Kernel Construction

\kappa_1, \kappa_2

valid kernels,

\phi

any feature map,

B

PSD matrix,

a > 0

, and

f

any real-valued function. The following are valid kernels:

\begin{aligned}\kappa(x, z) &= \kappa_1(x, z) + \kappa_2(x, z) \\\kappa(x, z) &= a \cdot \kappa_1(x, z) \\\kappa(x, z) &= \kappa_1(x, z) \cdot \kappa_2(x, z) \\\kappa(x, z) &= f(x) \cdot \kappa_1(x, z) \cdot f(z) \\\kappa(x, z) &= \kappa_1(\phi(x), \phi(z)) \\\kappa(x, z) &= x^\top B z \\\kappa(x, z) &= \kappa_1(x, x)^{-1/2} \cdot \kappa_1(x, z) \cdot \kappa_1(z, z)^{-1/2}\end{aligned}

The last one is called normalized kernel