For-all sparse recovery in near-optimal time

Anna C. Gilbert, Yi Li, Ely Porat, Martin J. Strauss

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

Abstract

An approximate sparse recovery system in ℓ1 norm consists of parameters k, ε, N, an m-by-N measurement Φ, and a recovery algorithm, R. Given a vector, x, the system approximates x by x̂ = R(Φx), which must satisfy ∥ x̂-x∥1 ≤ (1 + ε)∥x - x k∥1. We consider the "for all" model, in which a single matrix Φ is used for all signals x. The best existing sublinear algorithm by Porat and Strauss (SODA'12) uses O(ε-3 klog(N/k)) measurements and runs in time O(k1-α Nα) for any constant α>0. In this paper, we improve the number of measurements to O(ε-2 k log(N/k)), matching the best existing upper bound (attained by super-linear algorithms), and the runtime to O(k1+β poly(logN,1/ε)), with a modest restriction that k ≤ N 1-α and ε ≤ (logk/logN)γ, for any constants α, β,γ > 0. With no restrictions on ε, we have an approximation recovery system with m = O(k/εlog(N/k)((logN/logk) γ +1/ε)) measurements. The algorithmic innovation is a novel encoding procedure that is reminiscent of network coding and that reflects the structure of the hashing stages.

Original languageEnglish
Title of host publicationAutomata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings
PublisherSpringer Verlag
Pages538-550
Number of pages13
EditionPART 1
ISBN (Print)9783662439470
DOIs
StatePublished - 2014
Event41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark
Duration: 8 Jul 201411 Jul 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume8572 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference41st International Colloquium on Automata, Languages, and Programming, ICALP 2014
Country/TerritoryDenmark
CityCopenhagen
Period8/07/1411/07/14

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