Over-Parametrized Matrix Factorization in the Presence of Spurious Stationary Points

Motivated by the emerging role of interpolating machines in signal processing and machine learning, this work considers the computational aspects of over-parametrized matrix factorization. In this context, the optimization landscape may contain spurious stationary points (SSPs), which are proved to be full-rank matrices. The presence of these SSPs means that it is impossible to hope for any global guarantees in over-parametrized matrix factorization. For example, when initialized at an SSP, the gradient flow will be trapped there forever. Nevertheless, despite these SSPs, we establish in this work that the gradient flow of the corresponding merit function converges to a global minimizer, provided that its initialization is rank-deficient and sufficiently close to the feasible set of the optimization problem. We numerically observe that a heuristic discretization of the proposed gradient flow, inspired by primal-dual algorithms, is successful when initialized randomly. Our result is in sharp contrast with the local refinement methods which require an initialization close to the optimal set of the optimization problem. More specifically, we successfully avoid the traps set by the SSPs because the gradient flow remains rank-deficient at all times, and not because there are no SSPs nearby. The latter is the case for the local refinement methods. Moreover, the widely-used restricted isometry property plays no role in our main result.