update

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

@@ -421,6 +421,8 @@ 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};
+
 \visible<2->{
 \node [anchor=west] (layer01label) at ([xshift=1em]layer01.east) {第一层};
 }
@@ -468,7 +470,7 @@ cycle}
     \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);
 }
@@ -494,6 +496,15 @@ cycle}
 \end{frame}
 
 %%%------------------------------------------------------------------------------------------------------------
+%%% 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{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}
+\end{frame}
+
+%%%------------------------------------------------------------------------------------------------------------
 \subsection{参数学习 - 反向传播}
 
 \end{CJK}

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

+\frametitle{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}