Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization

TL;DR

This paper introduces a new nonconvex optimization algorithm for blind deconvolution that guarantees exact recovery under certain conditions, is robust to noise, and is computationally efficient, outperforming previous methods.

Contribution

The paper presents the first efficient, robust blind deconvolution algorithm with rigorous recovery guarantees under subspace assumptions.

Findings

Guaranteed exact recovery with near-optimal measurements

Algorithm converges geometrically and is noise-robust

Numerical experiments confirm theoretical and practical effectiveness

Abstract

We study the question of reconstructing two signals $f$ and $g$ from their convolution $y = f\ast g$. This problem, known as {\em blind deconvolution}, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on $f$ and $g$. The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework.