# 核技巧在SVM中的应用

根据[上文](https://windmising.gitbook.io/lihang-tongjixuexifangfa/11)可知，将核技巧应用于线性SVM中，可使线性SVM能够解决非线性问题。

根据[上文](https://windmising.gitbook.io/lihang-tongjixuexifangfa/7/8)可知，\
线性SVM最终要解决的是以下最优化问题：

$$
\begin{aligned}
W(a) = \frac{1}{2}\sum\_{i=1}^N\sum\_{j=1}^Na\_ia\_jy\_iy\_j\color{red}{(x\_i\cdot x\_j)} - \sum\_{i=1}^Na\_i && {1}
\end{aligned}
$$

以及最终的分类决策函数为：

$$
\begin{aligned}
f(x) = \text {sign}(\sum\_{i=1}^Na^*\_iy\_i\color{red}{x\_i^* \cdot x} + y\_j - \sum\_{i=1}^Ny\_ia^\*\_i\color{red}{(x\_i \cdot x\_j)}) && {2}
\end{aligned}
$$

在公式（1）和公式（2）中都只涉及x之间的内积，这些内积极都可以直接用$$K(x,z)$$代替\
公式（1）变为：

$$
\begin{aligned}
W(a) = \frac{1}{2}\sum\_{i=1}^N\sum\_{j=1}^Na\_ia\_jy\_iy\_j\color{red}{K(x\_i,x\_j)} - \sum\_{i=1}^Na\_i && {3}
\end{aligned}
$$

公式（2）变为：

$$
\begin{aligned}
f(x) = \text {sign}(\sum\_{i=1}^Na^**iy\_i\color{red}{K(x\_i,x)} + y\_j - \sum*{i=1}^Ny\_ia^*\_i\color{red}{K(x\_i, x\_j)}) && {4}
\end{aligned}
$$


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