On the degree of approximation by manifolds of finite pseudo-dimension

V. Maiorov, J. Ratsaby

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33 Scopus citations

Abstract

The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes. A class H of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we irtroduced a nonlinear approximation width ρn(F, Lq) = infHn dist(F, Hn, Lq) which measures the worst-case approximation error over all functions f ∈ F by the best manifold of pseudo-dimension n. In this paper we obtain tight upper and lower bounds on ρn(Wr,dp, Lq), both being a constant factor of n-r/d, for a Sobolev class Wr,dp. l ≤ p, q ≤ ∞. As this is also the estimate of the classical Alexandrov nonlinear n-width, our result proves that approximation of Wr,dp by the family of manifolds of pseudo-dimension n is as powerful as approximation by the family of all nonlinear manifolds with continuous selection operators.

Original languageEnglish
Pages (from-to)291-300
Number of pages10
JournalConstructive Approximation
Volume15
Issue number2
DOIs
StatePublished - 1999
Externally publishedYes

Keywords

  • Nonlinear widths
  • Pseudo-dimension
  • Sobolev class

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