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.
${x_1,,,,x_n} \subset X => K_0(x_i,x_j)=g(x_i)g(x_j)$
$=> k_0 =[g(x_1),..,g(x_n)]' *[g(x_1 ),..., g(x_n)]$

Thm:

Let F be a probalility measure on the half low Pat such that $0< \int _0^{\infin} s dF(s)<\infin$
and l(F,u)= $\int_0^{\infin} exp(-ts\phi)dF$ is P.D for all t>0;
example:
polynomial kernel.
RBF Gauss kernel.
two advantages: $1.low dimension-> \infin dimension$
$2.normalize.$

Levy distribution

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

Thm

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