{"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/a-riemannian-framework-for-statistical","title":"A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams","arxiv_id":"1605.08912","date":"2016-05-28","proceeding":null,"authors":["Rushil Anirudh","Vinay Venkataraman","Karthikeyan Natesan Ramamurthy","Pavan Turaga"],"abstract":"Topological data analysis is becoming a popular way to study high dimensional\nfeature spaces without any contextual clues or assumptions. This paper concerns\nitself with one popular topological feature, which is the number of\n$d-$dimensional holes in the dataset, also known as the Betti$-d$ number. The\npersistence of the Betti numbers over various scales is encoded into a\npersistence diagram (PD), which indicates the birth and death times of these\nholes as scale varies. A common way to compare PDs is by a point-to-point\nmatching, which is given by the $n$-Wasserstein metric. However, a big drawback\nof this approach is the need to solve correspondence between points before\ncomputing the distance; for $n$ points, the complexity grows according to\n$\\mathcal{O}($n$^3)$. Instead, we propose to use an entirely new framework\nbuilt on Riemannian geometry, that models PDs as 2D probability density\nfunctions that are represented in the square-root framework on a Hilbert\nSphere. The resulting space is much more intuitive with closed form expressions\nfor common operations. The distance metric is 1) correspondence-free and also\n2) independent of the number of points in the dataset. The complexity of\ncomputing distance between PDs now grows according to $\\mathcal{O}(K^2)$, for a\n$K \\times K$ discretization of $[0,1]^2$. This also enables the use of existing\nmachinery in differential geometry towards statistical analysis of PDs such as\ncomputing the mean, geodesics, classification etc. We report competitive\nresults with the Wasserstein metric, at a much lower computational load,\nindicating the favorable properties of the proposed approach.","url_abs":"http://arxiv.org/abs/1605.08912v1","url_pdf":"http://arxiv.org/pdf/1605.08912v1.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":"a-riemannian-framework-for-statistical","repo_url":"https://github.com/rushilanirudh/pdsphere","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"topological-data-analysis","task_name":"Topological Data Analysis"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}