Compressed Sensing: How Sharp Is the Restricted Isometry Property?
Compressed sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of CS measurements should be comparable to the information content of the vector, not simply N. CS combines directly the important task of compression with the...
| मुख्य लेखकों: | , , |
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| स्वरूप: | Journal article |
| भाषा: | English |
| प्रकाशित: |
2011
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| _version_ | 1826271990186508288 |
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| author | Blanchard, J Cartis, C Tanner, J |
| author_facet | Blanchard, J Cartis, C Tanner, J |
| author_sort | Blanchard, J |
| collection | OXFORD |
| description | Compressed sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of CS measurements should be comparable to the information content of the vector, not simply N. CS combines directly the important task of compression with the measurement task. Since its introduction in 2004ther e have been hundreds of papers on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS-exact reconstruction from seemingly undersampled measurements-it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The restricted isometry property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP that gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparison with approaches from high-dimensional geometry. © 2011 Society for Industrial and Applied Mathematics. |
| first_indexed | 2024-03-06T22:05:28Z |
| format | Journal article |
| id | oxford-uuid:4ffe10c8-4c77-4d09-a718-917c093c4c6b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:05:28Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:4ffe10c8-4c77-4d09-a718-917c093c4c6b2022-03-26T16:10:53ZCompressed Sensing: How Sharp Is the Restricted Isometry Property?Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4ffe10c8-4c77-4d09-a718-917c093c4c6bEnglishSymplectic Elements at Oxford2011Blanchard, JCartis, CTanner, JCompressed sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of CS measurements should be comparable to the information content of the vector, not simply N. CS combines directly the important task of compression with the measurement task. Since its introduction in 2004ther e have been hundreds of papers on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS-exact reconstruction from seemingly undersampled measurements-it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The restricted isometry property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP that gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparison with approaches from high-dimensional geometry. © 2011 Society for Industrial and Applied Mathematics. |
| spellingShingle | Blanchard, J Cartis, C Tanner, J Compressed Sensing: How Sharp Is the Restricted Isometry Property? |
| title | Compressed Sensing: How Sharp Is the Restricted Isometry Property? |
| title_full | Compressed Sensing: How Sharp Is the Restricted Isometry Property? |
| title_fullStr | Compressed Sensing: How Sharp Is the Restricted Isometry Property? |
| title_full_unstemmed | Compressed Sensing: How Sharp Is the Restricted Isometry Property? |
| title_short | Compressed Sensing: How Sharp Is the Restricted Isometry Property? |
| title_sort | compressed sensing how sharp is the restricted isometry property |
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