{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/generalized-bures-wasserstein-geometry-for","title":"Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry","arxiv_id":"2110.10464","date":"2021-10-20","proceeding":null,"authors":["Andi Han","Bamdev Mishra","Pratik Jawanpuria","Junbin Gao"],"abstract":"Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix $\\mathbf{M}$ such that when $\\mathbf{M} = \\mathbf{I}$, we recover the BW geometry. We provide a rigorous treatment to study various differential geometric notions on the proposed novel generalized geometry which makes it amenable to various machine learning applications. We also present experiments that illustrate the efficacy of the proposed GBW geometry over the BW geometry.","url_abs":"https://arxiv.org/abs/2110.10464v2","url_pdf":"https://arxiv.org/pdf/2110.10464v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"generalized-bures-wasserstein-geometry-for","repo_url":"https://github.com/andyjm3/gbw","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"riemannian-optimization","task_name":"Riemannian optimization"}],"methods":[{"method_slug":"procrustes","method_name":"Procrustes"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2110.10464","atlas_url":"https://app.syntology.ai/?focus=2110.10464","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}