Global Geometry of Multichannel Sparse Blind Deconvolution on the Sphere

Multichannel blind deconvolution is the problem of recovering an unknown signal f and multiple unknown channels x(i) 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 x(i) 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.