Phase retrieval of complex-valued objects via a randomized Kaczmarz method

TL;DR

This paper proves that a randomized Kaczmarz algorithm converges linearly for complex phase retrieval when sensing vectors are sufficiently numerous and well-initialized, extending previous real-valued results.

Contribution

It establishes the convergence of the randomized Kaczmarz method for complex phase retrieval, linking convergence to convexity and providing probabilistic guarantees.

Findings

Linear convergence with high probability for m > O(n log n)

Connection between convergence and convexity of an objective function

Algorithm effective with good initialization

Abstract

This paper investigates the convergence of the randomized Kaczmarz algorithm for the problem of phase retrieval of complex-valued objects. While this algorithm has been studied for the real-valued case}, its generalization to the complex-valued case is nontrivial and has been left as a conjecture. This paper establishes the connection between the convergence of the algorithm and the convexity of an objective function. Based on the connection, it demonstrates that when the sensing vectors are sampled uniformly from a unit sphere and the number of sensing vectors $m$ satisfies $m>O(n\log n)$ as $n, m\rightarrow\infty$, then this algorithm with a good initialization achieves linear convergence to the solution with high probability.