Phase retrieval with rank $d$ measurements -- \emph{descending} algorithms phase transitions

TL;DR

This paper extends a duality theory to analyze phase retrieval algorithms with rank d measurements, revealing phase transition phenomena in their success rates and validating predictions through simulations.

Contribution

It generalizes the Random duality theory to handle rank d positive definite measurements and characterizes phase transitions in the success of descending phase retrieval algorithms.

Findings

Identifies phase transition points in sample complexity for rank d measurements.

Theoretical phase transition locations match simulation results.

Demonstrates effectiveness of gradient descent variants in small-scale scenarios.

Abstract

Companion paper [118] developed a powerful \emph{Random duality theory} (RDT) based analytical program to statistically characterize performance of \emph{descending} phase retrieval algorithms (dPR) (these include all variants of gradient descents and among them widely popular Wirtinger flows). We here generalize the program and show how it can be utilized to handle rank $d$ positive definite phase retrieval (PR) measurements (with special cases $d=1$ and $d=2$ serving as emulations of the real and complex phase retrievals, respectively). In particular, we observe that the minimal sample complexity ratio (number of measurements scaled by the dimension of the unknown signal) which ensures dPR's success exhibits a phase transition (PT) phenomenon. For both plain and lifted RDT we determine phase transitions locations. To complement theoretical results we implement a log barrier gradient descent variant and observe that, even in small dimensional scenarios (with problem sizes on the order of 100), the simulated phase transitions are in an excellent agreement with the theoretical predictions.