Sparse signal estimation by maximally sparse convex optimization

Ivan W. Selesnick, Ilker Bayram

Research output: Contribution to journalArticlepeer-review


This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e.g., sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding non-convex optimization. For this purpose, this paper describes the design and use of non-convex penalty functions (regularizers) constrained so as to ensure the convexity of the total cost function F to be minimized. The method is based on parametric penalty functions, the parameters of which are constrained to ensure convexity of F. It is shown that optimal parameters can be obtained by semidefinite programming (SDP). This maximally sparse convex (MSC) approach yields maximally non-convex sparsity-inducing penalty functions constrained such that the total cost function F is convex. It is demonstrated that iterative MSC (IMSC) can yield solutions substantially more sparse than the standard convex sparsity-inducing approach, i.e., L1 norm minimization.

Original languageEnglish (US)
Article number6705656
Pages (from-to)1078-1092
Number of pages15
JournalIEEE Transactions on Signal Processing
Issue number5
StatePublished - Mar 1 2014


  • Convex optimization
  • L1 norm
  • basis pursuit
  • deconvolution
  • lasso
  • non-convex optimization
  • sparse optimization
  • sparse regularization
  • threshold function

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering


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