# Structured Sparsity Promoting Functions

arXiv: https://arxiv.org/abs/1809.06777 · HTML: https://mlsanity.com/paper/1809.06777/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2018.

## Built on by (most cited first, 4 of 4)

- [Sparse Deep Learning Models with the ℓ1 Regularization](https://mlsanity.com/paper/2408.02801/) (2024)
- [Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control](https://mlsanity.com/paper/2507.19895/) (2025)
- [A block-coordinate descent framework for non-convex composite optimization. Application to sparse precision matrix estimation](https://mlsanity.com/paper/2601.21467/) (2026)
- [MG-SpaIR: Multi-grade Sparse-guided Implicit Representation for Training-Data-Free Image Restoration](https://mlsanity.com/paper/2607.00138/) (2026)

## Abstract (arXiv)

Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we provide sparsity guarantees for the general family of functions. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and the linear combinations of them. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases.