Phase transition of \emph{descending} phase retrieval algorithms

TL;DR

This paper investigates the phase transition phenomena in descending phase retrieval algorithms using Random duality theory, revealing how sample complexity influences algorithm success and proposing a hybrid algorithmic approach.

Contribution

It introduces a theoretical framework for understanding phase transitions in descending phase retrieval algorithms and develops a practical hybrid algorithm based on these insights.

Findings

Identification of parametric manifold and funneling points as key to algorithm behavior

Discovery of a phase transition from failure to success with increasing sample complexity

Strong agreement between theoretical predictions and simulations in small dimensions

Abstract

We study theoretical limits of \emph{descending} phase retrieval algorithms. Utilizing \emph{Random duality theory} (RDT) we develop a generic program that allows statistical characterization of various algorithmic performance metrics. Through these we identify the concepts of \emph{parametric manifold} and its \emph{funneling points} as key mathematical objects that govern the underlying algorithms' behavior. An isomorphism between single funneling point manifolds and global convergence of descending algorithms is established. The structure and shape of the parametric manifold as well as its dependence on the sample complexity are studied through both plain and lifted RDT. Emergence of a phase transition is observed. Namely, as sample complexity increases, parametric manifold transitions from a multi to a single funneling point structure. This in return corresponds to a transition from the scenarios where descending algorithms generically fail to the scenarios where they succeed in solving phase retrieval. We also develop and implement a practical algorithmic variant that in a hybrid alternating fashion combines a barrier and a plain gradient descent. Even though the theoretical results are obtained for infinite dimensional scenarios (and consequently non-jittery parametric manifolds), we observe a strong agrement between theoretical and simulated phase transitions predictions for fairly small dimensions on the order of a few hundreds.