多项式核函数相关推导

定义 设 $x=(x_1,x_2,\dots,x_n)^T \in R^n$,则称乘积 $x_{j_1}x_{j_2}\dots x_{j_d}$为 $x$的一个 $d$阶多项式，其中 $j_1,j_2,\dots,j_d \in \{1,2,\dots,n\}$

有序齐次多项式

  考虑二维空间（ $x \in R^2$）的模式， $x = (x_1,x_2)^T$,其所有的二阶单项式为 $x_i^2,x_2^2,x_1x_2,x_2x_1$ 为有序单项式。

$C_2(x)=(x_1^2,x_2^2,x_1x_2,x_2x_1)^T$
$C_d(x)=(x_{j_1}x_{j_2}\dots x_{j_d}|_{j_1,j_2,\dots,j_d \in \{1,2,\dots,n\}})^T$

1、二次项推导

$令 x=(x_1,x_2)^T,y=(y_1,y_2)^T$
$\begin{aligned} C_2(x)·C_2(y) &=(x_1^2,x_2^2,x_1x_2,x_2x_1)^T · (y_1^2,y_2^2,y_1y_2,y_2y_1)^T\\ &=x_1^2y_1^2+x_2^2y_2^2+x_1x_2y_1y_2+x_2x_1y_2y_1\\ &=(x·y)^2 \end{aligned}$

2、多次幂推导

$令 x=(x_{j_1},x_{j_2},\dots,x_{j_d}|_{j_1,j_2,\dots,j_d \in \{1,2,\dots,n\}})^T$
$y=(y_{j_1},y_{j_2},\dots,y_{j_d}|_{j_1,j_2,\dots,j_d \in \{1,2,\dots,n\}})^T$
$\begin{aligned} C_d(x)·C_d(y) &=(x_{j_1},x_{j_2},\dots,x_{j_d}|_{j_1,j_2,\dots,j_d \in \{1,2,\dots,n\}})^T · (y_{j_1},y_{j_2},\dots,y_{j_d}|_{j_1,j_2,\dots,j_d \in \{1,2,\dots,n\}})^T\\ &=\sum_{j_1=1}^n\dots\sum_{j_d=1}^n x_{j_1}\dots x_{j_d} · y_{j_1}\dots y_{j_d}\\ &=\sum_{j_1=1}^n x_{j_1}y_{j_1} \dots \sum_{j_d=1}^n x_{j_d}y_{j_d}\\ &=(\sum_{j=1}^nx_iy_i)^d=(x·y)^d \end{aligned}$