The Proximity Operator of the Log-Sum Penalty

Ashley Prater-Bennette, Lixin Shen, Erin E. Tripp

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


The log-sum penalty is often adopted as a replacement for the ℓ pseudo-norm in compressive sensing and low-rank optimization. The proximity operator of the ℓ penalty, i.e., the hard-thresholding operator, 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 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. We present applications to solving compressive sensing problems and to mixed additive Gaussian white noise and impulse noise removal.

Original languageEnglish (US)
Article number67
JournalJournal of Scientific Computing
Issue number3
StatePublished - Dec 2022


  • Compressive sensing
  • Iteratively reweighted ℓ
  • Log-sum function
  • Low-rank regularization
  • Proximal mapping
  • Proximity operator

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software
  • Numerical Analysis
  • General Engineering
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


Dive into the research topics of 'The Proximity Operator of the Log-Sum Penalty'. Together they form a unique fingerprint.

Cite this