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 <inline-formula><math xmlns="h...
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MDPI AG
2021-09-01
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| Online Access: | https://www.mdpi.com/1099-4300/23/9/1214 |
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| author | Yihao Luo Shiqiang Zhang Yueqi Cao Huafei Sun |
| author_facet | Yihao Luo Shiqiang Zhang Yueqi Cao Huafei Sun |
| author_sort | Yihao Luo |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>P</mi><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>P</mi><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. The experimental results show the efficiency and robustness of our curvature-based methods. |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T07:41:20Z |
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| spelling | doaj.art-2b744b75e338452f8dba28072437de3c2023-11-22T12:58:26ZengMDPI AGEntropy1099-43002021-09-01239121410.3390/e23091214Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data ProcessingYihao Luo0Shiqiang Zhang1Yueqi Cao2Huafei Sun3School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, ChinaDepartment of Computing, Imperial College London, London SW7 2AZ, UKDepartment of Mathematics, Imperial College London, London SW7 2AZ, UKSchool of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, ChinaThe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>P</mi><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>P</mi><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. The experimental results show the efficiency and robustness of our curvature-based methods.https://www.mdpi.com/1099-4300/23/9/1214symmetric positive-definite matrixWasserstein metriccurvaturepoint cloud denoisingimage edge detecting |
| spellingShingle | Yihao Luo Shiqiang Zhang Yueqi Cao Huafei Sun Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing Entropy symmetric positive-definite matrix Wasserstein metric curvature point cloud denoising image edge detecting |
| title | Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing |
| title_full | Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing |
| title_fullStr | Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing |
| title_full_unstemmed | Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing |
| title_short | Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing |
| title_sort | geometric characteristics of the wasserstein metric on spd n and its applications on data processing |
| topic | symmetric positive-definite matrix Wasserstein metric curvature point cloud denoising image edge detecting |
| url | https://www.mdpi.com/1099-4300/23/9/1214 |
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