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Greedy Algorithms for Matrix-Valued Kernels

  • Dominik WittwarEmail author
  • Bernard Haasdonk
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

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 fP-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.

References

  1. 1.
    M. Alvarez, L. Rosasco, N.D. Lawrence, Kernels for vector-valued functions: a review. Found. Trends Mach. Learn. 4(3), 195–266 (2012)CrossRefGoogle Scholar
  2. 2.
    S. De Marchi, R. Schaback, H. Wendland, Near-optimal data-independent point locations for radial basis function interpolation. Adv. Comput. Math. 23(3), 317–330 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    C.A. Micchelli, M. Pontil, Kernels for multi-task learning, in Advances in Neural Information Processing Systems (MIT, Cambridge, 2004)Google Scholar
  4. 4.
    G. Santin, B. Haasdonk, Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation. Dolomites Res. Notes Approx. 10, 68–78 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    R. Schaback, J. Werner, Linearly constrained reconstruction of functions by kernes with applications to machine learning. Adv. Comput. Math. 25, 237–258 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Wittwar, G. Santin, B. Haasdonk, Interpolation with uncoupled separable matrix-valued kernels. arXiv 1807.09111Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of StuttgartInstitute of Applied Analysis and Numerical SimulationStuttgartGermany

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