On the Optimization Landscape of Burer-Monteiro Factorization: When do Global Solutions Correspond to Ground Truth?

In low-rank matrix recovery, the goal is to recover a low-rank matrix, given a limited number of linear and possibly noisy measurements. Low-rank matrix recovery is typically solved via a nonconvex method called Burer-Monteiro factorization (BM). If the rank of the ground truth is known, BM is free of sub-optimal local solutions, and its true solutions coincide with the global solutions -- that is, the true solutions are identifiable. When the rank of the ground truth is unknown, it must be over-estimated, giving rise to an over-parameterized BM. In the noiseless regime, it is recently shown that over-estimation of the rank leads to progressively fewer sub-optimal local solutions while preserving the identifiability of the true solutions. In this work, we show that with noisy measurements, the global solutions of the over-parameterized BM no longer correspond to the true solutions, essentially transmuting over-parameterization from blessing to curse. In particular, we study two classes of low-rank matrix recovery, namely matrix completion and matrix sensing. For matrix completion, we show that even if the rank is only slightly over-estimated and with very mild assumptions on the noise, none of the true solutions are local or global solutions. For matrix sensing, we show that to guarantee the correspondence between global and true solutions, it is necessary and sufficient for the number of samples to scale linearly with the over-estimated rank, which can be drastically larger than its optimal sample complexity that only scales with the true rank.