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

## TL;DR

This paper introduces a new non-convex optimization algorithm for estimating low-rank PSD matrices from underdetermined measurements, with applications to phase retrieval and analysis of oversampling effects on stability.

## Contribution

It proposes a novel algorithm for low-rank PSD matrix recovery and provides theoretical insights into the stability of phase retrieval under oversampling.

## Key findings

- The algorithm effectively estimates low-rank PSD matrices from limited data.
- Oversampling improves the stability of the phase retrieval problem.
- Theoretical analysis links oversampling to enhanced robustness in matrix recovery.

## Abstract

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.