Empirical Means on Pseudo-Orthogonal Groups

Wang, Jing; Sun, Huafei; Fiori, Simone

doi:10.3390/math7100940

Open AccessArticle

Empirical Means on Pseudo-Orthogonal Groups

by

Jing Wang

¹,

Huafei Sun

² and

Simone Fiori

^3,*

¹

School of Information, Beijing Wuzi University, Beijing 101149, China

²

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

³

Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, 60026 Ancona, Italy

^*

Author to whom correspondence should be addressed.

Mathematics 2019, 7(10), 940; https://doi.org/10.3390/math7100940

Submission received: 4 July 2019 / Revised: 22 September 2019 / Accepted: 1 October 2019 / Published: 11 October 2019

(This article belongs to the Special Issue Numerical Optimization: Mathematical Problems and Applications)

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Abstract

The present article studies the problem of computing empirical means on pseudo-orthogonal groups. To design numerical algorithms to compute empirical means, the pseudo-orthogonal group is endowed with a pseudo-Riemannian metric that affords the computation of the exponential map in closed forms. The distance between two pseudo-orthogonal matrices, which is an essential ingredient, is computed by both the Frobenius norm and the geodesic distance. The empirical-mean computation problem is solved via a pseudo-Riemannian-gradient-stepping algorithm. Several numerical tests are conducted to illustrate the numerical behavior of the devised algorithm.

Keywords: pseudo-orthogonal group; pseudo-Riemannian geometry; gradient-based function minimization on manifolds; geodesic stepping

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MDPI and ACS Style

Wang, J.; Sun, H.; Fiori, S. Empirical Means on Pseudo-Orthogonal Groups. Mathematics 2019, 7, 940. https://doi.org/10.3390/math7100940

AMA Style

Wang J, Sun H, Fiori S. Empirical Means on Pseudo-Orthogonal Groups. Mathematics. 2019; 7(10):940. https://doi.org/10.3390/math7100940

Chicago/Turabian Style

Wang, Jing, Huafei Sun, and Simone Fiori. 2019. "Empirical Means on Pseudo-Orthogonal Groups" Mathematics 7, no. 10: 940. https://doi.org/10.3390/math7100940

APA Style

Wang, J., Sun, H., & Fiori, S. (2019). Empirical Means on Pseudo-Orthogonal Groups. Mathematics, 7(10), 940. https://doi.org/10.3390/math7100940

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Empirical Means on Pseudo-Orthogonal Groups

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