new page finished

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

@@ -18,6 +18,7 @@
 \usepackage{changepage}
 \usepackage{pgfplots}
 \usepackage{subfigure}
+\usepackage{pgf-umlcd}
 
 \usetikzlibrary{arrows,decorations.pathreplacing}
 \usetikzlibrary{shadows} % LATEX and plain TEX when using Tik Z
@@ -66,6 +67,26 @@
 {\newcommand{\mycfont}{you}}
 {\newcommand{\mycfont}{gbsn}}
 
+% we use this function to plot a curve
+\def\curveh#1{0.2 * (#1 +0.4)^3 + 1.2 - 0.3 *(#1 + 0.8)^2}
+%\newcommand{\curvehh}[1]{0.2 * (#1 +0.4)^3 + 1.2 - 0.3 *(#1 + 0.8)^2}
+\newcommand{\curvehh}[1]{#1}
+\newcommand{\curvehhh}{1}
+\newcommand{\hish}{1}
+
+\renewcommand{\unidirectionalAssociation}[4]
+{
+  \draw[umlcd style, ->] (#1) -- (#4)% <------
+  node[near end, auto]{#2}
+  node[near end, auto,swap]{#3};
+}
+
+\newcommand{\curvehis}[3]
+{
+    \draw [-] (#1 - #3,0) -- (#1 - #3, 0.2 * (#2 +0.4)^3 + 1.2 - 0.3 *(#2 + 0.8)^2);
+}
+
+
 \begin{CJK}{UTF8}{\mycfont}
 \end{CJK}
 
@@ -502,7 +523,7 @@ cycle}
 %%% 两层神经网络可以逼近任何函数
 \begin{frame}{多层神经网络可以逼近任意函数}
 \begin{itemize}
-\item 以一个简单的两层网络为例（隐层激活函数：sigmoid）
+\item 以一个简单的三层网络为例（隐层激活函数：sigmoid）
 \end{itemize}
 
 \begin{center}
@@ -519,52 +540,162 @@ cycle}
 \draw [-,ublue] (n11.west) -- (n11.east);
 \node [anchor=north] (x1) at ([yshift=-6em]n11.south) {$x_1$};
 \node [anchor=north] (b) at ([yshift=-6em]n10.south) {$b$};
+\visible<1-10>{
+\draw [->,thick] (b.north) -- ([yshift=-0.1em]n10.south);
+\draw [->,thick] (x1.north) -- ([yshift=-0.1em]n10.290);
+}
+\visible<1>{
+\draw [->,thick] (b.north) -- ([yshift=-0.1em]n11.250);
+\draw [->,thick] (x1.north) -- ([yshift=-0.1em]n11.south);
+}
+
+\visible<11->{
 \draw [->,thick,red] (b.north) -- ([yshift=-0.1em]n10.south);
-\draw [->,thick,blue] (b.north) -- ([yshift=-0.1em]n11.250);
 \draw [->,thick,ugreen] (x1.north) -- ([yshift=-0.1em]n10.290);
+}
+
+\visible<2->{
+\draw [->,thick,blue] (b.north) -- ([yshift=-0.1em]n11.250);
 \draw [->,thick,purple] (x1.north) -- ([yshift=-0.1em]n11.south);
+}
+
+\visible<15->{
+\node [neuronnode] (n12) at (2.7,0) {\tiny{$f$}\\[-1ex] \tiny{$\sum$}};
+\node [neuronnode] (n13) at (3.8,0) {\tiny{$f$}\\[-1ex] \tiny{$\sum$}};
+\draw [-,ublue] (n12.west) -- (n12.east);
+\draw [-,ublue] (n13.west) -- (n13.east);
+\draw [->,thick] (b.north) -- ([yshift=-0.1em]n12.250);
+\draw [->,thick] (x1.north) -- ([yshift=-0.1em]n12.270);
+\draw [->,thick] (b.north) -- ([yshift=-0.1em]n13.230);
+\draw [->,thick] (x1.north) -- ([yshift=-0.1em]n13.250);
+}
+
+\visible<16->{
+\node [anchor=west] (morenodes) at (n13.east) {...};
+}
 
