License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/DagSemProc.07071.14
URN: urn:nbn:de:0030-drops-10634
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Guha, Sudipto ; Harb, Boulos

Nonlinear Approximation and Image Representation using Wavelets

07071.HarbBoulos.Paper.1063.pdf (0.8 MB)


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

  author =	{Guha, Sudipto and Harb, Boulos},
  title =	{{Nonlinear Approximation and Image Representation using Wavelets}},
  booktitle =	{Web Information Retrieval and Linear Algebra Algorithms},
  pages =	{1--18},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7071},
  editor =	{Andreas Frommer and Michael W. Mahoney and Daniel B. Szyld},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-10634},
  doi =		{10.4230/DagSemProc.07071.14},
  annote =	{Keywords: Nonlinear approximation, wavelets, approximation algorithms, streaming algorithms}

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

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