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\setbeamertemplate{navigation symbols}{\begin{CJK}{UTF8}{\mycfont} 第五章 神经网络和语言模型 \hspace*{2em} 肖桐\&朱靖波 \end{CJK} \hspace*{2em} \today \hspace*{2em} \insertframenumber{}/\inserttotalframenumber}

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\title{\Large{神经网络和语言模型}}
\author{\large{\textbf{肖桐\ \ 朱靖波}}}
\institute{
\blue{\url{xiaotong@mail.neu.edu.cn}} \black{} \\
\blue{\url{zhujingbo@mail.neu.edu.cn}} \black{} \\
\vspace{1.0em}
东北大学 自然语言处理实验室 \\
\blue{\underline{\url{http://www.nlplab.com}}} \black{} \\
\vspace{0.2cm}
\hspace{0.1cm} \includegraphics[scale=0.1]{../Figures/logo.pdf}
}
\date{}

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\section{为什么要谈神经网络}

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%%%------------------------------------------------------------------------------------------------------------
\section{神经语言模型}

%%%------------------------------------------------------------------------------------------------------------
\subsection{词嵌入}

%%%------------------------------------------------------------------------------------------------------------
%%% 线性代数基础
\begin{frame}{预热 - 线性代数知识}
\begin{itemize}
\item \textbf{矩阵}：我们用$a$表示一个标量(一个数)，用粗体$\textbf{a}$表示一个矩阵(或向量)，其中$a_{ij}$表示$\textbf{a}$第$i$行、第$j$列的元素\\
    \begin{displaymath}
    a = 5 \hspace{3em} \textbf{a} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
    \end{displaymath}
\item \textbf{向量}：一种特殊的矩阵，只有一行或者一列，这里默认使用行向量，比如$\textbf{a} = (a_1,a_2,a_3) = (10, 20, 30)$，$\textbf{a}$对应的列向量记为$\textbf{a}^T$
\item<2-> \textbf{代数运算}：矩阵可以按位进行+、-等代数运算，对于$\textbf{a} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$，$\textbf{b} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$，有$\textbf{a} + \textbf{b} = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$
\item<3-> \textbf{矩阵的微分}：按位进行，对于矩阵$\textbf{c}$和标量$x$有
    \begin{displaymath}
    \frac{\partial \textbf{c}}{\partial x} = \begin{pmatrix} \frac{\partial c_{11}}{\partial x} & \frac{\partial c_{12}}{\partial x} \\ \frac{\partial c_{21}}{\partial x} & \frac{\partial c_{22}}{\partial x} \end{pmatrix} \hspace{2em} \frac{\partial x}{\partial \textbf{c}} = \begin{pmatrix} \frac{\partial x}{\partial c_{11}} & \frac{\partial x}{\partial c_{12}} \\ \frac{\partial x}{\partial c_{21}} & \frac{\partial x}{\partial c_{22}} \end{pmatrix}
    \end{displaymath}
\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% 线性代数基础
\begin{frame}{预热 - 线性代数知识(续)}
\begin{itemize}


\item \textbf{矩阵的乘法}：对于$\textbf{a} \in \mathbb{R}^{n \times k}$和$\textbf{b} \in \mathbb{R}^{k \times m}$，用$\textbf{c} = \textbf{a} \textbf{b} \in \mathbb{R}^{n \times m}$表示\textbf{a}和\textbf{b}的矩阵乘法，其中
    \begin{displaymath}
    c_{pq} = \sum_{i = 1}^k a_{pi} b_{iq} 
    \end{displaymath}
    对于方程$\left\{ \begin{array}{l} 5x_{1} + 2x_{2} = y_{1} \\ 3x_{1} + x_{2} = y_{2}\end{array} \right.$，可以表示为$\textbf{a} \textbf{x}^T = \textbf{y}^T$ 其中$\textbf{a}=\begin{pmatrix} 5 & 2 \\ 3 & 1 \end{pmatrix}$，$\textbf{x}^T =\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$，$\textbf{y}^T =\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$
\item<2-> \textbf{其它}
    \begin{itemize}
    \item \textbf{单位矩阵}：方阵$\textbf{I}$，$I_{ij} = 1$当且仅当$i=j$，否则$I_{ij} = 0$
    \item \textbf{转置}：$\textbf{a}$的转置记为$\textbf{a}^T$，有$a^T_{ji}=a_{ij}$
    \item \textbf{逆矩阵}：方阵$\textbf{a}$的逆矩阵记为$\textbf{a}^{-1}$，有$\textbf{a} \textbf{a}^{-1} = \textbf{a}^{-1} \textbf{a} = \textbf{I}$
    \item \textbf{向量(矩阵)的范数}：$||\textbf{a}||_p = \big( \sum_i |a_i|^p \big)^{\frac{1}{p}}$ 
    \end{itemize}

\end{itemize}
\end{frame}

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%%% 深度学习带来的问题及思考 - 并不是无所不能


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