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# Nonlinear Approximation and Image Representation using Wavelets

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DagSemProc.07071.14.pdf
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## Cite As

Sudipto Guha and Boulos Harb. Nonlinear Approximation and Image Representation using Wavelets. In Web Information Retrieval and Linear Algebra Algorithms. Dagstuhl Seminar Proceedings, Volume 7071, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)
https://doi.org/10.4230/DagSemProc.07071.14

## 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.
##### Keywords
• Nonlinear approximation
• wavelets
• approximation algorithms
• streaming algorithms

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