Low rank phase retrieval

We study the problem of recovering a low-rank matrix, X, from phaseless measurements of random linear projections of its columns. We develop a novel solution approach, called AltMinTrunc, that consists of a two-step truncated spectral initialization step, followed by a three-step alternating minimization algorithm. We obtain sample complexity bounds for the AltMinTrunc initialization to provide a good approximation of the true X. When the rank of X is low enough, these are significantly smaller than what existing single vector phase retrieval algorithms need. Via extensive experiments, we demonstrate the same for the entire algorithm.