Work

Regularized gradient descent: a non-convex recipe for fast joint blind deconvolution and demixing

Year: 2017

Type: article

Abstract: We study the question of extracting a sequence of functions |$\{\boldsymbol{f}_{\!i}, \boldsymbol{g}_{i}\}_{i=1}^{s}$| from observing only the sum of their convolutions, i.e. from |$\boldsymbol{y} = \sum _{i=1}^{s} \boldsymbol{f}_{\!i}\ast \boldsymbol{g}_{i}$|⁠. While convex optimization techniques are able to solve this joint blind deconvolution–demixing problem provably and robustly under certain conditions, for medium-size or large-size problems we need computationally faster methods without sacrificing the benefits of mathematical rigor that come with convex methods. In this paper we present a non-convex algorithm which guarantees exact recovery under conditions that are competitive with convex optimization methods, with the additional advantage of being computationally much more efficient. Our two-step algorithm converges to the global minimum linearly and is also robust in the presence of additive noise. While the derived performance bounds are suboptimal in terms of the information-theoretic limit, numerical simulations show remarkable performance even if the number of measurements is close to the number of degrees of freedom. We discuss an application of the proposed framework in wireless communications in connection with the Internet-of-Things.

Source: Information and Inference A Journal of the IMA

DOI: 10.1093/imaiai/iax022

Authors: Shuyang Ling, Thomas Strohmer

Institutions: Courant Institute of Mathematical Sciences, New York University, University of California, Davis

Language: English

Topic: Sparse and Compressive Sensing Techniques

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Harmonic analysis, non-convex optimization, and large data sets, Methods and Algorithms from Harmonic Analysis for Threat Detection