 %% output layers
 \node [neuronnode] (n20) at (0.75,5em) {\scriptsize{$\sum$}};
-\draw [->,thick] ([yshift=0.1em]n10.north) -- ([yshift=-0.1em]n20.250);
-\draw [->,thick] ([yshift=0.1em]n11.north) -- ([yshift=-0.1em]n20.290);
+\visible<1-10>{\draw [->,thick] ([yshift=0.1em]n10.north) -- ([yshift=-0.1em]n20.250);}
+\visible<1-8>{\draw [->,thick] ([yshift=0.1em]n11.north) -- ([yshift=-0.1em]n20.290);}
+
+\visible<11->{\draw [->,thick,brown] ([yshift=0.1em]n10.north) -- ([yshift=-0.1em]n20.250);}
+\visible<9->{\draw [->,thick,orange] ([yshift=0.1em]n11.north) -- ([yshift=-0.1em]n20.290);}
 
 \node [] (y) at ([yshift=3em]n20.north) {$y$};
 \draw [->,thick] ([yshift=0.1em]n20.north) -- (y.south);
 
+\visible<15->{
+\draw [->,thick] ([yshift=0.1em]n12.north) -- ([yshift=-0.1em]n20.310);
+\draw [->,thick] ([yshift=0.1em]n13.north) -- ([yshift=-0.1em]n20.330);
+}
+
 %% weight and bias
-\node [anchor=center,rotate=90] (b0) at ([yshift=3em,xshift=-0.5em]b.north) {\tiny{$b=0$}};
-\node [anchor=center,rotate=59,fill=white,inner sep=1pt] (b1) at ([yshift=4.9em,xshift=2.2em]b.north) {\tiny{$b=0$}};
-\node [anchor=center,rotate=90] (w1) at ([yshift=3em,xshift=0.5em]x1.north) {\tiny{$w=1$}};
-\node [anchor=center,rotate=-59,fill=white,inner sep=1pt] (w2) at ([yshift=1.2em,xshift=-1.2em]x1.north) {\tiny{$w=0$}};
+\visible<11->{\node [anchor=center,rotate=90,fill=white,inner sep=1pt] (b0) at ([yshift=3em,xshift=-0.5em]b.north) {\tiny{$b=-6$}};}
+\visible<11->{\node [anchor=center,rotate=-59,fill=white,inner sep=1pt] (w2) at ([yshift=1.2em,xshift=-1.2em]x1.north) {\tiny{$w=100$}};}
+
+\visible<2-6>{\node [anchor=center,rotate=59,fill=white,inner sep=1pt] (b1) at ([yshift=4.9em,xshift=2.2em]b.north) {\tiny{$b=0$}};}
+\visible<7>{\node [anchor=center,rotate=59,fill=white,inner sep=1pt] (b1) at ([yshift=4.9em,xshift=2.2em]b.north) {\tiny{$b=-2$}};}
+\visible<8->{\node [anchor=center,rotate=59,fill=white,inner sep=1pt] (b1) at ([yshift=4.9em,xshift=2.2em]b.north) {\tiny{$b=-4$}};}
+\visible<2-4>{\node [anchor=center,rotate=90,fill=white,inner sep=1pt] (w1) at ([yshift=3em,xshift=0.5em]x1.north) {\tiny{$w=1$}};}
+\visible<5>{\node [anchor=center,rotate=90,fill=white,inner sep=1pt] (w1) at ([yshift=3em,xshift=0.5em]x1.north) {\tiny{$w=10$}};}
+\visible<6->{\node [anchor=center,rotate=90,fill=white,inner sep=1pt] (w1) at ([yshift=3em,xshift=0.5em]x1.north) {\tiny{$w=100$}};}
+
+\visible<11>{\node [anchor=center,rotate=62,fill=white,inner sep=1pt] (w21) at ([yshift=2em,xshift=0.5em]n10.north) {\tiny{$w'=0.7$}};}
+\visible<12->{\node [anchor=center,rotate=62,fill=white,inner sep=1pt] (w21) at ([yshift=2em,xshift=0.5em]n10.north) {\tiny{$w'=-0.7$}};}
+
+\visible<2-8>{\node [anchor=center,rotate=-62,fill=white,inner sep=1pt] (w22) at ([yshift=2em,xshift=-0.5em]n11.north) {\tiny{$w'=1$}};}
+\visible<9>{\node [anchor=center,rotate=-62,fill=white,inner sep=1pt] (w22) at ([yshift=2em,xshift=-0.5em]n11.north) {\tiny{$w'=0.9$}};}
+\visible<10->{\node [anchor=center,rotate=-62,fill=white,inner sep=1pt] (w22) at ([yshift=2em,xshift=-0.5em]n11.north) {\tiny{$w'=0.7$}};}
 
