Phaseless PCA: Low-Rank Matrix Recovery from Column-wise Phaseless Measurements

We study the following problem: recover a low-rank matrix from phaseless (magnitude-only) linear projections of each of its columns. In analogy with Robust PCA, we refer to this problem as "Phaseless PCA". It finds important applications in phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens. This work introduces the first provably correct solution, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), for this problem. Our guarantee for AltMinLowRaP shows that it can recover an $n \times q$ matrix of rank $r$ to $\epsilon$ accuracy using only about $nr^4 \log(1/\epsilon)$ measurements, as long as the matrix condition number is bounded by a numerical constant. This sample complexity is only $r^3$ times worse than the order-optimal value of $\max(n,q) r$. The only other assumption needed for our result is incoherence of the right singular vectors of the matrix. We demonstrate the practical advantage of AltMinLowRaP over existing work via extensive simulation, and some real-data, experiments. We also briefly study the dynamic extension of the above problem.