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Analysis of Regularized LS Reconstruction and Random Matrix Ensembles in Compressed Sensing
KTH, School of Electrical Engineering (EES), Communication Theory. Aalto Univ, Espoo 02150, Finland.
KTH, School of Electrical Engineering (EES), Communication Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.ORCID iD: 0000-0003-2638-6047
2016 (English)In: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 62, no 4, 2100-2124 p.Article in journal (Refereed) PublishedText
Abstract [en]

The performance of regularized least-squares estimation in noisy compressed sensing is analyzed in the limit when the dimensions of the measurement matrix grow large. The sensing matrix is considered to be from a class of random ensembles that encloses as special cases standard Gaussian, row-orthogonal, geometric, and so-called T-orthogonal constructions. Source vectors that have non-uniform sparsity are included in the system model. Regularization based on l(1)-norm and leading to LASSO estimation, or basis pursuit denoising, is given the main emphasis in the analysis. Extensions to l(2)-norm and zero-norm regularization are also briefly discussed. The analysis is carried out using the replica method in conjunction with some novel matrix integration results. Numerical experiments for LASSO are provided to verify the accuracy of the analytical results. The numerical experiments show that for noisy compressed sensing, the standard Gaussian ensemble is a suboptimal choice for the measurement matrix. Orthogonal constructions provide a superior performance in all considered scenarios and are easier to implement in practical applications. It is also discovered that for non-uniform sparsity patterns, the T-orthogonal matrices can further improve the mean square error behavior of the reconstruction when the noise level is not too high. However, as the additive noise becomes more prominent in the system, the simple row-orthogonal measurement matrix appears to be the best choice out of the considered ensembles.

Place, publisher, year, edition, pages
IEEE , 2016. Vol. 62, no 4, 2100-2124 p.
Keyword [en]
Compressed sensing, eigenvalues of random matrices, compressed sensing matrices, noisy linear measurements, l(1) minimization
National Category
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-185615DOI: 10.1109/TIT.2016.2525824ISI: 000372744300039ScopusID: 2-s2.0-84963759120OAI: oai:DiVA.org:kth-185615DiVA: diva2:924547
Funder
Swedish Research Council, 621-2011-1024
Note

QC 20160428

Available from: 2016-04-28 Created: 2016-04-25 Last updated: 2016-04-28Bibliographically approved

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Vehkaperä, MikkoKabashima, YoshiyukiChatterjee, Saikat
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