{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/62668"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/62668","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Algorithms and lower bounds for sparse recovery","abstract":"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.","abstract_html":"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, &lt;/-Cmink-sparse x&#x27; IIx - x&#x27;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&#x27; with IIlx&#x27; - xSII2 &lt;/- [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 &lt;/- 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.","abstract_has_math":false,"creators":["Price, Eric (Eric C.)"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.","school":null,"contributors":[],"advisors":["Piotr Indyk."],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-22T22:22:07Z","subjects":["Electrical Engineering and Computer Science."],"languages":["eng"],"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."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/62668","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Piotr Indyk."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."]},{"key":"dc:creator","label":"Author","values":["Price, Eric (Eric C.)"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2011-05-09T15:17:04Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2011-05-09T15:17:04Z"]},{"key":"dc:date.issued","label":"Date","values":["2010"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Electrical Engineering and Computer Science."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/62668"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.","Cataloged from PDF version of thesis.","Includes bibliographical references (p. 69-71)."]},{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["M.Eng."]},{"key":"dc:title","label":"Title","values":["Algorithms and lower bounds for sparse recovery"]}]}],"canonical_facts":{"dc:contributor.advisor":["Piotr Indyk."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science."],"dc:creator":["Price, Eric (Eric C.)"],"dc:date.accessioned":["2011-05-09T15:17:04Z"],"dc:date.available":["2011-05-09T15:17:04Z"],"dc:date.issued":["2010"],"dc:description":["Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.","Cataloged from PDF version of thesis.","Includes bibliographical references (p. 69-71)."],"dc:description.abstract":["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."],"dc:description.degree":["M.Eng."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/62668"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"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."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Electrical Engineering and Computer Science."],"dc:title":["Algorithms and lower bounds for sparse recovery"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:07Z"}