Entropy,Universal Coding,Approximation, and Bases Properties |
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Authors: | Gérard Kerkyacharian Dominique Picard |
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Institution: | (1) Université Paris X-Nanterre, 200 Avenue de la République, F 92001 Nanterre cedex, France and Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599, Université Paris VI et Université Paris VII, 16 rue de Clisson, F-75013 Paris, France |
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Abstract: | We shall present here results concerning the metric entropy of
spaces of linear and nonlinear approximation under very general conditions. Our
first result computes the metric entropy of the linear and m-terms
approximation classes according to a quasi-greedy basis verifying the
Temlyakov property. This theorem shows that the second index r is not visible
throughout the behavior of the metric entropy. However, metric entropy does
discriminate between linear and nonlinear approximation.
Our second result extends and refines a result
obtained in a Hilbertian framework by Donoho, proving that under orthosymmetry
conditions, m-terms approximation classes are characterized by the metric
entropy. Since these theorems are given under the general context of quasi-greedy
bases verifying the Temlyakov property, they have a large spectrum of
applications. For instance, it is proved in the last section that they can be
applied in the case of
L
p
norms for
R
d
for 1 < p < \infty.
We show that the lower bounds needed for this paper in fact follow
from quite simple
large deviation inequalities concerning hypergeometric or binomial distributions.
To prove the upper bounds, we provide a very simple universal coding
based on a thresholding-quantizing constructive procedure. |
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Keywords: | Entropy Coding Linear and nonlinear
approximation Term approximation Unconditional bases Quasi-greedy bases Wavelet bases |
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