Title: Phase Retrieval via Randomized Kaczmarz: Theoretical Guarantees,
Abstract: 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.