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URN: urn:nbn:de:0030-drops-10634
URL: http://drops.dagstuhl.de/opus/volltexte/2007/1063/
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Guha, Sudipto ; Harb, Boulos

Nonlinear Approximation and Image Representation using Wavelets

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Abstract

We address the problem of finding sparse wavelet representations of high-dimensional vectors. We present a lower-bounding technique and use it to develop an algorithm for computing provably-approximate instance-specific representations minimizing general $ell_p$ distances under a wide variety of compactly-supported wavelet bases. More specifically, given a vector $f in mathbb{R}^n$, a compactly-supported wavelet basis, a sparsity constraint $B in mathbb{Z}$, and $pin[1,infty]$, our algorithm returns a $B$-term representation (a linear combination of $B$ vectors from the given basis) whose $ell_p$ distance from $f$ is a $O(log n)$ factor away from that of the optimal such representation of $f$. Our algorithm applies in the one-pass sublinear-space data streaming model of computation, and it generalize to weighted $p$-norms and multidimensional signals. Our technique also generalizes to a version of the problem where we are given a bit-budget rather than a term-budget. Furthermore, we use it to construct a emph{universal representation} that consists of at most $B(log n)^2$ terms and gives a $O(log n)$-approximation under all $p$-norms simultaneously.

BibTeX - Entry

@InProceedings{guha_et_al:DSP:2007:1063,
  author =	{Sudipto Guha and Boulos Harb},
  title =	{Nonlinear Approximation and Image Representation using Wavelets},
  booktitle =	{Web Information Retrieval and Linear Algebra Algorithms},
  year =	{2007},
  editor =	{Andreas Frommer and Michael W. Mahoney and Daniel B. Szyld},
  number =	{07071},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2007/1063},
  annote =	{Keywords: Nonlinear approximation, wavelets, approximation algorithms, streaming algorithms}
}

Keywords: Nonlinear approximation, wavelets, approximation algorithms, streaming algorithms
Seminar: 07071 - Web Information Retrieval and Linear Algebra Algorithms
Issue Date: 2007
Date of publication: 28.06.2007


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