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 $y_k$ is an $m$ length vector, $A_k$ is an $n \times m$
matrix, and $'$ denotes matrix transpose. The question is when can we solve
LRPR with $m \ll n$? A reliable solution can enable fast and low-cost phaseless
dynamic imaging, e.g., Fourier ptychographic imaging of live biological
specimens. In this work, we develop the first provably correct approach for
solving this LRPR problem. Our proposed algorithm, Alternating Minimization for
Low-Rank Phase Retrieval (AltMinLowRaP), is an AltMin based solution and hence
is also provably fast (converges geometrically). Our guarantee shows that
AltMinLowRaP solves LRPR to $\epsilon$ accuracy, with high probability, as long
as $m q \ge C n r^4 \log(1/\epsilon)$, the matrices $A_k$ contain i.i.d.
standard Gaussian entries, and the right singular vectors of $X^*$ satisfy the
incoherence assumption from matrix completion literature. Here $C$ is a
numerical constant that only depends on the condition number of $X^*$ and on
its incoherence parameter. Its time complexity is only $ C mq nr
\log^2(1/\epsilon)$. Since even the linear (with phase) version of the above
problem is not fully solved, the above result is also the first complete
solution and guarantee for the linear case. Finally, we also develop a simple
extension of our results for the dynamic LRPR setting.

本文研究了低秩相位恢复问题，提出了一种基于 AltMin 算法的解决方案，并给出了理论保证。此方法能有效地实现低成本、高速的相位动态成像，对生物样本的傅里叶 ptychographic 成像具有重要应用价值。

可证明的低秩相位恢复

Provable Low Rank Phase Retrieval

We develop two iterative algorithms for solving the low rank phase retrieval
(LRPR) problem. LRPR refers to recovering a low-rank matrix $\X$ from
magnitude-only (phaseless) measurements of random linear projections of its
columns. Both methods consist of a spectral initialization step followed by an
iterative algorithm to maximize the observed data likelihood. We obtain sample
complexity bounds for our proposed initialization approach to provide a good
approximation of the true $\X$. When the rank is low enough, these bounds are
significantly lower than what existing single vector phase retrieval algorithms
need. Via extensive experiments, we show that the same is also true for the
proposed complete algorithms.

本文提出了两种迭代算法来解决低秩相位恢复问题，这些算法由谱初始化步骤和迭代算法组成，旨在最大化观测数据可能性，并得到了相应的样本复杂度界限。经过大量实验，表明这些算法在低秩相位恢复问题上具有良好的效果。