TY - JOUR
T1 - Global geometry of multichannel sparse blind deconvolution on the sphere
AU - Li, Yanjun
AU - Bresler, Yoram
N1 - Funding Information:
This work was supported in part by the National Science Foundation (NSF) under Grant IIS 14-47879. The authors would like to thank Ju Sun for helpful discussions about this paper. The manuscript benefited from constructive comments by the anonymous reviewers.
Publisher Copyright:
© 2018 Curran Associates Inc..All rights reserved.
PY - 2018
Y1 - 2018
N2 - Multichannel blind deconvolution is the problem of recovering an unknown signal f and multiple unknown channels xi from convolutional measurements yi = xi~f (i = 1, 2, . . ., N). We consider the case where the xi'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 yi ~ 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 xi 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.
AB - Multichannel blind deconvolution is the problem of recovering an unknown signal f and multiple unknown channels xi from convolutional measurements yi = xi~f (i = 1, 2, . . ., N). We consider the case where the xi'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 yi ~ 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 xi 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.
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M3 - Conference article
AN - SCOPUS:85064818946
SN - 1049-5258
VL - 2018-December
SP - 1132
EP - 1143
JO - Advances in Neural Information Processing Systems
JF - Advances in Neural Information Processing Systems
T2 - 32nd Conference on Neural Information Processing Systems, NeurIPS 2018
Y2 - 2 December 2018 through 8 December 2018
ER -