-\node [anchor=center,rotate=62,fill=white,inner sep=1pt] (w21) at ([yshift=2em,xshift=0.5em]n10.north) {\tiny{$w=0$}};
-\node [anchor=center,rotate=-62,fill=white,inner sep=1pt] (w22) at ([yshift=2em,xshift=-0.5em]n11.north) {\tiny{$w=0$}};
 
+%% sigmoid box
 \begin{scope}
+\visible<3->{
 \node [anchor=west] (flabel) at ([xshift=1.2in]y.east) {\footnotesize{sigmoid:}};
-\node [anchor=west,inner sep=2pt] (flabel2) at (flabel.east) {\footnotesize{$f(x)=1/(1+e^{-x})$}};
-\node [anchor=north west,inner sep=2pt] (flabel3) at (flabel2.south west) {\footnotesize{$s=x_1 \cdot w + b$}};
+\node [anchor=north east] (slabel) at ([xshift=0]flabel.south east) {\footnotesize{sum:}};
+
+\node [anchor=west,inner sep=2pt] (flabel2) at (flabel.east) {\footnotesize{$f(s)=1/(1+e^{-s})$}};
+\node [anchor=west,inner sep=2pt] (flabel3) at (slabel.east) {\footnotesize{$s=x_1 \cdot w + b$}};
 \draw [->,thick,dotted] ([yshift=-0.3em,xshift=-0.1em]n11.60)  .. controls +(east:1) and +(west:2) ..  ([xshift=-0.2em]flabel.west) ;
 
 \begin{pgfonlayer}{background}
+\visible<3->{
 \node [rectangle,inner sep=0.2em,fill=blue!20,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] [fit = (flabel) (flabel2) (flabel3)] (funcbox) {};
+}
 \end{pgfonlayer}
+}
 \end{scope}
 
