Sharp RIP bound for sparse signal and low-rank matrix recovery
This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix A satisfies the RIP condition δkA
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| Veröffentlicht in: | Applied and computational harmonic analysis 2013-07, Vol.35 (1), p.74-93 |
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| container_title | Applied and computational harmonic analysis |
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| creator | Cai, T. Tony Zhang, Anru |
| description | This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix A satisfies the RIP condition δkA |
| doi_str_mv | 10.1016/j.acha.2012.07.010 |
| format | Article |
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Tony ; Zhang, Anru</creator><creatorcontrib>Cai, T. Tony ; Zhang, Anru</creatorcontrib><description>This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix A satisfies the RIP condition δkA<1/3, then all k-sparse signals β can be recovered exactly via the constrained ℓ1 minimization based on y=Aβ. Similarly, if the linear map M satisfies the RIP condition δrM<1/3, then all matrices X of rank at most r can be recovered exactly via the constrained nuclear norm minimization based on b=M(X). Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.</description><identifier>ISSN: 1063-5203</identifier><identifier>EISSN: 1096-603X</identifier><identifier>DOI: 10.1016/j.acha.2012.07.010</identifier><language>eng</language><publisher>Elsevier Inc</publisher><subject>[formula omitted] minimization ; Beta ; Compressed sensing ; Constraints ; Dantzig selector ; Economic conditions ; Fourier analysis ; Low-rank matrix recovery ; Minimization ; Norms ; Nuclear norm minimization ; Optimization ; Recovery ; Restricted isometry ; Sparse signal recovery</subject><ispartof>Applied and computational harmonic analysis, 2013-07, Vol.35 (1), p.74-93</ispartof><rights>2012</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c443t-825c5305dd58c6bdc314e45d64e16d77f04e86d0fc9473b9075f4364f3cb8e473</citedby><cites>FETCH-LOGICAL-c443t-825c5305dd58c6bdc314e45d64e16d77f04e86d0fc9473b9075f4364f3cb8e473</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S1063520312001273$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65306</link.rule.ids></links><search><creatorcontrib>Cai, T. Tony</creatorcontrib><creatorcontrib>Zhang, Anru</creatorcontrib><title>Sharp RIP bound for sparse signal and low-rank matrix recovery</title><title>Applied and computational harmonic analysis</title><description>This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix A satisfies the RIP condition δkA<1/3, then all k-sparse signals β can be recovered exactly via the constrained ℓ1 minimization based on y=Aβ. Similarly, if the linear map M satisfies the RIP condition δrM<1/3, then all matrices X of rank at most r can be recovered exactly via the constrained nuclear norm minimization based on b=M(X). Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.</description><subject>[formula omitted] minimization</subject><subject>Beta</subject><subject>Compressed sensing</subject><subject>Constraints</subject><subject>Dantzig selector</subject><subject>Economic conditions</subject><subject>Fourier analysis</subject><subject>Low-rank matrix recovery</subject><subject>Minimization</subject><subject>Norms</subject><subject>Nuclear norm minimization</subject><subject>Optimization</subject><subject>Recovery</subject><subject>Restricted isometry</subject><subject>Sparse signal recovery</subject><issn>1063-5203</issn><issn>1096-603X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLxDAUhYMoOI7-AVdZumm9aV4tiCCDj4EBxQe4C2mSOh07TU06o_PvbRnXru7hcM6F8yF0TiAlQMTlKtVmqdMMSJaCTIHAAZoQKEQigL4fjlrQhGdAj9FJjCsAQhgvJuj6ZalDh5_nT7j0m9biygccOx2iw7H-aHWD9eA2_jsJuv3Ea92H-gcHZ_zWhd0pOqp0E93Z352it7vb19lDsni8n89uFolhjPZJnnHDKXBreW5EaQ0lzDFuBXNEWCkrYC4XFipTMEnLAiSvGBWsoqbM3WBN0cX-bxf818bFXq3raFzT6Nb5TVSEykwwTqUYotk-aoKPMbhKdaFe67BTBNQIS63UCEuNsBRINcAaSlf7khtGbGsXVDS1a42z9TC1V9bX_9V_AbEYcVg</recordid><startdate>201307</startdate><enddate>201307</enddate><creator>Cai, T. Tony</creator><creator>Zhang, Anru</creator><general>Elsevier Inc</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SP</scope><scope>8FD</scope><scope>L7M</scope></search><sort><creationdate>201307</creationdate><title>Sharp RIP bound for sparse signal and low-rank matrix recovery</title><author>Cai, T. Tony ; Zhang, Anru</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c443t-825c5305dd58c6bdc314e45d64e16d77f04e86d0fc9473b9075f4364f3cb8e473</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>[formula omitted] minimization</topic><topic>Beta</topic><topic>Compressed sensing</topic><topic>Constraints</topic><topic>Dantzig selector</topic><topic>Economic conditions</topic><topic>Fourier analysis</topic><topic>Low-rank matrix recovery</topic><topic>Minimization</topic><topic>Norms</topic><topic>Nuclear norm minimization</topic><topic>Optimization</topic><topic>Recovery</topic><topic>Restricted isometry</topic><topic>Sparse signal recovery</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cai, T. Tony</creatorcontrib><creatorcontrib>Zhang, Anru</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><collection>Electronics & Communications Abstracts</collection><collection>Technology Research Database</collection><collection>Advanced Technologies Database with Aerospace</collection><jtitle>Applied and computational harmonic analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cai, T. Tony</au><au>Zhang, Anru</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Sharp RIP bound for sparse signal and low-rank matrix recovery</atitle><jtitle>Applied and computational harmonic analysis</jtitle><date>2013-07</date><risdate>2013</risdate><volume>35</volume><issue>1</issue><spage>74</spage><epage>93</epage><pages>74-93</pages><issn>1063-5203</issn><eissn>1096-603X</eissn><abstract>This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix A satisfies the RIP condition δkA<1/3, then all k-sparse signals β can be recovered exactly via the constrained ℓ1 minimization based on y=Aβ. Similarly, if the linear map M satisfies the RIP condition δrM<1/3, then all matrices X of rank at most r can be recovered exactly via the constrained nuclear norm minimization based on b=M(X). Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.</abstract><pub>Elsevier Inc</pub><doi>10.1016/j.acha.2012.07.010</doi><tpages>20</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Applied and computational harmonic analysis, 2013-07, Vol.35 (1), p.74-93 |
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| language | eng |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | [formula omitted] minimization Beta Compressed sensing Constraints Dantzig selector Economic conditions Fourier analysis Low-rank matrix recovery Minimization Norms Nuclear norm minimization Optimization Recovery Restricted isometry Sparse signal recovery |
| title | Sharp RIP bound for sparse signal and low-rank matrix recovery |
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