Simultaneous Phase Retrieval and Blind Deconvolution via Convex Programming

TL;DR

This paper introduces a convex relaxation approach for simultaneous phase retrieval and blind deconvolution, providing the first theoretical guarantees for recovering signals from phaseless Fourier measurements of their convolution.

Contribution

The authors propose a novel convex relaxation method for joint phase retrieval and blind deconvolution, with proven recovery guarantees and an efficient ADMM implementation.

Findings

Successful recovery with m >> (k+n) log^2 m measurements

First theoretical guarantee for this problem using convex programming

Numerical experiments verify the effectiveness of the method

Abstract

We consider the task of recovering two real or complex $m$-vectors from phaseless Fourier measurements of their circular convolution. Our method is a novel convex relaxation that is based on a lifted matrix recovery formulation that allows a nontrivial convex relaxation of the bilinear measurements from convolution. We prove that if the two signals belong to known random subspaces of dimensions $k$ and $n$, then they can be recovered up to the inherent scaling ambiguity with $m \gg (k+n) \log^2 m$ phaseless measurements. Our method provides the first theoretical recovery guarantee for this problem by a computationally efficient algorithm and does not require a solution estimate to be computed for initialization. Our proof is based on Rademacher complexity estimates. Additionally, we provide an alternating direction method of multipliers (ADMM) implementation and provide numerical experiments that verify the theory.