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On Sparse Vector Recovery Performance in Structurally Orthogonal Matrices via LASSO

Wen, CK; Zhang, J; Wong, KK; Chen, JC; Yuen, C; (2016) On Sparse Vector Recovery Performance in Structurally Orthogonal Matrices via LASSO. IEEE Transactions on Signal Processing , 64 (17) pp. 4519-4533. 10.1109/TSP.2016.2569423. Green open access

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Abstract

In this paper, we consider the compressed sensing problem of reconstructing a sparse signal from an undersampled set of noisy linear measurements. The regularized least squares or least absolute shrinkage and selection operator (LASSO) formulation is used for signal estimation. The measurement matrix is assumed to be constructed by concatenating several randomly orthogonal bases, which we refer to as structurally orthogonal matrices. Such measurement matrix is highly relevant to large-scale compressive sensing applications because it facilitates rapid computation and parallel processing. Using the replica method in statistical physics, we derive the mean-squared-error (MSE) formula of reconstruction over the structurally orthogonal matrix in the large-system regime. Extensive numerical experiments are provided to verify the analytical result. We then consider the analytical result to investigate the MSE behaviors of the LASSO over the structurally orthogonal matrix, with an emphasis on performance comparisons with matrices with independent and identically distributed (i.i.d.) Gaussian entries. We find that structurally orthogonal matrices are at least as good as their i.i.d. Gaussian counterparts. Thus, the use of structurally orthogonal matrices is attractive in practical applications.

Type: Article
Title: On Sparse Vector Recovery Performance in Structurally Orthogonal Matrices via LASSO
Open access status: An open access version is available from UCL Discovery
DOI: 10.1109/TSP.2016.2569423
Publisher version: http://doi.org/10.1109/TSP.2016.2569423
Language: English
Additional information: Copyright © 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Keywords: Sparse matrices, Noise measurement, Standards, Signal processing algorithms, Discrete Fourier transforms, Compressed sensing, Approximation algorithms, signal reconstruction, compressed sensing, matrix algebra, mean square error methods, MSE behaviors, sparse vector recovery performance, structurally orthogonal matrices, LASSO formula, compressed sensing problem, sparse signal reconstruction, noisy linear measurements, the replica method, Compressed sensing, LASSO, orthogonal measurement matrix
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Electronic and Electrical Eng
URI: https://discovery.ucl.ac.uk/id/eprint/1505020
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