# 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.

## Notes

### Acknowledgements

We thank Gabriele Santin for fruitful discussions.

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