A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution
This addresses a signal processing problem for applications like imaging or communications, offering incremental improvements in efficiency and sample requirements.
The paper tackles the multichannel sparse blind deconvolution problem by proposing a nonconvex optimization approach using Riemannian gradient descent, which provably recovers the kernel and sparse signals with improved sample complexity and computational efficiency compared to state-of-the-art methods.
We study the multi-channel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel $\mathbf a$ and multiple sparse inputs $\{\mathbf x_i\}_{i=1}^p$ from their circulant convolution $\mathbf y_i = \mathbf a \circledast \mathbf x_i $ ($i=1,\cdots,p$). We formulate the task as a nonconvex optimization problem over the sphere. Under mild statistical assumptions 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 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 methods on both synthetic and real datasets.