definitions of tensors

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

@@ -105,62 +105,33 @@
 \subsection{神经网络的简单实现：张量计算}
 
 %%%------------------------------------------------------------------------------------------------------------
-%%% 张量
-\begin{frame}{如何描述神经网络 - 张量计算}
+%%% 张量的简单定义
+\begin{frame}{张量是什么}
 \begin{itemize}
-\item 对于神经网络，输入$\textbf{x}$和输出$\textbf{y}$的形式并不仅仅是向量
+\item \textbf{深度学习}中，张量被``简单"地定义为\alert{多维数组}
 \end{itemize}
+\end{frame}
 
-\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{$)$}};
-
-\visible<2->{
-\node [anchor=center,fill=yellow!30] (x2) at (x) {\LARGE{$\textbf{x}$}};
-\node [anchor=south] (xlabel) at ([xshift=-3em,yshift=1.5em]x.north) {\alert{向量？矩阵？...}};
-\draw [<-] ([yshift=0.2em,xshift=-0.5em]x2.north) -- ([xshift=1em]xlabel.south);
-
-\node [anchor=center,fill=red!20] (y2) at (y) {\LARGE{$\textbf{y}$}};
-\draw [<-] ([yshift=0.2em,xshift=0.5em]y2.north) -- ([xshift=-1em]xlabel.south);
-
-\node [anchor=center,fill=green!20] (w2) at (w) {\LARGE{$\textbf{w}$}};
-\node [anchor=north] (wlabel) at ([yshift=-1.0em]w.south) {矩阵 e.g.,};
-\draw [<-] ([yshift=-0.2em]w2.south) -- (wlabel.north);
-\node [anchor=west] (wsample) at ([xshift=-0.5em]wlabel.east) {\footnotesize{$\left(\begin{array}{c c} 1 & 2 \\ 3 & 4 \end{array}\right)$}};
-
-\node [anchor=center,fill=purple!20] (b2) at (b) {\LARGE{$\textbf{b}$}};
-\node [anchor=south] (blabel) at ([yshift=1.3em]b.north) {向量 e.g.,};
-\draw [<-] ([yshift=0.2em]b2.north) -- (blabel.south);
-\node [anchor=west] (bsample) at ([xshift=-0.5em]blabel.east) {\footnotesize{$(1, 3)$}};
-}
-
-\end{tikzpicture}
-\end{center}
-
+%%%------------------------------------------------------------------------------------------------------------
+%%% 张量是一个多维线性函数
+\begin{frame}{事实上，张量是个函数 - 别慌，了解一下 :)}
 \begin{itemize}
-\item<3-> $\textbf{x}$和$\textbf{y}$实际上是一个叫tensor的东西，即\textbf{张量}。比如，
+\item \textbf{非常负责任的说}，张量\alert{不是}向量和矩阵的简单扩展，甚至说，多维数组也\alert{不是}张量所必须的表达形式
+\item<2-> 严格意义上，张量是：
+    \begin{itemize}
+    \item<2-> \textbf{看不懂的定义}：由若干坐标系改变时满足一定坐标转化关系的抽象对象，它是一个不随参照系的坐标变换而变化的几何量（几何定义）
+    \item<3-> \textbf{还是看不懂的定义}：若干向量和协向量通过张量乘法定义的量（代数定义）
+    \item<4-> \textbf{还可以解释的定义}：\alert{张量是多重线性函数}，是定义在一些向量空间和笛卡儿积上的多重线性映射
+    \begin{itemize}
+        \item 这里把张量表示为$T(v_0,...,v_r)$，其中输入的是$r$个向量$\{v_0,...,v_r\}$
+        \item 多重线性是指，对于每个输入，函数都是线性的，比如，对于一个$v_i$，我们有
+        \begin{displaymath}
+        T(v_0,...,v_i+c \cdot u,...,v_r) = T(v_0,...,v_i,...,v_r) + c \cdot T(v_0,...,u,...,v_r)
+        \end{displaymath}
+        其中，$c$为任意数。这个性质非常重要，它可以推导出前面的其它定义。
+    \end{itemize}
+    \end{itemize}
 \end{itemize}
-
-\begin{center}
-\begin{tikzpicture}
-\begin{scope}
-\visible<4->{\node [anchor=west] (vector) at (0,0) {$\textbf{x} = (1,  3)$};}
-\visible<5->{\node [anchor=west] (matrix) at ([xshift=0.1in]vector.east) {$\textbf{x} = \left(\begin{array}{c c} -1 & 3 \\ 0.2 & 2 \end{array}\right)$};}
-\visible<6->{\node [anchor=west] (tensor3d) at ([xshift=0.1in]matrix.east) {啥？$\textbf{x} = \left(\begin{array}{c} \left(\begin{array}{c c} -1 & 3 \\ 0.2 & 2 \end{array}\right) \\ \left(\begin{array}{c c} -1 & 3 \\ 0.2 & 2 \end{array}\right) \end{array}\right)$};}
-\end{scope}
-\end{tikzpicture}
-\end{center}
-
 \end{frame}
 
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