前言

這個筆記是北大那位老師課程的學習筆記，講的概念淺顯易懂，非常有利于我們掌握基本的概念，從而掌握相關的技術。

basic concepts

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

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)=$${\int }_0^{\infty }exp(?ts?)dF$$ is P.D for all t>0;
example:
polynomial kernel.
RBF Gauss kernel.
two advantages:$$1.lowdimension?>\infty dimension$$
$$2.normalize.$$

Levy distribution

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

Thm

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

總結

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