Full length article: Approximation properties of certain operator-induced norms on Hilbert spaces

Authors:
Arash A. Amini;Martin J. Wainwright
Affiliations:
Department of Electrical Engineering and Computer Sciences, UC Berkeley, Berkeley, CA 94720, United States;Department of Statistics, UC Berkeley, Berkeley, CA 94720, United States and Department of Electrical Engineering and Computer Sciences, UC Berkeley, Berkeley, CA 94720, United States
Venue:
Journal of Approximation Theory
Year:
2012

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Abstract

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L^2 norm, and study their approximation properties over Hilbert subspaces of L^2. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysis of M-estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.