A note on Douglas-Rachford, gradients, and phase retrieval

TL;DR

This paper explores the connection between Douglas-Rachford algorithms, especially the relaxed-reflect-reflect variant, and gradient descent methods in phase retrieval, providing theoretical insights into their solutions and stability.

Contribution

It establishes that RRR can be viewed as gradient descent on a specific objective, offering new theoretical understanding of its solutions and properties.

Findings

RRR solutions coincide with critical points of an objective function

Local convexity around solutions is demonstrated

Stability guarantees for RRR are derived

Abstract

The properties of gradient techniques for the phase retrieval problem have received a considerable attention in recent years. In almost all applications, however, the phase retrieval problem is solved using a family of algorithms that can be interpreted as variants of Douglas-Rachford splitting. In this work, we establish a connection between Douglas-Rachford and gradient algorithms. Specifically, we show that in some cases a generalization of Douglas-Rachford, called relaxed-reflect-reflect (RRR), can be viewed as gradient descent on a certain objective function. The solutions coincide with the critical points of that objective, which---in contrast to standard gradient techniques---are not its minimizers. Using the objective function, we give simple proofs of some basic properties of the RRR algorithm. Specifically, we describe its set of solutions, show a local convexity around any solution, and derive stability guarantees. Nevertheless, in its present state, the analysis does not elucidate the remarkable empirical performance of RRR and its global properties.