Multi-channel sparse blind deconvolution, or convolutional sparse coding, refers to the problem of learning an unknown filter by observing its circulant convolutions with multiple input signals that are sparse. This problem finds numerous applications in signal processing, computer vision, and inverse problems. However, it is challenging to learn the filter efficiently due to the bilinear structure of the observations with respect to the unknown filter and inputs, as well as the sparsity constraint. In this paper, we propose a novel approach based on nonconvex optimization over the sphere manifold by minimizing a smooth surrogate of the sparsity-promoting loss function. It is demonstrated that manifold gradient descent with random initializations will probably recover the filter, up to scaling and shift ambiguity, as soon as the number of observations is sufficiently large under an appropriate random data model. Numerical experiments are provided to illustrate the performance of the proposed method with comparisons to existing ones.

中文翻译：

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流形梯度下降可证明且高效地解决多通道稀疏盲反卷积问题


多通道稀疏盲反卷积或卷积稀疏编码是指通过观察未知滤波器与多个稀疏输入信号的循环卷积来学习未知滤波器的问题。这个问题在信号处理、计算机视觉和反问题中有着广泛的应用。然而，由于相对于未知滤波器和输入的观测的双线性结构以及稀疏性约束，有效地学习滤波器是具有挑战性的。在本文中，我们提出了一种基于球流形非凸优化的新方法，通过最小化稀疏性促进损失函数的平滑代理。事实证明，只要在适当的随机数据模型下观察数量足够大，带有随机初始化的流形梯度下降就可能恢复滤波器，直至缩放和移动模糊度。提供数值实验来说明所提出方法的性能，并与现有方法进行比较。