Multichannel Sparse Blind Deconvolution on the Sphere

Abstract

Multichannel blind deconvolution is the problem of recovering an unknown signal <inline-formula> <tex-math notation= LaTeX >$f$ </tex-math></inline-formula> and multiple unknown channels <inline-formula> <tex-math notation= LaTeX >$x_{i}$ </tex-math></inline-formula> from their circular convolution <inline-formula> <tex-math notation= LaTeX >$y_{i}=x_{i} \\circledast f$ </tex-math></inline-formula> (<inline-formula> <tex-math notation= LaTeX >$i=1,2, {\\dots },N$ </tex-math></inline-formula>). We consider the case where the <inline-formula> <tex-math notation= LaTeX >$x_{i}$ </tex-math></inline-formula>’s are sparse, and convolution with <inline-formula> <tex-math notation= LaTeX >$f$ </tex-math></inline-formula> is invertible. Our nonconvex optimization formulation solves for a filter <inline-formula> <tex-math notation= LaTeX >$h$ </tex-math></inline-formula> on the unit sphere that produces sparse output <inline-formula> <tex-math notation= LaTeX >$y_{i}\\circledast h$ </tex-math></inline-formula>. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of <inline-formula> <tex-math notation= LaTeX >$f$ </tex-math></inline-formula> up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of <inline-formula> <tex-math notation= LaTeX >$f$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$x_{i}$ </tex-math></inline-formula> using a simple manifold gradient descent (MGD) algorithm. The same approach is also applicable to blind gain and phase calibration with a Fourier sensing matrix. Our algorithm and analysis require fewer assumptions than previous algorithms for the same problem. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods. Empirically, our algorithm has low computation cost (converging in a small number of iterations) and low memory footprint (solving only for the inverse filter of <inline-formula> <tex-math notation= LaTeX >$f$ </tex-math></inline-formula>).