机器学习导论（张志华）：正定核应用

前言

这个笔记是北大那位老师课程的学习笔记，讲的概念浅显易懂，非常有利于我们掌握基本的概念，从而掌握相关的技术。

basic concepts

If a function is positive definite，then matrix is P.S.D.
$$x_1,,,,x_n\subset X=>K_0(x_i,x_j)=g(x_i)g(x_j)$$
$$=>k_0=[g(x_1),..,g(x_n){]}^{\prime }\ast [g(x_1),...,g(x_n)]$$

Thm:

Let F be a probalility measure on the half low Pat such that $$0<{\int }_0^{\infty }sdF(s)<\infty$$
and l(F,u)=$${\int }_0^{\infty }exp(-ts\varphi )dF$$ is P.D for all t>0;
example:
polynomial kernel.
RBF Gauss kernel.
two advantages:$$1.lowdimension->\infty dimension$$
$$2.normalize.$$

Levy distribution

$$(B/2\ast pi{)}^1/2exp(sqrt(2B))|f(s)=sqrt(t/2\ast pi)u^{-}3/2exp(-t/2u)du$$
if $$\varphi (x)=K^{+1/2}$$
$$K^{\frac{1}{2}}\ast K^{\frac{1}{2}}=K^T$$

Thm

let $$kX\ast X->R$$ be a P.D kernel then exists a HILBERT space H and from x->H such that $$\varphi (H)$$
$$\forall x,y\subset x,K(x,y)=<\varphi (x),\varphi (y)>$$ three kernels.