Approximation Algorithms for Wavelet Transform Coding of Data Streams

• Published in: IEEE transactions on information theory Vol. 54; no. 2; pp. 811 - 830
• Main Authors: Guha, S., Harb, B.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive quantization; Algorithms; Applied sciences; Approximation; Approximation algorithms; Approximation methods; best basis selection; Coding, codes; compactly supported wavelets; Computational modeling; Dictionaries; Discrete wavelet transforms; Euclidean distance; Euclidean space; Exact sciences and technology; Image coding; Information theory; Information, signal and communications theory; Mathematical analysis; Miscellaneous; nonlinear approximation; Norms; Polynomials; Quantization; Representations; Sampling, quantization; Signal and communications theory; Signal processing; sparse representation; streaming algorithms; Streams; Telecommunications and information theory; Transform coding; universal representation; Wavelet; Wavelet transforms; Streaming; Dictionaries; compactly supported wavelets; best basis selection; nonlinear approximation; Polynomial approximation; Tree structure; Haar function; streaming algorithms; Algorithm; Information transmission; Polynomial time; Discrete function; Wavelet transformation; Adaptive quantization; universal representation; Signal quantization; Transform coding; Wavelet base; Signal processing; Sparse representation
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2007.913569

This paper addresses the problem of finding a B -term wavelet representation of a given discrete function fepsiR n whose distance from is minimized. The problem is well understood when we seek to minimize the Euclidean distance between f and its representation. The first-known algorithms for finding provably approximate representations minimizing general l p distances (including l infin ) 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 mu-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.