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

We study the multi-channel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel a and multiple sparse inputs $\left\{ {{{\mathbf{x}}_i}} \right\}_{i = 1}^p$ from their circulant convolution yi = a ⊛ xi (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 $\left\{ {{{\mathbf{x}}_i}} \right\}_{i = 1}^p$ up to a signed shift ambiguity. In comparison with state-of-the-art 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 method on synthetic datasets.