# Factor Group-Sparse Regularization for Efficient Low-Rank Matrix Recovery

arXiv: https://arxiv.org/abs/1911.05774 · HTML: https://mlsanity.com/paper/1911.05774/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Nov 2019.

## Built on by (most cited first, 15 of 15)

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- [Deep Distribution-preserving Incomplete Clustering with Optimal Transport](https://mlsanity.com/paper/2103.11424/) (2021)
- [Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization](https://mlsanity.com/paper/2406.15713/) (2024)
- [Tuning-Free Online Robust Principal Component Analysis through Implicit Regularization](https://mlsanity.com/paper/2409.07275/) (2024)
- [Tailed Low-Rank Matrix Factorization for Similarity Matrix Completion](https://mlsanity.com/paper/2409.19550/) (2024)
- [Simplifying Optimal Transport through Schatten-p Regularization](https://mlsanity.com/paper/2510.11910/) (2025)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)
- [Dynamic Proximal Gradient Algorithms for Schatten-p Quasi-Norm Regularized Problems](https://mlsanity.com/paper/2603.00333/) (2026)

## Abstract (arXiv)

This paper develops a new class of nonconvex regularizers for low-rank matrix recovery. Many regularizers are motivated as convex relaxations of the matrix rank function. Our new factor group-sparse regularizers are motivated as a relaxation of the number of nonzero columns in a factorization of the matrix. These nonconvex regularizers are sharper than the nuclear norm; indeed, we show they are related to Schatten-p norms with arbitrarily small 0 < p ≤ 1. Moreover, these factor group-sparse regularizers can be written in a factored form that enables efficient and effective nonconvex optimization; notably, the method does not use singular value decomposition. We provide generalization error bounds for low-rank matrix completion which show improved upper bounds for Schatten-p norm reglarization as p decreases. Compared to the max norm and the factored formulation of the nuclear norm, factor group-sparse regularizers are more efficient, accurate, and robust to the initial guess of rank. Experiments show promising performance of factor group-sparse regularization for low-rank matrix completion and robust principal component analysis.