Provable Low Rank Phase Retrieval and Compressive PCA

We study the Low Rank Phase Retrieval (LRPR) problem defined as follows: recover an $n \times q$ matrix $X^*$ of rank $r$ from a different and independent set of $m$ phaseless (magnitude-only) linear projections of each of its columns. To be precise, we need to recover $X^*$ from $y_k := |A_k{}' x^*_k|, k=1,2,\dots, q$ when the measurement matrices $A_k$ are mutually independent. Here $y_k \in \mathbb{R}^m$. The question is when can we solve LRPR with $m \ll n$? Our work introduces the first provably correct solution, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), for solving LRPR. We demonstrate its advantage over existing work via extensive simulation, and some partly real data, experiments. Our guarantee for AltMinLowRaP shows that it can solve LRPR to $\epsilon$ accuracy if $m q \ge C n r^4 \log(1/\epsilon)$, the matrices $A_k$ contain i.i.d. standard Gaussian entries, the condition number of $X^*$ is bounded by a numerical constant, and its right singular vectors satisfy the incoherence (denseness) assumption from matrix completion literature. Its time complexity is only $C mq nr \log^2(1/\epsilon)$. In the regime of small $r$, our sample complexity is much better than what standard PR methods need; and it is only about $r^3$ times worse than its order-optimal value of $(n + q) r$. Moreover, if we replace $m$ by its lower bound for each approach, then the same can be said for the time complexity comparison with standard PR. We also briefly study the dynamic extension of LRPR. The LRPR problem occurs in phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens, where acquiring measurements is expensive. We should point out that LRPR is a very different problem than its $A_k =A$ version, or its $A_k = A$ and with-phase (linear) version, both of which have been extensively studied in the literature.