nonlinear function examples

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

@@ -193,12 +193,70 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
 %%% 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
+\item 激活函数更多地是为了解决实际问题中的\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}
+
+\vspace{-1em}
+\begin{center}
+\begin{tikzpicture}
+\begin{scope}
+\draw [line width=3pt,ublue,-](0,0) -- (-2.0,1); 
+\node [anchor=north] (linelabel) at (-1.0,-0.5) {\footnotesize{我是一根筷子}};
+\end{scope}
+
+\begin{scope}[xshift=10em]
+\draw [line width=3pt,ublue,-,line cap=round](0,0) .. controls (-0.5,-0.25) and (-0.5,1).. (-1.3,0.3) .. controls (-2.3,-0.3) and (-1.1,1.8).. (-2.0,1); 
+\node [] at (-2,1) {\white{$\cdot$}};
+\node [anchor=north] (linelabel) at (-1.0,-0.5) {\footnotesize{我是一只蚯蚓}};
+\end{scope}
+\end{tikzpicture}
+\end{center}
+
+\begin{itemize}
+\item<2-> 简单的非线性函数
+\end{itemize}
+
+\vspace{-1em}
+
+\visible<2->{
+\begin{center}
+\begin{tikzpicture}
+\begin{scope}[]
+\draw [->,thick] (-1.5,0) -- (1.5,0);
+\draw [->,thick] (0,-0.1) -- (0,1.5);
+\draw [-,very thick,ublue,domain=-1.2:1.2,samples=100] plot (\x,{0.5 * (\x -0.3)^2 + 0.2});
+\node [anchor=west] (ylabel) at (0,1.3) {$y$};
+\node [anchor=north] (xlabel) at (1.3,0) {$x$};
+\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = \frac{1}{2}  (x - 0.3)^2 + 0.2$}};
+\node [anchor=south west] (flabel) at (func.north west) {\footnotesize{Quadratic:}};
+\end{scope}
+
+\begin{scope}[xshift=9.5em]
+\draw [->,thick] (-1.5,0) -- (1.5,0);
+\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});
+\node [anchor=west] (ylabel) at (0,1.3) {$y$};
+\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=south west] (flabel) at ([xshift=-1.8em]func.north west) {\footnotesize{Exponential:}};
+\end{scope}
+
+\begin{scope}[xshift=19em]
+\draw [->,thick] (-1.5,0) -- (1.5,0);
+\draw [->,thick] (0,-0.1) -- (0,1.5);
+\draw [-,very thick,ublue,domain=-1.1:1.2,samples=100] plot (\x,{abs(\x -0.2) + 0.1});
+\node [anchor=west] (ylabel) at (0,1.3) {$y$};
+\node [anchor=north] (xlabel) at (1.3,0) {$x$};
+\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = |x - 0.3| + 0.1$}};
+\node [anchor=south west] (flabel) at ([xshift=-0.4em]func.north west) {\footnotesize{Absolute:}};
+\end{scope}
+\end{tikzpicture}
+\end{center}
+}
+
 \end{frame}
 
 %%%------------------------------------------------------------------------------------------------------------

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

@@ -947,14 +947,73 @@ T(\alpha \textbf{a}) & = & \alpha T(\textbf{a}) \nonumber
 %%% 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
+\item 激活函数更多地是为了解决实际问题中的\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}
+
+\vspace{-1em}
+\begin{center}
+\begin{tikzpicture}
+\begin{scope}
+\draw [line width=3pt,ublue,-](0,0) -- (-2.0,1); 
+\node [anchor=north] (linelabel) at (-1.0,-0.5) {\footnotesize{我是一根筷子}};
+\end{scope}
+
+\begin{scope}[xshift=10em]
+\draw [line width=3pt,ublue,-,line cap=round](0,0) .. controls (-0.5,-0.25) and (-0.5,1).. (-1.3,0.3) .. controls (-2.3,-0.3) and (-1.1,1.8).. (-2.0,1); 
+\node [] at (-2,1) {\white{$\cdot$}};
+\node [anchor=north] (linelabel) at (-1.0,-0.5) {\footnotesize{我是一只蚯蚓}};
+\end{scope}
+\end{tikzpicture}
+\end{center}
+
+\begin{itemize}
+\item<2-> 简单的非线性函数
+\end{itemize}
+
+\vspace{-1em}
+
+\visible<2->{
+\begin{center}
+\begin{tikzpicture}
+\begin{scope}[]
+\draw [->,thick] (-1.5,0) -- (1.5,0);
+\draw [->,thick] (0,-0.1) -- (0,1.5);
+\draw [-,very thick,ublue,domain=-1.2:1.2,samples=100] plot (\x,{0.5 * (\x -0.3)^2 + 0.2});
+\node [anchor=west] (ylabel) at (0,1.3) {$y$};
+\node [anchor=north] (xlabel) at (1.3,0) {$x$};
+\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = \frac{1}{2}  (x - 0.3)^2 + 0.2$}};
+\node [anchor=south west] (flabel) at (func.north west) {\footnotesize{Quadratic:}};
+\end{scope}
+
+\begin{scope}[xshift=9.5em]
+\draw [->,thick] (-1.5,0) -- (1.5,0);
+\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});
+\node [anchor=west] (ylabel) at (0,1.3) {$y$};
+\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=south west] (flabel) at ([xshift=-1.8em]func.north west) {\footnotesize{Exponential:}};
+\end{scope}
+
+\begin{scope}[xshift=19em]
+\draw [->,thick] (-1.5,0) -- (1.5,0);
+\draw [->,thick] (0,-0.1) -- (0,1.5);
+\draw [-,very thick,ublue,domain=-1.1:1.2,samples=100] plot (\x,{abs(\x -0.2) + 0.1});
+\node [anchor=west] (ylabel) at (0,1.3) {$y$};
+\node [anchor=north] (xlabel) at (1.3,0) {$x$};
+\node [anchor=north] (func) at (0,-0.8) {\footnotesize{$y = |x - 0.3| + 0.1$}};
+\node [anchor=south west] (flabel) at ([xshift=-0.4em]func.north west) {\footnotesize{Absolute:}};
+\end{scope}
+\end{tikzpicture}
+\end{center}
+}
+
 \end{frame}
 
+
 %%%------------------------------------------------------------------------------------------------------------
 %%% 常用的激活函数
 \begin{frame}{常用的激活函数}