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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 onrecent advances in non-convex optimization schemes. We apply this in particularto the phase retrieval problem for Fourier data, which can be formulated as arank 1 PSD matrix recovery problem. Moreover, we provide theory for howoversampling affects the stability of the lifted inverse problem.

On phase retrieval via matrix completion and the estimation of low rank PSD matrices