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A Riemannian submersion‐based approach to the Wasserstein barycenter of positive definite matrices
Author(s) -
Li Mingming,
Sun Huafei,
Li Didong
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6247
Subject(s) - positive definite matrix , mathematics , metric (unit) , computation , wasserstein metric , matrix (chemical analysis) , gradient descent , space (punctuation) , pure mathematics , algorithm , eigenvalues and eigenvectors , computer science , artificial intelligence , operations management , physics , materials science , quantum mechanics , economics , composite material , operating system , artificial neural network
In this paper, we introduce a novel geometrization on the space of positive definite matrices, derived from the Riemannian submersion from the general linear group to the space of positive definite matrices, resulting in easier computation of its geometric structure. The related metric is found to be the same as a particular Wasserstein metric. Based on this metric, the Wasserstein barycenter problem is studied. To solve this problem, some schemes of the numerical computation based on gradient descent algorithms are proposed and compared. As an application, we test the k‐means clustering of positive definite matrices with different choices of matrix mean.