Escaping Spurious Local Minima of Low-Rank Matrix Factorization Through Convex Lifting

This work proposes a rapid global solver for nonconvex low-rank matrix factorization (MF) problems that we name MF-Global. Through convex lifting steps, our method efficiently escapes saddle points and spurious local minima ubiquitous in noisy real-world data, and is guaranteed to always converge to the global optima. Moreover, the proposed approach adaptively adjusts the rank for the factorization and provably identifies the optimal rank for MF automatically in the course of optimization through tools of manifold identification, and thus it also spends significantly less time on parameter tuning than existing MF methods, which require an exhaustive search for this optimal rank. On the other hand, when compared to methods for solving the lifted convex form only, MF-Global leads to significantly faster convergence and much shorter running time. Experiments on real-world large-scale recommendation system problems confirm that MF-Global can indeed effectively escapes spurious local solutions at which existing MF approaches stuck, and is magnitudes faster than state-of-the-art algorithms for the lifted convex form.