On the Degree of Approximation by Manifolds of Finite Pseudo-Dimension |
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Authors: | V Maiorov J Ratsaby |
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Institution: | (1) Department of Mathematics Technion Haifa 32000 Israel maiorov@tx.technion.ac.il, IL;(2) Department of Electrical Engineering Technion Haifa 32000 Israel jer@ee.technion.ac.il, IL |
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Abstract: | The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes.
A class of finite pseudo-dimension possesses a useful statistical smoothness property. In 10] we introduced a nonlinear approximation
width = which measures the worst-case approximation error over all functions by the best manifold of pseudo-dimension n . In this paper we obtain tight upper and lower bounds on ρ
n
(W
r,d
p
, L
q
) , both being a constant factor of n
-r/d
, for a Sobolev class W
r,d
p
, . As this is also the estimate of the classical Alexandrov nonlinear n -width, our result proves that approximation of W
r,d
p
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.
March 12, 1997. Dates revised: August 26, 1997, October 24, 1997, March 16, 1998, June 15, 1998. Date accepted: June 25, 1998. |
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Keywords: | , Nonlinear widths, Pseudo-Dimension, Sobolev class, AMS Classification, 41A45, 41A46, 42A10, 42A61, |
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