# Sparse Signal Estimation by Maximally Sparse Convex Optimization

arXiv: https://arxiv.org/abs/1302.5729 · HTML: https://mlsanity.com/paper/1302.5729/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2013.

## Built on by (most cited first, 3 of 3)

- [Self-concordant smoothing in proximal quasi-Newton algorithms for large-scale convex composite optimization](https://mlsanity.com/paper/2309.01781/) (2023)
- [Proximal Operators of Sorted Nonconvex Penalties](https://mlsanity.com/paper/2506.15315/) (2025)
- [A block-coordinate descent framework for non-convex composite optimization. Application to sparse precision matrix estimation](https://mlsanity.com/paper/2601.21467/) (2026)

## Abstract (arXiv)

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.