# On a Variational Norm Tailored to Variable-Basis Approximation Schemes

Gnecco, Giorgio and Sanguineti, Marcello On a Variational Norm Tailored to Variable-Basis Approximation Schemes. IEEE Transactions on Information Theory, 57 (1). pp. 549-558. ISSN 0018-9448 (2011)

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## Abstract

A variational norm associated with sets of computational units and used in function approximation, learning from data, and infinite-dimensional optimization is investigated. For sets Gk obtained by varying a vector y of parameters in a fixed-structure computational unit K(-,y) (e.g., the set of Gaussians with free centers and widths), upper and lower bounds on the GK -variation norms of functions having certain integral representations are given, in terms of the £1-norms of the weighting functions in such representations. Families of functions for which the two norms are equal are described.

Item Type: Article 10.1109/TIT.2010.2090198 ${cal L}_1$-norm, Approximation schemes, convex hulls, infinite-dimensional optimization, upper and lower bounds, variation with respect to a set Q Science > QA Mathematics > QA75 Electronic computers. Computer science Computer Science and Applications Giorgio Gnecco 13 Sep 2013 11:34 16 Sep 2013 12:03 http://eprints.imtlucca.it/id/eprint/1719