# Sparse Regularization via Convex Analysis

arXiv: https://arxiv.org/abs/1803.06765 · HTML: https://mlsanity.com/paper/1803.06765/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2018.

## Built on by (most cited first, 17 of 17)

- [UNO: Unlimited Sampling Meets One-Bit Quantization](https://mlsanity.com/paper/2301.10155/) (2022)
- [Optimal Regularization for a Data Source](https://mlsanity.com/paper/2212.13597/) (2022)
- [Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields](https://mlsanity.com/paper/2109.08666/) (2021)
- [Multivariate Fields of Experts for Convergent Image Reconstruction](https://mlsanity.com/paper/2508.06490/) (2025)
- [DuRIN: A Deep-unfolded Sparse Seismic Reflectivity Inversion Network](https://mlsanity.com/paper/2104.04704/) (2021)
- [A Sparsity Algorithm with Applications to Corporate Credit Rating](https://mlsanity.com/paper/2107.10306/) (2021)
- [Mathematical Algorithm Design for Deep Learning under Societal and Judicial Constraints: The Algorithmic Transparency Requirement](https://mlsanity.com/paper/2401.10310/) (2024)
- [WEEP: A Differentiable Nonconvex Sparse Regularizer via Weakly-Convex Envelope](https://mlsanity.com/paper/2507.20447/) (2025)
- [Nonconvex Regularization for Feature Selection in Reinforcement Learning](https://mlsanity.com/paper/2509.15652/) (2025)
- [RINS-T: Robust Implicit Neural Solvers for Time Series Linear Inverse Problems](https://mlsanity.com/paper/2510.17396/) (2025)
- [NuSPAN: A Proximal Average Network for Nonuniform Sparse Model -- Application to Seismic Reflectivity Inversion](https://mlsanity.com/paper/2105.00003/) (2021)
- [Low-Rank Tensor Completion Based on Bivariate Equivalent Minimax-Concave Penalty](https://mlsanity.com/paper/2201.12709/) (2022)
- [Playing the Lottery With Concave Regularizers for Sparse Trainable Neural Networks](https://mlsanity.com/paper/2501.11135/) (2025)
- [Nonconvex Latent Optimally Partitioned Block-Sparse Recovery via Log-Sum and Minimax Concave Penalties](https://mlsanity.com/paper/2603.01304/) (2026)
- [Framelet-Based Blind Image Restoration with Minimax Concave Regularization](https://mlsanity.com/paper/2604.19314/) (2026)
- [Nonlinear mixture model motivated subspace clustering](https://mlsanity.com/paper/2606.29261/) (2026)
- [Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions](https://mlsanity.com/paper/2607.04990/) (2026)

## Abstract (arXiv)

Sparse approximate solutions to linear equations are classically obtained via L1 norm regularized least squares, but this method often underestimates the true solution. As an alternative to the L1 norm, this paper proposes a class of non-convex penalty functions that maintain the convexity of the least squares cost function to be minimized, and avoids the systematic underestimation characteristic of L1 norm regularization. The proposed penalty function is a multivariate generalization of the minimax-concave (MC) penalty. It is defined in terms of a new multivariate generalization of the Huber function, which in turn is defined via infimal convolution. The proposed sparse-regularized least squares cost function can be minimized by proximal algorithms comprising simple computations.