Phase Retrieval via Randomized Kaczmarz: Theoretical Guarantees
This addresses a fundamental challenge in signal processing and imaging for researchers, offering a theoretical foundation for an efficient algorithm, though it is incremental as it builds on prior numerical work.
The paper tackles the problem of phase retrieval by providing the first theoretical convergence guarantee for a randomized Kaczmarz method, showing it works with as many Gaussian measurements as the dimension up to a constant factor.
We consider the problem of phase retrieval, i.e. that of solving systems of quadratic equations. A simple variant of the randomized Kaczmarz method was recently proposed for phase retrieval, and it was shown numerically to have a computational edge over state-of-the-art Wirtinger flow methods. In this paper, we provide the first theoretical guarantee for the convergence of the randomized Kaczmarz method for phase retrieval. We show that it is sufficient to have as many Gaussian measurements as the dimension, up to a constant factor. Along the way, we introduce a sufficient condition on measurement sets for which the randomized Kaczmarz method is guaranteed to work. We show that Gaussian sampling vectors satisfy this property with high probability; this is proved using a chaining argument coupled with bounds on VC dimension and metric entropy.