Year: 2024
Author: Jianfeng Cai, Ke Wei
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 3 : pp. 755–783
Abstract
A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y.$ The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2207-m2021-0247
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 3 : pp. 755–783
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: Phaseless equations Riemannian gradient descent Manifold of rank-1 and positive semidefinite matrices Optimal sampling complexity.
Author Details
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A Preconditioned Riemannian Gradient Descent Algorithm for Low-Rank Matrix Recovery
Bian, Fengmiao
Cai, Jian-Feng
Zhang, Rui
SIAM Journal on Matrix Analysis and Applications, Vol. 45 (2024), Iss. 4 P.2075
https://doi.org/10.1137/23M1570442 [Citations: 0]