# Low-Rank Mirror-Prox for Nonsmooth and Low-Rank Matrix Optimization Problems

arXiv: https://arxiv.org/abs/2206.11523 · HTML: https://mlsanity.com/paper/2206.11523/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 48 references that are in the map.
- First posted on arXiv 23 Jun 2022.

## Builds on (most cited first, 26 of 26)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [Non-convex Robust PCA](https://mlsanity.com/paper/1410.7660/) (2014)
- [Dropping Convexity for Faster Semi-definite Optimization](https://mlsanity.com/paper/1509.03917/) (2015)
- [Fast Algorithms for Robust PCA via Gradient Descent](https://mlsanity.com/paper/1605.07784/) (2016)
- [Near-optimal bounds for phase synchronization](https://mlsanity.com/paper/1703.06605/) (2017)
- [Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence](https://mlsanity.com/paper/1904.10020/) (2019)
- [Tightness of the maximum likelihood semidefinite relaxation for angular synchronization](https://mlsanity.com/paper/1411.3272/) (2014)
- [A Unified Approach to Error Bounds for Structured Convex Optimization Problems](https://mlsanity.com/paper/1512.03518/) (2015)
- [Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs](https://mlsanity.com/paper/1804.02008/) (2018)
- [Projection Efficient Subgradient Method and Optimal Nonsmooth Frank-Wolfe Method](https://mlsanity.com/paper/2010.01848/) (2020)
- [Generalized Nonconvex Nonsmooth Low-Rank Minimization](https://mlsanity.com/paper/1404.7306/) (2014)
- [Large-Scale Low-Rank Matrix Learning with Nonconvex Regularizers](https://mlsanity.com/paper/1708.00146/) (2017)
- [Stochastic Frank-Wolfe for Composite Convex Minimization](https://mlsanity.com/paper/1901.10348/) (2019)
- [On the Convergence of Projected-Gradient Methods with Low-Rank Projections for Smooth Convex Minimization over Trace-Norm Balls and Related Problems](https://mlsanity.com/paper/1902.01644/) (2019)
- [Spectral Frank-Wolfe Algorithm: Strict Complementarity and Linear Convergence](https://mlsanity.com/paper/2006.01719/) (2020)
- [Scalable Robust Matrix Recovery: Frank-Wolfe Meets Proximal Methods](https://mlsanity.com/paper/1403.7588/) (2014)
- [Linear Convergence of Frank-Wolfe for Rank-One Matrix Recovery Without Strong Convexity](https://mlsanity.com/paper/1912.01467/) (2019)
- [kFW: A Frank-Wolfe style algorithm with stronger subproblem oracles](https://mlsanity.com/paper/2006.16142/) (2020)
- [Fast Stochastic Algorithms for Low-rank and Nonsmooth Matrix Problems](https://mlsanity.com/paper/1809.10477/) (2018)
- [Frank-Wolfe Works for Non-Lipschitz Continuous Gradient Objectives: Scalable Poisson Phase Retrieval](https://mlsanity.com/paper/1602.00724/) (2016)
- [On the Convergence of Stochastic Gradient Descent with Low-Rank Projections for Convex Low-Rank Matrix Problems](https://mlsanity.com/paper/2001.11668/) (2020)
- [On the Efficient Implementation of the Matrix Exponentiated Gradient Algorithm for Low-Rank Matrix Optimization](https://mlsanity.com/paper/2012.10469/) (2020)
- [Low-Rank Extragradient Method for Nonsmooth and Low-Rank Matrix Optimization Problems](https://mlsanity.com/paper/2202.04026/) (2022)

## Abstract (arXiv)

Low-rank and nonsmooth matrix optimization problems capture many fundamental tasks in statistics and machine learning. While significant progress has been made in recent years in developing efficient methods for textitsmooth low-rank optimization problems that avoid maintaining high-rank matrices and computing expensive high-rank SVDs, advances for nonsmooth problems have been slow paced. In this paper we consider standard convex relaxations for such problems. Mainly, we prove that under a textitstrict complementarity condition and under the relatively mild assumption that the nonsmooth objective can be written as a maximum of smooth functions, approximated variants of two popular textitmirror-prox methods: the Euclidean textitextragradient method and mirror-prox with textitmatrix exponentiated gradient updates, when initialized with a "warm-start", converge to an optimal solution with rate O(1/t), while requiring only two textitlow-rank SVDs per iteration. Moreover, for the extragradient method we also consider relaxed versions of strict complementarity which yield a trade-off between the rank of the SVDs required and the radius of the ball in which we need to initialize the method. We support our theoretical results with empirical experiments on several nonsmooth low-rank matrix recovery tasks, demonstrating both the plausibility of the strict complementarity assumption, and the efficient convergence of our proposed low-rank mirror-prox variants.