Departmental Papers (CIS)

Date of this Version

February 2008

Document Type

Journal Article

Comments

Copyright 2008 IEEE. Reprinted from IEEE Transactions on Information Theory, Volume 54, Issue 2, February 2008, pages 811-830.

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Abstract

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.

Keywords

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

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Date Posted: 20 March 2008

This document has been peer reviewed.