Universal multi-task kernels
Andreas Caponnetto, Charles Micchelli, Massimiliano Pontil and Yiming Ying
In this paper we are concerned with reproducing kernel Hilbert spaces HK of functions
from an input space into a Hilbert space Y, an environment appropriate for multi-task
learning. The reproducing kernel K associated to HK has its values as operators on Y.
Our primary goal here is to derive conditions which ensure that the kernel K is universal.
This means that on every compact subset of the input space, every continuous function with
values in Y can be uniformly approximated by sections of the kernel. We provide various
characterizations of universal kernels and highlight them with several concrete examples
of some practical importance. Our analysis uses basic principles of functional analysis and
especially the useful notion of vector measures which we describe in sufficient detail to
clarify our results.