Implicit regularization and solution uniqueness in over-parameterized matrix sensing
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We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-r positive semi-definite (PSD) matrices in R^n × n, with a sensing mechanism that satisfies the restricted isometry property (RIP). The algorithm we study is that of factored gradient descent, where we model the low-rankness and PSD constraints with the factorization UU^, where U ∈R^n × r. Surprisingly, recent work argues that the choice of r ≤ n is not pivotal: even setting U ∈R^n × n is sufficient for factored gradient descent to find the rank-r solution, which suggests that operating over the factors leads to an implicit regularization. In this note, we provide a different perspective. We show that, in the noiseless case, under certain conditions, the PSD constraint by itself is sufficient to lead to a unique rank-r matrix recovery, without implicit or explicit low-rank regularization. I.e., under assumptions, the set of PSD matrices, that are consistent with the observed data, is a singleton, irrespective of the algorithm used.
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