For multichannel sparse blind deconvolution, Huber-loss Riemannian gradient descent with random initialization plus an LP-rounding step provably recovers the kernel and sparse signals up to a signed shift, with sample complexity O~(n^4).
Multichannel Sparse Blind Deconvolution on the Sphere
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abstract
Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from their circular convolution $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). 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 $y_i\circledast h$. 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 (MGD) algorithm. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.
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A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution
For multichannel sparse blind deconvolution, Huber-loss Riemannian gradient descent with random initialization plus an LP-rounding step provably recovers the kernel and sparse signals up to a signed shift, with sample complexity O~(n^4).