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Published in:
2019 | OriginalPaper | Chapter
Greedy Algorithms for Matrix-Valued Kernels
Authors : Dominik Wittwar, Bernard Haasdonk
Published in: Numerical Mathematics and Advanced Applications ENUMATH 2017
Publisher: Springer International Publishing
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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.