  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 \textit{smooth}
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 \textit{strict 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 \textit{mirror-prox} methods: the Euclidean \textit{extragradient
method} and mirror-prox with \textit{matrix exponentiated gradient updates},
when initialized with a "warm-start", converge to an optimal solution with rate
$O(1/t)$, while requiring only two \textit{low-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.
