Complexity of Gaussian-radial-basis networks approximating smooth functions

Authors:
Paul C. Kainen;Vra Krková;Marcello Sanguineti
Affiliations:
Department of Mathematics, Georgetown University, Washington, DC, 20057-1233, USA;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 Genoa, Via Opera Pia 13, 16145 Genova, Italy
Venue:
Journal of Complexity
Year:
2009

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Abstract

Complexity of Gaussian-radial-basis-function networks, with varying widths, is investigated. Upper bounds on rates of decrease of approximation errors with increasing number of hidden units are derived. Bounds are in terms of norms measuring smoothness (Bessel and Sobolev norms) multiplied by explicitly given functions a(r,d) of the number of variables d and degree of smoothness r. Estimates are proven using suitable integral representations in the form of networks with continua of hidden units computing scaled Gaussians and translated Bessel potentials. Consequences on tractability of approximation by Gaussian-radial-basis function networks are discussed.