Optimization-Based AMP for Phase Retrieval: The Impact of Initialization and $\ell_{2}$ Regularization

Abstract

We consider an 2-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting m, n → ∞, m/n → δ and obtain sharp performance bounds, where m is the number of measurements and n is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only m = 64 π 2 − 4 n ≈ 2.5n measurements. Also, we provide sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: (i) Adding 2 regularization to the nonconvex loss function can be beneficial. (ii) Spectral initialization has marginal impact on the performance of the algorithm. The sharp analyses in this paper, not only enable us to compare the performance of our method with other phase recovery schemes, but also shed light on designing better iterative algorithms for other non-convex optimization problems.

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

“…Recall that E[Y ] = 2x x + x 2 2 I n has a bounded spectral norm, then (136) implies that, to keep the deviation between Y and E[Y ] well-controlled, we must at least have (n log m)/m 1.…”

“…A few other nonconvex matrix factorization algorithms have been left out due to space, including but not limited to normalized iterative hard thresholding (NIHT) [129], atomic decomposition for minimum rank approximation (Admira) [130], composite optimization (e.g. prox-linear algorithm) [131][132][133][134], approximate message passing [135][136][137], block coordinate descent [19], coordinate descent [138], and conjugate gradient [139]. The readers are referred to these papers for detailed descriptions.…”

“…Leaving out this boundedness issue, a more severe matter is that these parameters often depend on the problem size. For example, recall that in [27,91] vanilla GD [18, 26, 58-60, 75, 90, 201, 202] [ 46,48,49,95,165,166,203,204] Phase retrieval regularized GD [20,24,79,80,87,87,205] and quadratic sensing alternating minimization [118][119][120]206] approximate message passing [136,137] matrix completion [29,159,160,207] vanilla gradient descent [26] [1, 11, 13, 208-211] regularized Grassmanian GD [16,65] projected / regularized GD [19,21,63] alternating minimization [17,123,124] blind deconvolution / demixing vanilla GD [26,62] [5, 6] (subspace model) regularized GD [22,66,212] robust PCA [159] low-rank projection + thresholding [25] [7, 8, 213, 214] GD + thresholding…”

Gradient descent with random initialization: fast global convergence for nonconvex phase retrieval

“…Almost all of these nonconvex methods require carefully-designed initialization to guarantee a sufficiently accurate initial point. One exception is the approximate message passing algorithm proposed in [MXM18], which works as long as the correlation between the truth and the initial signal is bounded away from zero. This, however, does not accommodate the case when the initial signal strength is vanishingly small (like random initialization).…”

Optimal Spectral Initialization for Signal Recovery With Applications to Phase Retrieval

“…An active line of recent work studies nonconvex optimization algorithms for solving the classical phase retrieval problem (see, e.g., [1]- [8]). Compared to methods using convex relaxation [9]- [12], the nonconvex approaches tend to require much lower computational complexity and memory footprints.…”