ICML 2025

Guarantees of a Preconditioned Subgradient Algorithm for
Overparameterized Asymmetric Low-rank Matrix Recovery
Paris Giampouras 1 HanQin Cai 2 Ren´e Vidal 3
Abstract
In this paper, we focus on a matrix factorization-
based approach to recover low-rank asymmet-
ric matrices from corrupted measurements. We
propose an Overparameterized Preconditioned
Subgradient Algorithm (OPSA) and provide, for
the first time in the literature, linear convergence
rates independent of the rank of the sought asym-
metric matrix in the presence of gross corrup-
tions. Our work goes beyond existing results in
preconditioned-type approaches addressing their
current limitation, i.e., the lack of convergence
guarantees in the case of asymmetric matrices
of unknown rank. By applying our approach to
(robust) matrix sensing, we highlight its merits
when the measurement operator satisfies a mixed-
norm restricted isometry property. Lastly, we
present extensive numerical experiments that val-
idate our theoretical results and demonstrate the
effectiveness of our approach for different levels
of overparameterization and outlier corruptions.
1. Introduction
Low-rank matrix recovery has been a ubiquitous problem
showing up in numerous applications in the fields of sig-
nal/image processing, machine learning, and data science
(Recht et al., 2010; Chen et al., 2013; Davenport & Romberg,
2016; Cai et al., 2021a; Smith et al., 2024; Wang et al., 2024).
For instance, problems such as matrix sensing (Jain et al.,
2013; Wei et al., 2016; Park et al., 2017; Li et al., 2020),
matrix completion (Candes & Recht, 2008; Nie et al., 2012;
1Department of Computer Science, University of Warwick,
Coventry, England, UK 2Department of Statistics and Data Sci-
ence and Department of Computer Science, University of Central
Florida, Orlando, FL, USA 3Center for Innovation in Data Engi-
neering and Science (IDEAS), Departments of ESE and Radiology,
University of Pennsylvania, Philadelphia, PA, USA. Correspon-
dence to: Re...