Replicant compression coding in Besov spaces

Gérard Kerkyacharian; Dominique Picard

ESAIM: Probability and Statistics (2003)

  • Volume: 7, page 239-250
  • ISSN: 1292-8100

Abstract

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We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space B π , q s on a regular domain of ℝ d . The result is: if s - d ( 1 / π - 1 / p ) + > 0 , then the Kolmogorov metric entropy satisfies H ( ϵ ) ∼ ϵ - d / s . This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound, we provide a new universal coding based on a thresholding-quantizing procedure using replication.

How to cite

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Kerkyacharian, Gérard, and Picard, Dominique. "Replicant compression coding in Besov spaces." ESAIM: Probability and Statistics 7 (2003): 239-250. <http://eudml.org/doc/244843>.

@article{Kerkyacharian2003,
abstract = {We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B^s_\{\pi ,q\}$ on a regular domain of $\{\mathbb \{R\}\}^d.$ The result is: if $s-d(1/\pi -1/p)_+$$&gt; 0,$ then the Kolmogorov metric entropy satisfies $H(\epsilon ) \sim \epsilon ^\{-d/s\}$. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound, we provide a new universal coding based on a thresholding-quantizing procedure using replication.},
author = {Kerkyacharian, Gérard, Picard, Dominique},
journal = {ESAIM: Probability and Statistics},
keywords = {entropy; coding; Besov spaces; wavelet bases; replication},
language = {eng},
pages = {239-250},
publisher = {EDP-Sciences},
title = {Replicant compression coding in Besov spaces},
url = {http://eudml.org/doc/244843},
volume = {7},
year = {2003},
}

TY - JOUR
AU - Kerkyacharian, Gérard
AU - Picard, Dominique
TI - Replicant compression coding in Besov spaces
JO - ESAIM: Probability and Statistics
PY - 2003
PB - EDP-Sciences
VL - 7
SP - 239
EP - 250
AB - We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B^s_{\pi ,q}$ on a regular domain of ${\mathbb {R}}^d.$ The result is: if $s-d(1/\pi -1/p)_+$$&gt; 0,$ then the Kolmogorov metric entropy satisfies $H(\epsilon ) \sim \epsilon ^{-d/s}$. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound, we provide a new universal coding based on a thresholding-quantizing procedure using replication.
LA - eng
KW - entropy; coding; Besov spaces; wavelet bases; replication
UR - http://eudml.org/doc/244843
ER -

References

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