Local saddle structure in relaxed averaged alternating reflections Algorithms on phase retrieval

TL;DR

This paper analyzes the local saddle structure of the RAAR algorithm in phase retrieval, revealing how penalty parameters influence attractor basins and help avoid stagnation at non-global minima.

Contribution

It provides a theoretical analysis of the local saddle structure in RAAR, showing how penalty parameters affect convergence to true solutions in phase retrieval.

Findings

RAAR acts as a continuation algorithm searching for local Nash equilibria.

Proper penalty parameters enlarge attractor basins for true solutions.

The dual iteration approximates a gradient ascent flow in a primal-dual space.

Abstract

Phase retrieval can be expressed as one non-convex constrained optimization problem to identify one phase minimizer in the primal space. Many iterative transform techniques have been proposed to identify the minimizer, e.g., relaxed averaged alternating reflections(RAAR) algorithms. RAAR algorithm is one alternating direction method of multipliers(ADMM) with one penalty parameter. Pairing with multipliers (dual vectors), phase vectors are lifted to higher dimensional vectors, RAAR algorithms actually is one continuation algorithm, which searches for local Nash equilibria in a primal-dual space. The dual iteration approximates one gradient ascent flow, which drives the corresponding local minimizers in a positive-definite Hessian region. The penalty parameter, which is the reciprocal of the RAAR parameter, plays a role of altering the size of the attractor basin for each stationary point. Under proper penalty parameters, the RAAR avoids the stagnation of these corresponding local minimizers in the primal space and thus screens out many stationary points corresponding to non-local minimizers.