# The Proximity Operator of the Log-Sum Penalty

arXiv: https://arxiv.org/abs/2103.02681 · HTML: https://mlsanity.com/paper/2103.02681/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2021.

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

- [Power of ℓ1-Norm Regularized Kaczmarz Algorithms for High-Order Tensor Recovery](https://mlsanity.com/paper/2405.08275/) (2024)
- [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)

The log-sum penalty is often adopted as a replacement for the ℓ0 pseudo-norm in compressive sensing and low-rank optimization. The hard-thresholding operator, i.e., the proximity operator of the ℓ0 penalty, plays an essential role in applications; similarly, we require an efficient method for evaluating the proximity operator of the log-sum penalty. Due to the nonconvexity of this function, its proximity operator is commonly computed through the iteratively reweighted ℓ1 method, which replaces the log-sum term with its first-order approximation. This paper reports that the proximity operator of the log-sum penalty actually has an explicit expression. With it, we show that the iteratively reweighted ℓ1 solution disagrees with the true proximity operator of the log-sum penalty in certain regions. As a by-product, the iteratively reweighted ℓ1 solution is precisely characterized in terms of the chosen initialization. We also give the explicit form of the proximity operator for the composition of the log-sum penalty with the singular value function, as seen in low-rank applications. These results should be useful in the development of efficient and accurate algorithms for optimization problems involving the log-sum penalty.