Approximation Algorithms for Wavelet Transform Coding of Data Streams

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adaptive quantization
best basis selection
compacted supported wavelets
nonlinear approximation
sparse representation
streaming algorithms
transform coding
universal representation
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Harb, Boulos

This paper addresses the problem of finding a B-term wavelet representation of a given discrete function ƒ ∈ Rn whose distance from ƒ is minimized. The problem is well understood when we seek to minimize the Euclidean distance between ƒ and its representation. The first-known algorithms for finding provably approximate representations minimizing general lp distances (including l∞) under a wide variety of compactly supported wavelet bases are presented in this paper. For the Haar basis, a polynomial time approximation scheme is demonstrated. These algorithms are applicable in the one-pass sublinear-space data stream model of computation. They generalize naturally to multiple dimensions and weighted norms. A universal representation that provides a provable approximation guarantee under all p-norms simultaneously; and the first approximation algorithms for bit-budget versions of the problem, known as adaptive quantization, are also presented. Further, it is shown that the algorithms presented here can be used to select a basis from a tree-structured dictionary of bases and find a B-term representation of the given function that provably approximates its best dictionary-basis representation.

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Copyright 2008 IEEE. Reprinted from IEEE Transactions on Information Theory, Volume 54, Issue 2, February 2008, pages 811-830. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to By choosing to view this document, you agree to all provisions of the copyright laws protecting it.
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