Abstract

 We study the convolutional phase retrieval problem, which considers recovery of an unknown signal from m measurements consisting of the magnitude of its cyclic convolution with a known kernel a of length m. This model is motivated by applications to channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is di?cult or impossible to acquire. We show that when a is random and m is suficiently large, x can be eficiently recovered up to a global phase using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator; we overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis for alternating minimizing methods.