# Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence

## TL;DR

This paper demonstrates that nonsmooth formulations for low-rank matrix recovery are better conditioned and enable faster, dimension-independent convergence of optimization algorithms, also offering robustness to outliers.

## Contribution

It shows that nonsmooth penalty formulations avoid ill-conditioning in low-rank matrix recovery, leading to rapid convergence and robustness, unifying several key problems.

## Key findings

- Nonsmooth formulations have better conditioning than smooth ones.
- Standard algorithms converge rapidly and dimension-independently.
- Nonsmooth methods are robust to outliers.

## Abstract

The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, nonsmooth penalty formulations do not suffer from the same type of ill-conditioning. Consequently, standard algorithms for nonsmooth optimization, such as subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. Moreover, nonsmooth formulations are naturally robust against outliers. Our framework subsumes such important computational tasks as phase retrieval, blind deconvolution, quadratic sensing, matrix completion, and robust PCA. Numerical experiments on these problems illustrate the benefits of the proposed approach.