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

a two-class problem can be assumped as a Bernoulli distribution

Z‾M(z∣θ,n1+nm)\overline Z ~~ M(z|\theta,n_1+n_m)ZM(z∣θ,n1​+nm​)

reproducing kernels

1 Cover’s theorem

A complex pattern-classification problem cast in a high-dimensional space nonlincedy is more likely to be linearly seperable than in a low-dimension space.

kernel

Def 1.1 Let x⊆RPx \subseteq R^Px⊆RP be a non empty set: A function K:x*x -> R is called a kerne; over x.

Def 1.2A kernel k is called symmetric of k(xi,xj)=k(xj,xi)k(x_i,x_j)=k(x_j,x_i)k(xi​,xj​)=k(xj​,xi​) for any xiandxj⊆Xx_i and x_j \subseteq Xxi​andxj​⊆X

Def1.3 A symmetric kernel is positive definite if ∑j,k=1nαjαkK(Xi,Xk)≥0\sum_{j,k=1}^n\alpha_j\alpha_kK(X_i,X_k) \geq 0 j,k=1∑n​αj​αk​K(Xi​,Xk​)≥0

for any n⊂Nn \subset Nn⊂N and α1,...,αn{{ \alpha_1,...,\alpha_n}}α1​,...,αn​.

we call the symmetric K a conditional.P.D if +

∑j,k=1nαjαkK(Xi,Xk)≥0\sum_{j,k=1}^n\alpha_j\alpha_kK(X_i,X_k) \geq 0 j,k=1∑n​αj​αk​K(Xi​,Xk​)≥0

foralln≥zfor \quad all \quad n \geq z foralln≥z,x1,xk⊂X{x_1,x_k}\subset Xx1​,xk​⊂X

Def 1.4 if k is C.P.D then we call -k negative definite.

k(xi,xj)≥0k(x_i,x_j) \geq 0 k(xi​,xj​)≥0

ϕ(x,y)=∣∣x−y∣∣22\phi(x,y)=||x-y||^2_2 ϕ(x,y)=∣∣x−y∣∣22​