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9排版+改图

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Chapter9/Figures/figure-activate-2.tex 0 → 100644
%%%------------------------------------------------------------------------------------------------------------
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\node[black,anchor=south,scale=0.8] at (0,1.6) {\small $y ={\textrm e}^{-x^2}$};
\node [anchor=south east,inner sep=1pt,scale=0.8] (labele) at (0.8,-2) {\small{(b) Gaussian}};
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%%%------------------------------------------------------------------------------------------------------------
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\node[black,anchor=south,scale=0.8] at (0,1.6) {\small $y =x$};
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\ No newline at end of file
Chapter9/Figures/figure-activate.tex
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\node[black,anchor=south] at (0,1.5) {\small $y = \ln(1+{\textrm e}^x)$};
\node [anchor=south east,inner sep=1pt] (labela) at (0.8,-2) {\small{(a) Softplus}};
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%%%------------------------------------------------------------------------------------------------------------
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\node[black,anchor=south] at (0,1.5) {\small $y = \frac{1}{1+{\textrm e}^{-x}}$};
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%%%------------------------------------------------------------------------------------------------------------
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%%%------------------------------------------------------------------------------------------------------------
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\node [anchor=south east,inner sep=1pt] (labeld) at (0.8,-2) {\small{(d) ReLU}};
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%%%------------------------------------------------------------------------------------------------------------
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\node[black,anchor=south] at (0,1.4) {\small $y =e^{-x^2}$};
\node [anchor=south east,inner sep=1pt] (labele) at (0.8,-2) {\small{(e) Gaussian}};
\end{scope}
%%%------------------------------------------------------------------------------------------------------------
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\node[black,anchor=south] at (0,1.4) {\small $y =x$};
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%%%------------------------------------------------------------------------------------------------------------
\ No newline at end of file
Chapter9/chapter9.tex
......@@ -697,14 +697,6 @@ x_1\cdot w_1+x_2\cdot w_2+x_3\cdot w_3 & = & 0\cdot 1+0\cdot 1+1\cdot 1 \nonumbe
\parinterval 在神经网络中，对于输入向量$ {\mathbi{x}}\in {\mathbb R}^m $，一层神经网络首先将其经过线性变换映射到$ {\mathbb R}^n $，再经过激活函数变成${\mathbi{y}}\in {\mathbb R}^n $。还是上面天气预测的例子，每个神经元获得相同的输入，权重矩阵$ {\mathbi{W}} $是一个$ 2\times 3 $矩阵，矩阵中每个元素$ w_{ij} $代表第$ j $个神经元中$ x_{i} $对应的权重值，假设编号为1的神经元负责预测温度，则$ w_{i1} $的含义为预测温度时输入$ x_{i} $对其影响程度。此外所有神经元的偏置$ b_{1} $，$ b_{2} $，$ b_{3} $组成了最终的偏置向量$ {\mathbi{b}}$。在该例中则有，权重矩阵$ {\mathbi{W}}=\begin{pmatrix} w_{11} & w_{12} & w_{13}\\ w_{21} & w_{22} & w_{23}\end{pmatrix} $，偏置向量$ {\mathbi{b}}=(b_1,b_2,b_3) $。
%----------------------------------------------
\begin{figure}[htp]
\centering
\input{./Chapter9/Figures/figure-translation}
\caption{线性变换示意图}
\label{fig:9-13}
\end{figure}
%-------------------------------------------
\parinterval 那么，线性变换的本质是什么？图\ref{fig:9-13}正是线性变换的简单示意。
......@@ -719,8 +711,21 @@ x_1\cdot w_1+x_2\cdot w_2+x_3\cdot w_3 & = & 0\cdot 1+0\cdot 1+1\cdot 1 \nonumbe
\end{eqnarray}
这样，矩形区域由第一象限旋转90度到了第四象限，如图\ref{fig:9-13}第一步所示。公式$ {\mathbi{x}}\cdot {\mathbi{W}}+{\mathbi{b}}$中的公式中的${\mathbi{b}}$相当于对其进行平移变换。其过程如图\ref{fig:9-13} 第二步所示，偏置矩阵$ {\mathbi{b}}=\begin{pmatrix} 0.5 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix} $将矩形区域沿$x$轴向右平移了一段距离。
%----------------------------------------------
\begin{figure}[htp]
\centering
\input{./Chapter9/Figures/figure-translation}
\caption{线性变换示意图}
\label{fig:9-13}
\end{figure}
%-------------------------------------------
\vspace{0.5em}
\end{itemize}
\parinterval 线性变换提供了对输入数据进行空间中旋转、平移的能力。线性变换也适用于更加复杂的情况，这也为神经网络提供了拟合不同函数的能力。比如可以利用线性变换将三维图形投影到二维平面上，或者将二维平面上的图形映射到三维空间。如图\ref{fig:9-14}所示，通过一个简单的线性变换，可以将三维图形投影到二维平面上。
%----------------------------------------------
\begin{figure}[htp]
\centering
......@@ -729,6 +734,9 @@ x_1\cdot w_1+x_2\cdot w_2+x_3\cdot w_3 & = & 0\cdot 1+0\cdot 1+1\cdot 1 \nonumbe
\label{fig:9-14}
\end{figure}
%-------------------------------------------
\parinterval 那激活函数又是什么？一个神经元在接收到经过线性变换的结果后，通过激活函数的处理，得到最终的输出$ y $。激活函数的目的是解决实际问题中的非线性变换，线性变换只能拟合直线，而激活函数的加入，使神经网络具有了拟合曲线的能力。 特别是在实际问题中，很多现象都无法用简单的线性关系描述，这时可以使用非线性激活函数来描述更加复杂的问题。常见的非线性激活函数有Sigmoid、ReLU、Tanh等。图\ref{fig:9-15}和\ref{fig:9-15-2}中列举了几种激活函数的形式。
%----------------------------------------------
\begin{figure}[htp]
\centering
......@@ -737,10 +745,14 @@ x_1\cdot w_1+x_2\cdot w_2+x_3\cdot w_3 & = & 0\cdot 1+0\cdot 1+1\cdot 1 \nonumbe
\label{fig:9-15}
\end{figure}
%-------------------------------------------
\parinterval 线性变换提供了对输入数据进行空间中旋转、平移的能力。线性变换也适用于更加复杂的情况，这也为神经网络提供了拟合不同函数的能力。比如可以利用线性变换将三维图形投影到二维平面上，或者将二维平面上的图形映射到三维空间。如图\ref{fig:9-14}所示，通过一个简单的线性变换，可以将三维图形投影到二维平面上。
\parinterval 那激活函数又是什么？一个神经元在接收到经过线性变换的结果后，通过激活函数的处理，得到最终的输出$ y $。激活函数的目的是解决实际问题中的非线性变换，线性变换只能拟合直线，而激活函数的加入，使神经网络具有了拟合曲线的能力。 特别是在实际问题中，很多现象都无法用简单的线性关系描述，这时可以使用非线性激活函数来描述更加复杂的问题。常见的非线性激活函数有Sigmoid、ReLU、Tanh等。图\ref{fig:9-15}中列举了几种激活函数的形式。
%----------------------------------------------
\begin{figure}[htp]
\centering
\input{./Chapter9/Figures/figure-activate-2}
\caption{几种常见的激活函数（补）}
\label{fig:9-15-2}
\end{figure}
%-------------------------------------------
%----------------------------------------------------------------------------------------
% NEW SUBSUB-SECTION
......
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