Linear Convergence of An Iterative Phase Retrieval Algorithm with Data Reuse

TL;DR

This paper proves the linear convergence of the randomized Kaczmarz method for phase retrieval without assuming independence between data and updates, bridging the gap between empirical success and theoretical understanding.

Contribution

It introduces a novel analysis that removes the independence assumption, establishing the first rigorous proof of linear convergence for data-reusing phase retrieval algorithms.

Findings

Proves linear convergence of the randomized Kaczmarz method without independence assumptions.

Derives bounds on the reduction of mean squared error during iterations.

Fills theoretical gaps for efficient phase retrieval algorithms with data reuse.

Abstract

Phase retrieval has been an attractive but difficult problem rising from physical science, and there has been a gap between state-of-the-art theoretical convergence analyses and the corresponding efficient retrieval methods. Firstly, these analyses all assume that the sensing vectors and the iterative updates are independent, which only fits the ideal model with infinite measurements but not the reality, where data are limited and have to be reused. Secondly, the empirical results of some efficient methods, such as the randomized Kaczmarz method, show linear convergence, which is beyond existing theoretical explanations considering its randomness and reuse of data. In this work, we study for the first time, without the independence assumption, the convergence behavior of the randomized Kaczmarz method for phase retrieval. Specifically, beginning from taking expectation of the squared estimation error with respect to the index of measurement by fixing the sensing vector and the error in the previous step, we discard the independence assumption, rigorously derive the upper and lower bounds of the reduction of the mean squared error, and prove the linear convergence. This work fills the gap between a fast converging algorithm and its theoretical understanding. The proposed methodology may contribute to the study of other iterative algorithms for phase retrieval and other problems in the broad area of signal processing and machine learning.