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

@@ -862,7 +862,104 @@ GPT-2 (Transformer) & Radford et al. & 2019 & \alert{35.7}
 %%% 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
+\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}
+
+\begin{center}
+\begin{tikzpicture}
+
+\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);
+
+\begin{pgfonlayer}{background}
+\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}};
+\draw [<-] ([yshift=-0.2em]linear.south) -- (linearlabel.north);
+\end{pgfonlayer}
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+
+%%%------------------------------------------------------------------------------------------------------------
+%%% 线性变换
+\begin{frame}{线性变换}
+\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->{
+\begin{center}
+\begin{tikzpicture}
+
+\node [anchor=west] (x) at (0,0) {\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{$\textbf{b}$}};
+
+\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);
+}
+
+\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);
+}
+
+\end{tikzpicture}
+\end{center}
+}
+
+\end{frame}
+
+%%%------------------------------------------------------------------------------------------------------------
+%%% 线性变换：更复杂的实例
+\begin{frame}{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
+\end{itemize}
+\end{frame}
+
+%%%------------------------------------------------------------------------------------------------------------
+%%% 激活函数
+\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}