-\begin{scope}[xshift=2.8in,yshift=0.2in]
-\draw [->,thick] (-2,0) -- (2,0);
+%% output illustration
+\begin{scope}[xshift=2.8in,yshift=0.1in]
+\visible<4->{
+\draw [->,thick] (-2.2,0) -- (2.2,0);
 \draw [->,thick] (0,0) -- (0,2);
 \draw [-] (-0.05,1) -- (0.05,1);
 \node [anchor=east,inner sep=1pt] (label1) at (0,1) {\tiny{1}};
 \node [anchor=south east,inner sep=1pt] (label2) at (0,0) {\tiny{0}};
-\draw [-,very thick,ublue,domain=-2:2,samples=100] plot (\x,{1/(1+exp(-2*\x))});
-\draw [-,very thick,ublue,domain=-2:2,samples=100] plot (\x,{1/(1+exp(-4*\x))});
-\draw [-,very thick,ublue,rounded corners=0.2em] (-2,0) -- (0,0) -- (0,1) -- (2,1);
-\draw [-,very thick,ublue,rounded corners=0.2em] (-2,0) -- (0.25,0) -- (0.25,1) -- (2,1);
-\draw [-,very thick,ublue,rounded corners=0.2em] (-2,0) -- (0.5,0) -- (0.5,1) -- (2,1);
+}
+
+\visible<4>{\draw [-,very thick,ublue,domain=-2:2,samples=100] plot (\x,{1/(1+exp(-2*\x))});}
+\visible<5>{\draw [-,very thick,ublue,domain=-2:2,samples=100] plot (\x,{1/(1+exp(-4*\x))});}
+\visible<6>{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0,0) -- (0,1) -- (2,1);}
+\visible<7>{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0.25,0) -- (0.25,1) -- (2,1);}
+\visible<8>{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0.5,0) -- (0.5,1) -- (2,1);}
+\visible<9>{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0.5,0) -- (0.5,0.9) -- (2,0.9);}
+\visible<10>{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0.5,0) -- (0.5,0.7) -- (2,0.7);}
+\visible<11>{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0.5,0) -- (0.5,0.7) -- (0.7,0.7) -- (0.7,1.4) -- (2,1.4);}
+\visible<12->{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0.5,0) -- (0.5,0.7) -- (0.7,0.7) -- (0.7,0) -- (2,0);}
+\visible<15->{\draw [-,very thick,ublue,rounded corners=0.1em] (-2,0) -- (0.7,0) -- (0.7,0.6) -- (0.9,0.6) -- (0.9,0) -- (2,0);}
+
+\visible<14>{\draw [->,dashed] (0.6,-0.05) -- (0.6,-0.96in);}
+\visible<15->{\draw [->,dashed] (0.8,-0.05) -- (0.8,-0.98in);}
+
+\visible<4>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\scriptsize{$w_1=1$}\\[-0ex] \scriptsize{\ $b_1=0$}};}
+\visible<5>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\alert{\scriptsize{$w_1=10$}}\\[-0ex] \scriptsize{\ $b_1=0$}};}
+\visible<6>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\alert{\scriptsize{$w_1=100$}}\\[-0ex] \scriptsize{\ $b_1=0$}};}
+\visible<7>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\scriptsize{$w_1=100$}\\[-0ex] \alert{\scriptsize{\ $b_1=-2$}}};}
+\visible<8>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\scriptsize{$w_1=100$}\\[-0ex] \alert{\scriptsize{\ $b_1=-4$}}};}
+\visible<9>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\alert{\scriptsize{$w'_1=0.9$}}};}
+\visible<10>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\alert{\scriptsize{$w'_1=0.7$}}};}
+\visible<11>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\alert{\scriptsize{$w_2=100$}}\\[-0ex] \alert{\scriptsize{\ $b_2=-6$}}\\[-0ex] \alert{\scriptsize{\ $w'_2=0.7$}}};}
+\visible<12>{\node [anchor=north west,align=left] (wblabel) at (-2,2) {\scriptsize{$w_2=100$}\\[-0ex] \scriptsize{\ $b_2=-6$}\\[-0ex] \alert{\scriptsize{\ $w'_2=-0.7$}}};}
+\visible<13->{\node [anchor=north west,align=left] (wblabel) at (-2.5,2) {\scriptsize{这是一个}\\[-1ex] \scriptsize{step function}};}
+\end{scope}
+
+\begin{scope}[xshift=2.8in,yshift=-1.2in]
+
+\visible<13->{
+\draw [->,thick] (-2.2,0) -- (2.2,0);
+\draw [->,thick] (0,0) -- (0,2);
+\draw [-,very thick,red,domain=-1.98:2,samples=100] plot (\x,{0.2 * (\x +0.4)^3 + 1.2 - 0.3 *(\x + 0.8)^2});
+}
+
+\visible<14->{
+\foreach \n in {0.5}{
+    \pgfmathsetmacro{\result}{0.2 * (\n + 0.1 + 0.4)^3 + 1.2 - 0.3 *(\n + 0.1 + 0.8)^2}%
+    \draw [-,ublue,thick] (\n,0) -- (\n, \result) -- (\n + 0.2, \result) -- (\n + 0.2, 0);
+}
+}
+
+\visible<15->{
+\foreach \n in {0.7}{
+    \pgfmathsetmacro{\result}{0.2 * (\n + 0.1 + 0.4)^3 + 1.2 - 0.3 *(\n + 0.1 + 0.8)^2}%
+    \draw [-,ublue,thick] (\n,0) -- (\n, \result) -- (\n + 0.2, \result) -- (\n + 0.2, 0);
+}
+}
+
+\visible<16->{
+\foreach \n in {-1.9,-1.7,...,1.9}{
+    \pgfmathsetmacro{\result}{0.2 * (\n + 0.1 + 0.4)^3 + 1.2 - 0.3 *(\n + 0.1 + 0.8)^2}%
+    \draw [-,ublue,thick] (\n,0) -- (\n, \result) -- (\n + 0.2, \result) -- (\n + 0.2, 0);
+}
+}
+
+\visible<14>{\node [anchor=north west,align=left] (wblabel) at (-2.5,2.5) {\scriptsize{函数的每一段都可}\\[-1ex] \scriptsize{由step function}\\[-1ex] \scriptsize{近似}};}
+\visible<15>{\node [anchor=north west,align=left] (wblabel) at (-2.5,2.5) {\scriptsize{增加因层神经元}\\[-1ex] \scriptsize{可以拟合更多的}\\[-1ex] \scriptsize{部分}};}
+\visible<16>{\node [anchor=north west,align=left] (wblabel) at (-2.5,2.5) {\scriptsize{理论上足够多的}\\[-1ex] \scriptsize{隐层神经元可以}\\[-1ex] \scriptsize{拟合\alert{任意函数}}};}
 
-\node [anchor=north west,align=left] (wblabel) at (-2,2) {\scriptsize{$w_1=1$}\\[-0ex] \scriptsize{\ $b_1=0$}};
 \end{scope}
 
 \end{scope}