A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution

TL;DR

This paper introduces a nonconvex optimization method using Riemannian gradient descent for multichannel sparse blind deconvolution, achieving provable recovery of signals and kernel with improved efficiency and sample complexity.

Contribution

It proves that vanilla RGD can reliably recover signals and kernel in MCS-BD under mild assumptions, with better sample complexity and computational efficiency than existing methods.

Findings

RGD recovers signals and kernel up to signed shift.

Significant reduction in sample complexity.

Numerical experiments confirm superior performance.

Abstract

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.