Abstract
Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from their circular convolution $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent (MGD) algorithm. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.
Abstract (translated)
多通道盲反褶积是从循环卷积中恢复未知信号$f$和多个未知通道$x_i$的问题，循环卷积为$y_i=x_icircledast f$（$i=1,2，点，n$）。我们考虑这样一种情况，即$x_i$是稀疏的，与$f$的卷积是可逆的。我们的非凸优化公式解决了单位球面上的过滤器$H$的问题，该过滤器产生稀疏输出$Y_iCircledast H$。在一些技术假设下，我们证明了目标函数的所有局部极小值都对应于$F$的逆滤波器，达到了一个固有的符号和移位模糊度，并且所有鞍点都有严格的负曲率。这种几何结构允许使用简单的流形梯度下降（mgd）算法成功地恢复$f$和$x_i$。我们的理论研究结果得到了数值实验的补充，这证明了所提出的方法比以前的方法有更好的性能。
URL
https://arxiv.org/abs/1805.10437
