Greedy Algorithms for Matrix-Valued Kernels

Abstract

We are interested in approximating vector-valued functions on a compact set $\varOmega \subset \mathbb {R}^d$. We consider reproducing kernel Hilbert spaces of $\mathbb {R}^m$-valued functions which each admit a unique matrix-valued reproducing kernel k. These spaces seem promising, when modelling correlations between the target function components. The approximation of a function is a linear combination of matrix-valued kernel evaluations multiplied with coefficient vectors. To guarantee a fast evaluation of the approximant the expansion size, i.e. the number of centers n is desired to be small. We thus present three different greedy algorithms by which a suitable set of centers is chosen in an incremental fashion: First, the P-Greedy which requires no function evaluations, second and third, the f-Greedy and f∕P-Greedy which require function evaluations but produce centers tailored to the target function. The efficiency of the approaches is investigated on some data from an artificial model.