  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 $\mathbb{R}^{n
\times n}$, with a sensing mechanism that satisfies restricted isometry
properties (RIP). The algorithm we study is \emph{factored gradient descent},
where we model the low-rankness and PSD constraints with the factorization
$UU^\top$, for $U \in \mathbb{R}^{n \times r}$. Surprisingly, recent work
argues that the choice of $r \leq n$ is not pivotal: even setting $U \in
\mathbb{R}^{n \times n}$ 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 contribution, we provide a different
perspective to the problem of implicit regularization. We show that 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, regardless of the
algorithm used.
