Fixed Point Analysis of Douglas-Rachford Splitting for Ptychography and Phase Retrieval

TL;DR

This paper analyzes the fixed point properties of Douglas-Rachford Splitting methods for phase retrieval and ptychography, showing boundedness, convergence conditions, and optimal step size, improving understanding and performance of these algorithms.

Contribution

It provides a fixed point analysis of DRS methods for phase retrieval, deriving explicit step size formulas and demonstrating convergence in blind ptychography.

Findings

Fixed points are attracting if and only if they are regular solutions.

Explicit optimal step size depends on the spectral gap of an underlying matrix.

Alternating Minimization with DRS inner loops converges geometrically in blind ptychography.

Abstract

Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent. Fixed point analysis also leads to a simple, explicit expression for the optimal step size in terms of the spectral gap of an underlying matrix. When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, Alternating Minimization with the DRS inner loops, even with a far from optimal step size, converges geometrically under the nearly minimum conditions established in the uniqueness theory.