Algorithms and lower bounds for sparse recovery

We consider the following k-sparse recovery problem: design a distribution of m x n matrix A, such that for any signal x, given Ax with high probability we can efficiently recover x satisfying IIx - x l, </-Cmink-sparse x' IIx - x'II. It is known that there exist such distributions with m = O(k log(n/k)) rows; in this thesis, we show that this bound is tight. We also introduce the set query algorithm, a primitive useful for solving special cases of sparse recovery using less than 8(k log(n/k)) rows. The set query algorithm estimates the values of a vector x [epsilon] Rn over a support S of size k from a randomized sparse binary linear sketch Ax of size O(k). Given Ax and S, we can recover x' with IIlx' - xSII2 </- [theta]IIx - xsII2 with probability at least 1 - k-[omega](1). The recovery takes O(k) time. While interesting in its own right, this primitive also has a number of applications. For example, we can: * Improve the sparse recovery of Zipfian distributions O(k log n) measurements from a 1 + [epsilon] approximation to a 1 + o(1) approximation, giving the first such approximation when k </- O(n1-[epsilon]). * Recover block-sparse vectors with O(k) space and a 1 + [epsilon] approximation. Previous algorithms required either w(k) space or w(1) approximation.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2010

Author dc:creator

• Price, Eric (Eric C.)

Advisor dc:contributor.advisor

• Piotr Indyk.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/62668

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/62668