Sparse Signal Estimation by Maximally Sparse Convex Optimization

Abstract

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.

13 pages, 9 figures

References in corpus (3)

• A fast approach for overcomplete sparse decomposition based on smoothed L0 norm
• One-step sparse estimates in nonconcave penalized likelihood models
• Group-Sparse Signal Denoising: Non-Convex Regularization, Convex Optimization

Cited by in corpus (20)

• Sparse Regularization via Convex Analysis
• Group-Sparse Signal Denoising: Non-Convex Regularization, Convex Optimization
• Sparsity-based Algorithm for Detecting Faults in Rotating Machines
• Total Variation Denoising via the Moreau Envelope
• Enhanced Low-Rank Matrix Approximation
• On the Convergence of the Iterative Shrinkage/Thresholding Algorithm With a Weakly Convex Penalty
• Repetitive Transients Extraction Algorithm for Detecting Bearing Faults
• Convex Denoising using Non-Convex Tight Frame Regularization
• Enhanced Sparsity by Non-Separable Regularization
• Improved Sparse Low-Rank Matrix Estimation
• Homotopy based algorithms for ${\ell }_0$-regularized least-squares
• A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal
• SPOQ ${\ell }_p$-Over-${\ell }_q$ Regularization for Sparse Signal Recovery applied to Mass Spectrometry
• Sparsity-based Correction of Exponential Artifacts
• Cell Detection by Functional Inverse Diffusion and Non-negative Group Sparsity$-$Part I: Modeling and Inverse Problems
• The Douglas-Rachford Algorithm for Weakly Convex Penalties
• Sparsity Within and Across Overlapping Groups
• Determined BSS based on time-frequency masking and its application to harmonic vector analysis
• A Non-Convex Optimization Technique for Sparse Blind Deconvolution -- Initialization Aspects and Error Reduction Properties
• A Convex-Nonconvex Framework for Enhancing Minimization Induced Penalties