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 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.
This paper considered multichannel sparse blind deconvolution(MSBD) and proposed to solve a nonconvex optimization problem for recovering the underlying signal and filters. Under some technical assumptions, the authors showed that all local minima of the objective function correspond to the inverse filter of the ground truth signal up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures.  Finally, the authors proposed to solve this nonconvex optimization problem via a Riemannian gradient descend method.  My main comments are as follows.  1)  The authors did not clarify why they use the nonconvex optimization in (P1), what is the main idea behind this, and intuitively why this formulation will lead to correct recovery.  2) Compared to the reference [48], the difference is that this paper considered ‘multichannel’ (i.e., a bunch of channels (#N) to convolve with the ground truth signal and measure a bunch of (#N) observations). Despite this difference, these two papers share similarities: the objective is the same (except an additional summation appeared in this paper) and the main purpose is to analyze the geometry of the corresponding nonconvex optimization problem. However, it is unclear what are the technical difficulties and challenges for the multichannel case. It could be just a direct extension of the result in [48] and thus it is hard to assess the value of contribution and significance of this paper.   3) The Bernoulli-Rademacher model for the channels is too strong and is far away to real-world applications. The authors should clarify why using this assumption for the channels.    %%%%%% after rebuttal  %%%%%% I thank the authors for the response to my concerns. It is very good to see that all the results can be extended to Bernoulli-subGaussian model. The Bernoulli-subGaussian model would make the results more general and applicable. So I suggest to  at least add discussion about this in the final version. Also the explanation of the equivalence between maximizing \ell_4^4 and minimizing l0 norm could be in more detail. It is not very clear to see this relationship from the plot of \ell_4 norm.