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We study the multi-channel sparse blind deconvolution (MCS-BD) problem, whosetask is to simultaneously recover a kernel $\mathbf a$ and multiple sparseinputs $\{\mathbf x_i\}_{i=1}^p$ from their circulant convolution $\mathbf y_i= \mathbf a \circledast \mathbf x_i $ ($i=1,\cdots,p$). We formulate the taskas a nonconvex optimization problem over the sphere. Under mild statisticalassumptions of the data, we prove that the vanilla Riemannian gradient descent(RGD) method, with random initializations, provably recovers both the kernel$\mathbf a$ and the signals $\{\mathbf x_i\}_{i=1}^p$ up to a signed shiftambiguity. In comparison with state-of-the-art results, our work showssignificant improvements in terms of sample complexity and computationalefficiency. Our theoretical results are corroborated by numerical experiments,which demonstrate superior performance of the proposed approach over theprevious methods on both synthetic and real datasets.

A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution