Convex Phase Retrieval without Lifting via PhaseMax
Tom Goldstein, Christoph Studer
Abstract
Semidefinite relaxation methods transform a variety of non-convex optimization problems into convex problems, but square the number of variables. We study a new type of convex relaxation for phase retrieval problems, called PhaseMax, that convexifies the underlying problem without lifting. The resulting problem formulation can be solved using standard convex optimization routines, while still working in the original, low-dimensional variable space. We prove, using a random spherical distribution measurement model, that PhaseMax succeeds with high probability for a sufficiently large number of measurements. We compare our approach to other phase retrieval methods and demonstrate that our theory accurately predicts the success of PhaseMax.
BibTeX
@InProceedings{pmlr-v70-goldstein17a,
title = {Convex Phase Retrieval without Lifting via {P}hase{M}ax},
author = {Tom Goldstein and Christoph Studer},
booktitle = {Proceedings of the 34th International Conference on Machine Learning},
pages = {1273--1281},
year = {2017},
editor = {Precup, Doina and Teh, Yee Whye},
volume = {70},
series = {Proceedings of Machine Learning Research},
month = {06--11 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v70/goldstein17a/goldstein17a.pdf},
url = {https://proceedings.mlr.press/v70/goldstein17a.html},
abstract = {Semidefinite relaxation methods transform a variety of non-convex optimization problems into convex problems, but square the number of variables. We study a new type of convex relaxation for phase retrieval problems, called PhaseMax, that convexifies the underlying problem without lifting. The resulting problem formulation can be solved using standard convex optimization routines, while still working in the original, low-dimensional variable space. We prove, using a random spherical distribution measurement model, that PhaseMax succeeds with high probability for a sufficiently large number of measurements. We compare our approach to other phase retrieval methods and demonstrate that our theory accurately predicts the success of PhaseMax.}
}