Low Complexity Regularized Phase Retrieval

TL;DR

This work develops a unified, geometry-based analysis for low-complexity regularized phase retrieval, providing both exact recovery in the noiseless setting and stable recovery under noise. Central to the theory are deterministic conditions on the descent cone of the regularizer and probabilistic bounds for Gaussian measurement maps, expressed via the Gaussian width of the descent cone. The results cover decomposable regularizers (e.g., Lasso, group Lasso), frame-analytic priors, and total variation, with explicit sample complexity scaling in the intrinsic dimension rather than the ambient dimension. The paper also studies two practical recovery formulations (constrained Mozorov and penalized Tikhonov), establishes convergence and rate results for the penalized problem, and supports the theory with numerical experiments using a Bregman Proximal Gradient algorithm. Overall, it extends phase retrieval beyond sparsity to a broad class of structured priors, offering nearly optimal sample complexity under Gaussian measurements and guiding algorithmic choices for robust reconstruction.

Abstract

In this paper, we study the phase retrieval problem in the situation where the vector to be recovered has an a priori structure that can encoded into a regularization term. This regularizer is intended to promote solutions conforming to some notion of simplicity or low complexity. We investigate both noiseless recovery and stability to noise and provide a very general and unified analysis framework that goes far beyond the sparse phase retrieval mostly considered in the literature. In the noiseless case we provide sufficient conditions under which exact recovery, up to global sign change, is possible. For Gaussian measurement maps, we also provide a sample complexity bound for exact recovery. This bound depends on the Gaussian width of the descent cone at the soughtafter vector which is a geometric measure of the complexity of the latter. In the noisy case, we consider both the constrained (Mozorov) and penalized (Tikhonov) formulations. We provide sufficient conditions for stable recovery and prove linear convergence for sufficiently small noise. For Gaussian measurements, we again give a sample complexity bound for linear convergence to hold with high probability. This bound scales linearly in the intrinsic dimension of the sought-after vector but only logarithmically in the ambient dimension.

Figures (5)

• Figure 1: Phase retrieval with the Lasso ($\ell_1-$norm) regularizer.
• Figure 2: Stable phase retrieval with the Lasso (left) and TV (right).
• Figure 3: Phase retrieval with the group Lasso ($\ell_1-\ell_2$ norm) regularizer.
• Figure 4: Phase retrieval with the TV semi-norm.
• Figure 5: Phase retrieval with the wavelet-synthesis prior formulation.

Theorems & Definitions (53)

• Remark 2.1
• Definition 2.2: Support function
• Definition 2.3: Polar set
• Definition 2.4: Gauge
• Definition 2.5: Asymptotic cone
• Definition 2.7
• Definition 2.8
• Proposition 2.9
• Proposition 2.10
• Proposition 3.1