{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1720","name":"Nonlinear Approximation and Image Representation using Wavelets","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"],"author":[{"@type":"Person","name":"Guha, Sudipto","givenName":"Sudipto","familyName":"Guha"},{"@type":"Person","name":"Harb, Boulos","givenName":"Boulos","familyName":"Harb"}],"position":14,"pageStart":1,"pageEnd":18,"dateCreated":"2007-06-28","datePublished":"2007-06-28","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Guha, Sudipto","givenName":"Sudipto","familyName":"Guha"},{"@type":"Person","name":"Harb, Boulos","givenName":"Boulos","familyName":"Harb"}],"copyrightYear":"2007","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07071.14","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume641","volumeNumber":7071,"name":"Dagstuhl Seminar Proceedings, Volume 7071","dateCreated":"2007-06-28","datePublished":"2007-06-28","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1720","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume641"}}}