Blind Deconvolution by a Steepest Descent Algorithm on a Quotient Manifold

TL;DR

This paper introduces a Riemannian steepest descent algorithm on a quotient manifold for blind deconvolution, providing theoretical guarantees for exact recovery and demonstrating superior empirical performance over existing methods.

Contribution

It develops a novel Riemannian steepest descent approach with a quotient structure that simplifies analysis and improves efficiency in blind deconvolution tasks.

Findings

Exact recovery with high probability under near-minimal measurements

Fewer operations needed compared to existing algorithms

Higher success rate in synthetic and image deblurring tests

Abstract

In this paper, we propose a Riemannian steepest descent method for solving a blind deconvolution problem. We prove that the proposed algorithm with an appropriate initialization will recover the exact solution with high probability when the number of measurements is, up to log-factors, the information-theoretical minimum scaling. The quotient structure in our formulation yields a simpler penalty term in the cost function compared to [LLSW16], which eases the convergence analysis and yields a natural implementation. Empirically, the proposed algorithm has better performance than the Wirtinger gradient descent algorithm and an alternating minimization algorithm in the sense that i) it needs fewer operations, such as DFTs and matrix-vector multiplications, to reach a similar accuracy, and ii) it has a higher probability of successful recovery in synthetic tests. An image deblurring problem is also used to demonstrate the efficiency and effectiveness of the proposed algorithm.