Local saddles of relaxed averaged alternating reflections algorithms on phase retrieval

Abstract

Phase retrieval can be expressed as a non-convex constrained optimization problem to identify one phase minimizer one a torus. Many iterative transform techniques have been proposed to identify the minimizer, e.g., relaxed averaged alternating reflections (RAAR) algorithms. In this paper, we present one optimization viewpoint on the RAAR algorithm. RAAR algorithm is one alternating direction method of multipliers with one penalty parameter. Pairing with multipliers (dual vectors), phase vectors on the primal space are lifted to higher dimensional vectors, RAAR algorithm is one continuation algorithm, which searches for local saddles in the primal-dual space. The dual iteration approximates one gradient ascent flow, which drives the corresponding local minimizers in a positive-definite Hessian region. Altering 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.