Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces
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We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields n points at one sitting for a given integer n. In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the P-greedy algorithm, which is known to provide nearly optimal points.
KeywordsReproducing kernel Hilbert space Kernel interpolation Point set Power function Optimal design Second-order cone programming
Mathematics Subject Classification (2010)65D05 65D15 41A05 46E22
The author thanks Takanori Maehara for his valuable comment about the MATLAB programs used in the numerical experiments. Thanks to the comment, much faster execution of the proposed algorithms has been realized than that in the initially submitted version of this article. Furthermore, the author gives thanks to the anonymous reviewers for their valuable suggestions about this article.
The author is supported by the grant-in-aid of Japan Society of the Promotion of Science with KAKENHI Grant Number 17K14241.
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