Implicit Regularization of Sub-Gradient Method in Robust Matrix Recovery: Don't be Afraid of Outliers

It is well-known that simple short-sighted algorithms, such as gradient descent, generalize well in the over-parameterized learning tasks, due to their implicit regularization. However, it is unknown whether the implicit regularization of these algorithms can be extended to robust learning tasks, where a subset of samples may be grossly corrupted with noise. In this work, we provide a positive answer to this question in the context of robust matrix recovery problem. In particular, we consider the problem of recovering a low-rank matrix from a number of linear measurements, where a subset of measurements are corrupted with large noise. We show that a simple sub-gradient method converges to the true low-rank solution efficiently, when it is applied to the over-parameterized l1-loss function without any explicit regularization or rank constraint. Moreover, by building upon a new notion of restricted isometry property, called sign-RIP, we prove the robustness of the sub-gradient method against outliers in the over-parameterized regime. In particular, we show that, with Gaussian measurements, the sub-gradient method is guaranteed to converge to the true low-rank solution, even if an arbitrary fraction of the measurements are grossly corrupted with noise.