Provable Low Rank Phase Retrieval

Abstract

We study the "Low Rank Phase Retrieval (LRPR)" problem defined as follows: recover an n × 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, . . . , q when the measurement matrices A k are mutually independent. Here y k is an m length vector and denotes transpose. The question is when can we solve LRPR with m 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 accuracy if mq ≥ Cnr 4 log(1/ ), 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 Cmqnr log 2 (1/ ). 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.for each k. If we assume κ is a constant, up to constant factors, (4) also implies (3). Thus, up to constant factors, requiring right incoherence is the same as requiring that the maximum energy of any signal x * k is within constant factors of the average.

Sample-Efficient Low Rank Phase Retrieval

“…We also showed extensive numerical experiments that demonstrated the practical power of AltMinLowRaP. Our guarantee from [14] showed that, if right incoherence holds, if a new set of samples was used in each iteration, and for each update of U and B * (sample-splitting), and if the total number of samples mq ≥ C κ,µ •nr 4 •log(1/ǫ), and if m ≥ C max(r, log q, log n), one can recover X * to ǫ accuracy by using AltMinLowRaP. Here C κ,µ = Cκ 10 µ 4 .…”

“…Low rank is another common assumption. As explained in [13], [14], the practical way to impose it is to consider joint recovery of a set q of correlated signals, that together form an (exactly or approximately) low rank matrix, from m different phaseless linear projections of each of the q signals. This model is useful to enable fast and low-cost dynamic phaseless imaging applications, such as dynamic Fourier ptychography, where measurement acquisition is slow or expensive [15].…”

“…3) Related Work and Our Contributions: LRPR was first studied in [13] where we introduced an alternating minimization (AltMin) algorithm and analyzed its initialization step. In recent work [14] (and shorter version in [18]), we developed the first provably correct LRPR solution, that we called AltMinLowRaP (AltMin for Low Rank PR). We also showed extensive numerical experiments that demonstrated the practical power of AltMinLowRaP.…”

“…See Lemma 3.7. (ii) Secondly, in [14], we only considered real-valued Gaussian measurements. In this work, we extend our result to also handle complex-valued Gaussian measurements.…”

“…The linear version of LRPR, PCA via random projections or "compressive PCA", has received little attention until recently. As explained in [14], there have been some older attempts to develop a solution and try to analyze sub-parts of it [20], [21], [22]. AltMinLowRaP [14], [18] can be understood as the first provably correct algorithm for this problem.…”