  Given underdetermined measurements of a Positive Semi-Definite (PSD) matrix
$X$ of known low rank $K$, we present a new algorithm to estimate $X$ based on
recent advances in non-convex optimization schemes. We apply this in particular
to the phase retrieval problem for Fourier data, which can be formulated as a
rank 1 PSD matrix recovery problem. Moreover, we provide theory for how
oversampling affects the stability of the lifted inverse problem.
