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

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 e...

Full description

Saved in:
Bibliographic Details
Published inEntropy (Basel, Switzerland) Vol. 23; no. 9; p. 1214
Main Authors Luo, Yihao, Zhang, Shiqiang, Cao, Yueqi, Sun, Huafei
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 14.09.2021
MDPI
Subjects
Algorithms
Artificial intelligence
Curvature
Data processing
Decomposition
Eigenvalues
Geodesy
Geometry
image edge detecting
Mathematical analysis
Matrices (mathematics)
point cloud denoising
symmetric positive-definite matrix
Symmetry
Wasserstein metric
Online AccessGet full text

Cover

More Information
Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
content type line 23
ISSN:1099-4300
1099-4300
DOI:10.3390/e23091214