Exact and Efficient Multi-Channel Sparse Blind Deconvolution - A Nonconvex Approach

Qing Qu, Xiao Li, Zhihui Zhu

Abstract: We study the multi-channel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel a and multiple sparse inputs {x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> }pi“1 from their circulant convolution yi = a ⊙x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (i = 1, . . . , p). We formulate the task as a nonconvex optimization problem over the sphere. Under mild assumptions of the data, we prove that the vanilla Riemannian gradient descent (RGD) method, with random initializations, provably recovers both the kernel a and the signals {xi}pi“1 up to a signed shift ambiguity. In comparison with state-of-theart results, our work shows significant improvements in terms of sample complexity and computational efficiency. Our theoretical results are corroborated by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods on synthetic datasets.