# Smoothed Low Rank and Sparse Matrix Recovery by Iteratively Reweighted Least Squares Minimization

arXiv: https://arxiv.org/abs/1401.7413 · HTML: https://mlsanity.com/paper/1401.7413/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2014.

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

- [Optimistic Optimisation of Composite Objective with Exponentiated Update](https://mlsanity.com/paper/2208.04065/) (2022)
- [Structure-Preserving Progressive Low-rank Image Completion for Defending Adversarial Attacks](https://mlsanity.com/paper/2103.02781/) (2021)
- [Stable and Compact Face Recognition via Unlabeled Data Driven Sparse Representation-Based Classification](https://mlsanity.com/paper/2111.02847/) (2021)

## Abstract (arXiv)

This work presents a general framework for solving the low rank and/or sparse matrix minimization problems, which may involve multiple non-smooth terms. The Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables and their weights. However, the traditional IRLS can only solve a sparse only or low rank only minimization problem with squared loss or an affine constraint. This work generalizes IRLS to solve joint/mixed low rank and sparse minimization problems, which are essential formulations for many tasks. As a concrete example, we solve the Schatten-p norm and ℓ2,q-norm regularized Low-Rank Representation (LRR) problem by IRLS, and theoretically prove that the derived solution is a stationary point (globally optimal if p,q≥1). Our convergence proof of IRLS is more general than previous one which depends on the special properties of the Schatten-p norm and ℓ2,q-norm. Extensive experiments on both synthetic and real data sets demonstrate that our IRLS is much more efficient.