A DC-Programming Algorithm for Kernel Selection

by Andreas Argyriou, Raphael Hauser, Charles A. Micchelli and Massimiliano Pontil

There are several points in this paper:
One single kernel might not be desirable, since if we know nothing about the data, the kernel is selected empirically from a set of known kernels.
Given a set of parameterized kernels, how is it possible to select one that is the convex combination of them? The authors mention a finite representation in Learning convex combinations of continuously parameterized basic kernels (in COTL 2005).

A procedure for calculating the kernel selection is illustrated in the following algorithm:

The first step is carried out in convex optimization algorithm and the second is a DC programming problem. The stopping criterion ensures the procedure will increase the object function.

But it is a little strange why Step 2 is a DC programming problem. The authors cite a result from On functions representable as a difference of convex functions by Hartman, which implies:

• Every two continuously differentiable function on Omega is DC;
• Every continuous function on Omega is the limit of a sequence of DC functions that converges uniformly on Omega.

I have to check these things.