Kernels

Question 1

What is a kernel?

Answer 1

A Kernels is a function of two original vectors (x, y) that computes the dot product of the expansion of those vectors according to a certain basis function without ever having to compute the actual expansion.

Question 2

Why are kernels useful?

Answer 2

This reduces the amount of computation required which increases speed of calcualtions.

Question 3

What is the kernel for polynomials?

Answer 3

k(x,y) = (1 + x^Ty)^s

Question 4

What is the kernel for Gaussian?

Answer 4

k(x,y) = exp(-||x-y||^2 / 2σ)

Question 5

What is the kernel for sigmoid?

Answer 5

k(x,y) = tanh(Kx^Ty - δ)

Question 6

How do we apply kernels to maximisation of L(λ)?

Answer 6

max L(λ) = ∑(n=1)^N λ_n - 0.5 ∑(n=1)^N ∑_(m=1)^N (λ_n λ_m t_n t_m k(x_n,x_m))