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修改第2章习题的图
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docs/chapter2/2-1-Convex-Hull.png

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docs/chapter2/chapter2.md

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**凸壳**
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**定义1:**设集合$S \subset R^n$,是由$R^n$中的$k$个点所组成的集合,即$S=\{x_1,x_2,\cdots, x_k\}$。定义$S$的凸壳$\text{conv}(S)$为:$$\text{conv}(S) = \left\{ x = \sum_{i=1}^k \lambda_i x_i \Big| \sum_{i=1}^k \lambda_i=1,\lambda_i \geqslant 0, i=1,2,\cdots, k \right\}$$说明:凸壳是一个集合,对于所有可能的$\lambda_i,i=1,2,\cdots,k$只要满足$\displaystyle \sum_{i=1}^k \lambda_i = 1$,那么$\displaystyle x = \sum_{i=1}^k$即为凸壳中的元素,凸壳可以用二维的图形表示如下:
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<br/><center>
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<img style="border-radius: 0.3125em;box-shadow: 0 2px 4px 0 rgba(34,36,38,.12),0 2px 10px 0 rgba(34,36,38,.08);" src="../images/2-1-Convex-Hull.png"><br><div style="color:orange; border-bottom: 1px solid #d9d9d9;display: inline-block;color: #000;padding: 2px;">图2.1 凸壳</div></center>
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<img style="border-radius: 0.3125em;box-shadow: 0 2px 4px 0 rgba(34,36,38,.12),0 2px 10px 0 rgba(34,36,38,.08);" src="2-1-Convex-Hull.png"><br><div style="color:orange; border-bottom: 1px solid #d9d9d9;display: inline-block;color: #000;padding: 2px;">图2.1 凸壳</div></center>
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**线性可分**
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**定义2:**给定一个数据集$$T=\{(x_1,y_1), (x_2,y_2), \cdots, (x_n,y_n)\}$$其中$x_i \in \mathcal{X}=R_n, y_i \in \mathcal{Y} = \{+1, -1\}, i=1,2,\cdots, n$,如果存在某个超平面$S:w \cdot x + b = 0$,能够将数据集的正实例点和负实例点完全正确划分到超平面的两侧,即对所有的正实例点即$y_i=+1$的实例$i$,有$w \cdot x_i + b > 0$,对所有的负实例点即$y_i = -1$的实例$i$,有$w \cdot x_i + b < 0$,则称数据集$T$线性可分,否则称数据集$T$线性不可分。

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