Sample-Efficient Low Rank Phase Retrieval

Abstract

In this paper we obtain an improved guarantee for the Low Rank Phase Retrieval (LRPR) problem: recover an n × q rank-r matrix X * from y. . , q, when each y k is an m-length vector containing the m phaseless linear projections of column x * k . We study the AltMinLowRaP algorithm which is a fast non-convex LRPR solution developed in our previous work. We show that, as long as the right singular vectors of X * satisfy the incoherence assumption, and the measurements' matrix A k consists of i.i.d. real or complex standard Gaussian entries, we can recover X * to ǫ accuracy if mq ≥ C nr 3 log(1/ǫ) and m ≥ C max(r, log q, log n) log(1/ǫ). Thus the per column (per signal) sample complexity is only order max( n q r 3 , r, log q, log n) log(1/ǫ). This improves upon our previous work by a factor of r; moreover our previous result only allowed real-valued Gaussian measurements. Finally, the above result also provides an immediate corollary for the linear (with phase) version of the LRPR problem, often referred to in literature as "PCA via random projections" or "compressive PCA".