inner sep=1pt] (wlabel) at ([yshift=-0.7em]w.south) {\small{旋转(rotation)}};
\draw [->] (x1.east) -- (neuron02.160) node [pos=0.1,below] {\tiny{$w_{12}$}}; \draw [<-] ([yshift=-0.2em]w2.south) -- (wlabel.north);
\draw [->] (x2.east) -- (neuron02.180) node [pos=0.3,below] {\tiny{$b_{2}$}};
\draw [->] (neuron02.east) -- (y2.west);
} }
\visible<4->{ \visible<4->{
\node [anchor=east,align=left] (inputlabel) at ([xshift=-0.1em]x1.west) {输入向量:\\\small{$\textbf{x}=(x_0,x_1)$}}; \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)};
\visible<5->{ \draw [<-] ([xshift=0.2em]b2.east) -- (blabel.west);
\node [anchor=west,align=left] (outputlabel) at ([xshift=0.1em]y1.east) {输出向量:\\\small{$\textbf{y}=(y_0,y_1,y_2)$}};
} }
\begin{pgfonlayer}{background} \end{tikzpicture}
\visible<6->{ \end{center}
\node [rectangle,inner sep=0.4em,fill=red!20] [fit = (neuron00) (neuron01) (neuron02)] (layer) {};
\node [anchor=south] (layerlabel) at ([yshift=0.2em]layer.north) {一层神经元};
} }
\visible<4->{ \end{frame}
\node [rectangle,inner sep=0.1em,fill=ugreen!20] [fit = (x0) (x1)] (inputshadow) {};
}
\visible<5->{
\node [rectangle,inner sep=0.1em,fill=blue!20] [fit = (y0) (y1) (y2)] (outputshadow) {};
}
\end{pgfonlayer}
\visible<7->{ %%%------------------------------------------------------------------------------------------------------------
\node [anchor=north west] (wlabel) at ([yshift=-1em,xshift=-7em]x2.south) {参数(矩阵):$\textbf{w} = \Big( \begin{array}{lll} w_{01} & w_{01} & w_{02} \\ w_{11} & w_{11} & w_{12} \end{array} \Big)$}; %%% 线性变换：更复杂的实例
} \begin{frame}{线性变换（续）}
\visible<8->{ \begin{itemize}
\node [anchor=west] (blabel) at (wlabel.east) {参数(向量):$\textbf{b} = (b_0, b_1, b_2)$}; \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
} \end{itemize}
\end{frame}
\end{scope} %%%------------------------------------------------------------------------------------------------------------
\end{tikzpicture} %%% 激活函数
\end{center} \begin{frame}{激活函数}
\begin{itemize}
\item 激活函数的设计更多的是为了进行\alert{非线性}变换
\begin{itemize}
\item 很多实际问题都是非线性的
\item 非线性部分提供了拟合任意函数的能力（稍后介绍）
\end{itemize}
\end{itemize}
\end{frame}
%%%------------------------------------------------------------------------------------------------------------
%%% 常用的激活函数
\begin{frame}{常用的激活函数}
\begin{itemize}
\item 好多好多，列举不全 ...
\end{itemize}
\end{frame} \end{frame}
%%%------------------------------------------------------------------------------------------------------------ %%%------------------------------------------------------------------------------------------------------------
......
Section05-Neural-Networks-and-Language-Modeling/section05.tex
...@@ -717,27 +717,27 @@ GPT-2 (Transformer) & Radford et al. & 2019 & \alert{35.7} ...@@ -717,27 +717,27 @@ GPT-2 (Transformer) & Radford et al. & 2019 & \alert{35.7}
\node [anchor=west] (eq) at (y.east) {\LARGE{$=$}}; \node [anchor=west] (eq) at (y.east) {\LARGE{$=$}};
\node [anchor=west] (func) at (eq.east) {\LARGE{$f$}}; \node [anchor=west] (func) at (eq.east) {\LARGE{$f$}};
\node [anchor=west] (brace01) at (func.east) {\LARGE{$($}}; \node [anchor=west] (brace01) at (func.east) {\LARGE{$($}};
\node [anchor=west] (x) at (brace01.east) {\LARGE{$\textbf{x}$}}; \node [anchor=west] (x) at (brace01.east) {\LARGE{$\textbf{x}$}};
\node [anchor=west] (dot) at (x.east) {\LARGE{$\cdot$}}; \node [anchor=west] (dot) at (x.east) {\LARGE{$\cdot$}};
\node [anchor=west] (w) at (dot.east) {\LARGE{$\textbf{w}$}}; \node [anchor=west] (w) at (dot.east) {\LARGE{$\textbf{w}$}};
\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{$\textbf{b}$}};
\node [anchor=west] (brace02) at (b.east) {\LARGE{$)$}}; \node [anchor=west] (brace02) at (b.east) {\LARGE{$)$}};
\visible<2->{ \visible<2->{
\node [anchor=center,fill=yellow!30] (x2) at (x) {\LARGE{$\textbf{x}$}}; \node [anchor=center,fill=yellow!30] (x2) at (x) {\LARGE{$\textbf{x}$}};
\node [anchor=south] (xlabel) at ([yshift=1.5em]x.north) {输入}; \node [anchor=south] (xlabel) at ([yshift=1.5em]x.north) {输入};
\draw [<-] ([yshift=0.2em]x2.north) -- (xlabel.south); \draw [<-] ([yshift=0.2em]x2.north) -- (xlabel.south);
} }
\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] (wlabel) at ([yshift=-1.5em]w.south) {参数(权重)}; \node [anchor=north] (wlabel) at ([yshift=-1.5em]w.south) {参数(权重)};
\draw [<-] ([yshift=-0.2em]w2.south) -- (wlabel.north); \draw [<-] ([yshift=-0.2em]w2.south) -- (wlabel.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=south] (blabel) at ([yshift=1.3em]b.north) {偏移}; \node [anchor=south] (blabel) at ([yshift=1.3em]b.north) {偏移};
\draw [<-] ([yshift=0.2em]b2.north) -- (blabel.south); \draw [<-] ([yshift=0.2em]b2.north) -- (blabel.south);
} }
...@@ -859,6 +859,14 @@ GPT-2 (Transformer) & Radford et al. & 2019 & \alert{35.7} ...@@ -859,6 +859,14 @@ GPT-2 (Transformer) & Radford et al. & 2019 & \alert{35.7}
\end{frame} \end{frame}
%%%------------------------------------------------------------------------------------------------------------ %%%------------------------------------------------------------------------------------------------------------
%%% 神经网络的作用
\begin{frame}{神经网络：线性变换 + 激活函数}
\begin{itemize}
\item 对于向量$\textbf{x} \in \mathbb{R}^m$，一层神经网络实际上就是把
\end{itemize}
\end{frame}
%%%------------------------------------------------------------------------------------------------------------
\subsection{多层神经网络} \subsection{多层神经网络}
%%%------------------------------------------------------------------------------------------------------------ %%%------------------------------------------------------------------------------------------------------------
......
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