Phase Retrieval via Sensor Network Localization

TL;DR

This paper introduces a novel approach to phase retrieval by linking it to sensor network localization, enabling the development of an efficient two-stage algorithm with improved measurement requirements and faster reconstruction.

Contribution

It establishes a new connection between phase retrieval and sensor network localization, leading to a provably effective two-stage recovery algorithm and a new phase retrieval formulation related to complex rigidity theory.

Findings

The proposed algorithm improves measurement efficiency in both sparse and dense cases.

Numerical results validate the theoretical guarantees and demonstrate the algorithm's efficiency.

A new phase retrieval problem formulation is introduced, connecting to complex rigidity theory.

Abstract

The problem of phase retrieval is revisited and studied from a fresh perspective. In particular, we establish a connection between the phase retrieval problem and the sensor network localization problem, which allows us to utilize the vast theoretical and algorithmic literature on the latter to tackle the former. Leveraging this connection, we develop a two-stage algorithm for phase retrieval that can provably recover the desired signal. In both sparse and dense settings, our proposed algorithm improves upon prior approaches simultaneously in the number of required measurements for recovery and the reconstruction time. We present numerical results to corroborate our theory and to demonstrate the efficiency of the proposed algorithm. As a side result, we propose a new form of phase retrieval problem and connect it to the complex rigidity theory proposed by Gortler and Thurston.