Phaseless PCA: Low-Rank Matrix Recovery from Column-wise Phaseless Measurements
Phaseless PCA: Low-Rank Matrix Recovery from Column-wise Phaseless Measurements
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发表时间：
2019-02
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通讯作者：
Seyedehsara Nayer;Namrata Vaswani;Yonina C. Eldar
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作者：
Seyedehsara Nayer;Namrata Vaswani;Yonina C. Eldar
我们研究了以下问题：恢复一个低秩矩阵从无相位（幅度只有）线性投影的每一列。与鲁棒PCA类似，我们将此问题称为“无相位PCA”。它在无相位动态成像中有重要的应用，例如，活体生物标本的傅立叶重叠关联成像。这项工作为该问题引入了第一个可证明正确的解决方案，即低秩相位检索的交替最小化（AltMinLowRaP）。我们对AltMinLowRaP的保证表明，只要矩阵条件数受数值常数的限制，它就可以使用大约nr^4\log（1/\log）$的测量值将秩为r$的$n \times q$矩阵恢复到$\n $精度。这个样本复杂度只比$\max（n，q）r$的阶最优值差$r^3$倍。我们的结果所需的唯一其他假设是矩阵的右奇异向量的不相干性。我们通过大量的模拟和一些真实的数据，实验证明了AltMinLowRaP在现有工作中的实际优势。我们还简要地研究了上述问题的动态扩展。
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.