MULTICHANNEL SPARSE BLIND DECONVOLUTION ON THE SPHERE

Multichannel blind deconvolution is the problem of recovering an unknown signal f and multiple unknown channels xi from convolutional measurements y(i) = x(i) circle star f (i = 1, 2, ... N). We consider the case where the x(i)'s are sparse, and convolution with f is invertible. Our nonconvex optimization formulation solves for a filter h on the unit sphere that produces sparse output y(i) circle star h. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of f up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of f and xi using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.