Analysis of Regularized LS Reconstruction and Random Matrix Ensembles in Compressed Sensing

Mikko Vehkapera, Yoshiyuki Kabashima, Saikat Chatterjee

Research output: Contribution to journalArticleScientificpeer-review

38 Citations (Scopus)


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 $\ell-{1}$-norm and leading to LASSO estimation, or basis pursuit denoising, is given the main emphasis in the analysis. Extensions to $\ell-{2}$-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.

Original languageEnglish
Article number7399380
Pages (from-to)2100-2124
Number of pages25
JournalIEEE Transactions on Information Theory
Issue number4
Publication statusPublished - 1 Apr 2016
MoE publication typeA1 Journal article-refereed


  • '1 minimization
  • Compressed sensing
  • compressed sensing matrices
  • eigenvalues of random matrices
  • noisy linear measurements


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