{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T14:00:05Z","timestamp":1760623205473,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T00:00:00Z","timestamp":1631577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["No. 2020YFC2006201"],"award-info":[{"award-number":["No. 2020YFC2006201"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e23091214","type":"journal-article","created":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T23:34:11Z","timestamp":1631662451000},"page":"1214","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing"],"prefix":"10.3390","volume":"23","author":[{"given":"Yihao","family":"Luo","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China"}]},{"given":"Shiqiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Computing, Imperial College London, London SW7 2AZ, UK"}]},{"given":"Yueqi","family":"Cao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Imperial College London, London SW7 2AZ, UK"}]},{"given":"Huafei","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,14]]},"reference":[{"key":"ref_1","unstructured":"Hosseini, R., and Sra, S. 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