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Global Geometry of Multichannel Sparse Blind Deconvolution on the Sphere
Yanjun Li · Yoram Bresler

Tue Dec 04 07:20 AM -- 07:25 AM (PST) @ Room 517 CD
Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from convolutional measurements $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). 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 algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

#### Author Information

##### Yanjun Li (UIUC)

Yanjun Li received the Bachelor of Engineering degree in Automation from Tsinghua University (Beijing, China) in 2012, with the highest honor. He received the M.S. degree and the Ph.D. degree in Electrical and Computer Engineering from the University of Illinois at Urbana-Champaign in 2015 and 2018, respectively. His research interests center around low-dimensional and parsimonious structures in signal processing and machine learning. His research directions include bilinear inverse problems, image and video restoration, dimensionality reduction, kernel approximation, and various applications in imaging, image processing, and computer vision. He is the recipient of a Best Student Paper Prize at SPARS 2015. He received the Robert T. Chien Memorial Award in 2016, the Hong, McCully, and Allen Fellowship, and a Scott Dissertation Completion Fellowship in 2017, from the University of Illinois at Urbana-Champaign.