



@article {2258,
	title = {Implicit regularization and solution uniqueness in over-parameterized matrix sensing},
	year = {2018},
	abstract = {<p>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 Rn\&times;n, with a sensing mechanism that satisfies the restricted isometry property (RIP). The algorithm we study is that of \emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization UU⊤, where U\&isin;Rn\&times;r. Surprisingly, recent work argues that the choice of r\&le;n is not pivotal: even setting U\&isin;Rn\&times;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. \emph{I.e.}, under assumptions, the set of PSD matrices, that are consistent with the observed data, is a singleton, irrespective of the algorithm used.</p>},
	url = {https://arxiv.org/abs/1806.02046},
	author = {Anastasios Kyrillidis and Amir Kalev}
}
