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.
During the preparation of this manuscript, we became aware of independent simultaneous work done by Jeong and G "u nt "u rk. They also studied the randomized Kaczmarz method adapted to phase retrieval, and obtained almost the same result that we did (see @cite_14 and Theorem 1.1 therein). In order to prove their guarantee, they use a stopping time argument similar to ours, but replace the ACW condition with a stronger condition called . They prove that measurement systems comprising vectors drawn independently and uniformly from the sphere satisfy this property with high probability, and the main tools they use in their proof are hyperplane tessellations and a net argument together with Lipschitz relaxation of indicator functions.
@cite_14
2040192961
Abstract Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles yield stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4 M − 4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M = 2 , 3 . Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer–Rao lower bounds to identify stability with stronger versions of the injectivity characterizations.