Sparse Signal Estimation by Maximally Sparse Convex Optimization

• Published in: IEEE transactions on signal processing Vol. 62; no. 5; pp. 1078 - 1092
• Main Authors: Selesnick, Ivan W., Bayram, Ilker
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; basis pursuit; Constraints; Convex optimization; Convexity; Cost function; deconvolution; Detection, estimation, filtering, equalization, prediction; Economic models; Exact sciences and technology; Information, signal and communications theory; Iterative methods; L1 norm; lasso; Least squares method; non-convex optimization; Norms; Optimization; Penalty function; Signal and communications theory; Signal processing; Signal, noise; sparse optimization; sparse regularization; Sparsity; Telecommunications and information theory; threshold function; Non convex programming; lasso; L1 approximation; Signal estimation; Iterative method; sparse regularization; basis pursuit; Convex programming; Optimization; Penalty function; non-convex optimization; Deconvolution; sparse optimization; Least squares method; Convex optimization; Signal processing; Cost function; Convex function; L1 norm; Convexity; Threshold function
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2014.2298839

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.