Gradient projection methods for the $n$-coupling problem
J. Korean Math. Soc. 2019 Vol. 56, No. 4, 1001-1016
Published online July 1, 2019
Sangho Kum, Sangwoon Yun
Chungbuk National University; Sungkyunkwan University
Abstract : We are concerned with optimization methods for the $L^2$-Wasserstein least squares problem of Gaussian measures (alternatively the $n$-coupling problem). Based on its equivalent form on the convex cone of positive definite matrices of fixed size and the strict convexity of the variance function, we are able to present an implementable (accelerated) gradient method for finding the unique minimizer. Its global convergence rate analysis is provided according to the derived upper bound of Lipschitz constants of the gradient function.
Keywords : $L^2$-Wasserstein least squares problem, $n$-coupling problem, Gaussian measure, positive definite matrix, Nesterov-Todd scaling, gradient projection method
MSC numbers : Primary 49M30, 90C25
Supported by : The work of the first author was supported by Basic Science Research Program through NRF Grant No.NRF-2017R1A2B1002008 and NRF-2014R1A1A2056635. The work of the second author was supported by Science Research Center Program through NRF funded by the Ministry of Science, ICT & Future Planning No.NRF-2016R1A5A1008055 and NRF-2016R1D1A1B03934371.
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