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 Vapnik–Chervonenkis (VC) dimension and metric entropy.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Apr 03, 2018
- Source ID
- 10.1093/imaiai/iay005
Entities
People
- Roman Vershynin
- Yan Shuo Tan
Organizations
- National Science Foundation Division of Mathematical Sciences
- United States Air Force
- University of California
- University of Michigan