SIAM Journal on Imaging Sciences, Volume 13, Issue 3, Page 1630-1652, January 2020.
We study the multichannel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel $a$ and multiple sparse inputs $\{x_i\}_{i=1}^p$ from their circulant convolution $y_i = a \;\circledast \;x_i $ ($i=1,\dots,p$). 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 $a$ and the signals $\{x_i\}_{i=1}^p$ 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 the superior performance of the proposed approach over the previous methods on both synthetic and real datasets.

中文翻译：

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通过梯度下降精确恢复多通道稀疏卷积反卷积

SIAM影像科学杂志，第13卷，第3期，第1630-1652页，2020年1月。
我们研究了多通道稀疏盲反卷积（MCS-BD）问题，其任务是从循环卷积$ y_i同时恢复内核$ a $和多个稀疏输入$ \ {x_i \} _ {i = 1} ^ p $ = a \; \ circledast \; x_i $（$ i = 1，\ dots，p $）。我们将任务表述为球面上的非凸优化问题。在数据的轻微统计假设下，我们证明了采用随机初始化的香草黎曼梯度下降（RGD）方法可证明地恢复了内核$ a $和信号$ \ {x_i \} _ {i = 1} ^ p $直到签署的转移歧义。与最新结果相比，我们的工作显示出样本复杂性和计算效率方面的显着提高。数值实验证实了我们的理论结果，