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Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

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School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
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Department of Computing, Imperial College London, London SW7 2AZ, UK
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Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
Author to whom correspondence should be addressed.
Academic Editor: Carlos M. Travieso-González
Entropy 2021, 23(9), 1214; https://doi.org/10.3390/e23091214
Received: 5 August 2021 / Revised: 9 September 2021 / Accepted: 10 September 2021 / Published: 14 September 2021
The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods. View Full-Text
Keywords: symmetric positive-definite matrix; Wasserstein metric; curvature; point cloud denoising; image edge detecting symmetric positive-definite matrix; Wasserstein metric; curvature; point cloud denoising; image edge detecting
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MDPI and ACS Style

Luo, Y.; Zhang, S.; Cao, Y.; Sun, H. Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing. Entropy 2021, 23, 1214. https://doi.org/10.3390/e23091214

AMA Style

Luo Y, Zhang S, Cao Y, Sun H. Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing. Entropy. 2021; 23(9):1214. https://doi.org/10.3390/e23091214

Chicago/Turabian Style

Luo, Yihao, Shiqiang Zhang, Yueqi Cao, and Huafei Sun. 2021. "Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing" Entropy 23, no. 9: 1214. https://doi.org/10.3390/e23091214

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