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 yi = x_i~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 ~ 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.