[Paper Review] Blind Deconvolutional Phase Retrieval via Convex Programming

Ali Ahmed, Alireza Aghasi+1 more|arXiv (Cornell University)|Jun 21, 2018

Sparse and Compressive Sensing Techniques2 references4 citations

TL;DR

This paper proposes a convex optimization framework for blind deconvolutional phase retrieval, recovering two signals from phaseless Fourier measurements of their circular convolution. By lifting the bilinear convolutional measurements into a matrix recovery problem and applying a novel convex relaxation, the method provably recovers signals from $ m \gg (k+n)\log^2 m $ measurements when both signals lie in known random subspaces of dimensions $ k $ and $ n $, without requiring initialization.

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 $m >> (k+n) \log^2 m$ 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 Rademacher complexity estimates. Additionally, we provide an ADMM implementation of the method and provide numerical experiments that verify the theory.

Motivation & Objective

• To address the ill-posed problem of recovering two unknown signals from phaseless Fourier measurements of their circular convolution.
• To develop a computationally efficient algorithm that does not require an initial guess or solution estimate for recovery.
• To provide the first theoretical recovery guarantee for blind deconvolutional phase retrieval using a convex relaxation approach.
• To establish conditions under which exact recovery is possible when signals reside in known low-dimensional subspaces.

Proposed method

• The method lifts the bilinear convolutional measurements into a matrix recovery problem using a lifted matrix formulation.
• It applies a novel convex relaxation to the bilinear measurements, transforming the non-convex problem into a semidefinite program.
• The approach assumes both signals lie in known random subspaces, parameterized via known matrices $ \boldsymbol{B} \in \mathbb{H}^{m \times k} $ and $ \boldsymbol{C} \in \mathbb{H}^{m \times n} $.
• The algorithm uses an ADMM-based implementation to solve the resulting convex program efficiently.
• The ADMM scheme includes updates for the lifted matrix variable, dual variables, and a projection step onto the positive semidefinite cone.
• The $ \boldsymbol{u} $-update step involves solving a constrained optimization problem via KKT conditions, reducing to a quartic polynomial equation for each component.

Experimental results

Research questions

• RQ1 Can blind deconvolutional phase retrieval be solved via a convex relaxation with provable recovery guarantees?
• RQ2 What is the minimum number of phaseless Fourier measurements required for exact recovery when signals lie in known subspaces?
• RQ3 Does the proposed method eliminate the need for initialization in blind deconvolution problems with phaseless measurements?
• RQ4 Can the bilinear convolutional structure be effectively relaxed into a convex program while preserving recovery fidelity?
• RQ5 How does the Rademacher complexity framework support theoretical recovery bounds in this non-linear, bilinear measurement model?

Key findings

• The method provably recovers two signals from $ m \gg (k+n)\log^2 m $ phaseless Fourier measurements when both signals lie in known random subspaces of dimensions $ k $ and $ n $.
• The recovery is exact up to the inherent scaling ambiguity, and no initialization is required.
• The theoretical recovery guarantee is established using Rademacher complexity estimates, providing a non-asymptotic bound on the number of measurements.
• Numerical experiments confirm the theoretical predictions, showing successful recovery of blurred images from magnitude-only Fourier measurements.
• The ADMM implementation efficiently solves the semidefinite program, with each $ \boldsymbol{u} $-update solved via a quartic polynomial equation derived from KKT conditions.
• The method enables recovery in applications such as X-ray crystallography, diffraction imaging, and visible light communication, where phase information is lost.