Convolutional Phase Retrieval via Gradient Descent

We study the convolutional phase retrieval problem, which considers recovering an unknown signal $\mathbf x \in \mathbb C^n $ from $m$ measurements consisting of the magnitude of its cyclic convolution with a known kernel $\mathbf a \in \mathbb C^m $. This model is motivated by applications such as channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when $\mathbf a$ is random and the sample number $m$ is sufficiently large, with high probability $\mathbf x$ can be efficiently 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 minimization methods.