Provably accurate recovery for low-rank matrix phase retrieval with sublinear measurements

Construct efficient algorithms for low-rank matrix phase retrieval that achieve provable accuracy when the number of measurements satisfies m < n1 n2, and devise initialization strategies with theoretical guarantees enabling global recovery from random quadratic measurements.

Background

Matrix phase retrieval aims to recover a low-rank matrix from quadratic (phaseless) measurements. Known results require strong initialization or large sample regimes; the authors note the absence of efficient, provably accurate methods when the sample size is below the ambient product n1 n2.

This gap affects practical applicability in high-dimensional regimes, and the paper highlights that even providing a suitable initializer is currently unresolved in the low-rank matrix case, indicating a broader challenge for developing globally convergent approaches.