Low-Rank Regularization And Solution Uniqueness In Over-Parameterized Matrix Sensing

We consider the question whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit low-rank regularization. We focus on the noiseless matrix sensing scenario over low-rank positive semi-definite (PSD) matrices over the reals, with a sensing mechanism that satisfies restricted isometry properties. Surprisingly, it was recently argued that for recovery of PSD matrices, gradient descent over a squared, full-rank factorized space introduces implicit low-rank regularization. Thus, a clever choice of the recovery algorithm avoids the need for explicit low-rank regularization. In this contribution, we prove that in fact, under certain conditions, the PSD constraint by itself is sufficient to lead to a unique low-rank matrix recovery, without explicit or implicit regularization. Therefore, under these conditions, the set of PSD matrices that are consistent with the observed data, is a singleton, regardless of the algorithm used. Our numerical study indicates that this result is general and extends to cases beyond the those covered by the proof.