Sample-Efficient Low Rank Phase Retrieval

TL;DR

This paper advances the theoretical understanding of low rank phase retrieval by providing improved guarantees for an alternating minimization algorithm, demonstrating geometric convergence and stability in noisy settings.

Contribution

It offers a significantly improved sample complexity analysis for AltMinLowRaP, extending guarantees to noisy scenarios and refining convergence conditions.

Findings

Geometric convergence of AltMinLowRaP under certain measurement conditions

Improved sample complexity by a factor of r^2 for the algorithm

Stability of the method in the presence of small additive noise

Abstract

This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an $n \times q$ rank-$r$ matrix $X^*$ from $y_k = |A_k^\top x^*_k|$, $k=1, 2,..., q$, when each $y_k$ is an m-length vector containing independent phaseless linear projections of $x^*_k$. The different matrices $A_k$ are i.i.d. and each contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was introduced and studied in our recent work. As long as the right singular vectors of $X^*$ satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to $X^*$ if the total number of measurements $mq \gtrsim nr^2 (r + \log(1/\epsilon))$. In addition, we also need $m \gtrsim max(r, \log q, \log n)$ because of the specific asymmetric nature of our problem. Compared to our recent work, we improve the sample complexity of the AltMin iterations by a factor of $r^2$, and that of the initialization by a factor of $r$. We also extend our result to the noisy case; we prove stability to corruption by small additive noise.