  We study the robust recovery of a low-rank matrix from sparsely and grossly
corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank.
We consider the robust matrix factorization approach. We employ a robust
$\ell_1$ loss function and deal with the challenge of the unknown rank by using
an overspecified factored representation of the matrix variable. We then solve
the associated nonconvex nonsmooth problem using a subgradient method with
diminishing stepsizes. We show that under a regularity condition on the sensing
matrices and corruption, which we call restricted direction preserving property
(RDPP), even with rank overspecified, the subgradient method converges to the
exact low-rank solution at a sublinear rate. Moreover, our result is more
general in the sense that it automatically speeds up to a linear rate once the
factor rank matches the unknown rank. On the other hand, we show that the RDPP
condition holds under generic settings, such as Gaussian measurements under
independent or adversarial sparse corruptions, where the result could be of
independent interest. Both the exact recovery and the convergence rate of the
proposed subgradient method are numerically verified in the overspecified
regime. Moreover, our experiment further shows that our particular design of
diminishing stepsize effectively prevents overfitting for robust recovery under
overparameterized models, such as robust matrix sensing and learning robust
deep image prior. This regularization effect is worth further investigation.
