Estimates of covering numbers of convex sets with slowly decaying orthogonal subsets

Authors:
Vra Krková;Marcello Sanguineti
Affiliations:
Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou ví2, 182 07, Prague 8, Czech Republic;Department of Communications, Computer, and System Sciences (DIST), University of Genova, Via Opera Pia 13, 16145 Genova, Italy
Venue:
Discrete Applied Mathematics
Year:
2007

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Abstract

Covering numbers of precompact symmetric convex subsets of Hilbert spaces are investigated. Lower bounds are derived for sets containing orthogonal subsets with norms of their elements converging to zero sufficiently slowly. When these sets are convex hulls of sets with power-type covering numbers, the bounds are tight. The arguments exploit properties of generalized Hadamard matrices. The results are illustrated by examples from machine learning, neurocomputing, and nonlinear approximation.