{"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/persistence-fisher-kernel-a-riemannian","title":"Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams","arxiv_id":"1802.03569","date":"2018-02-10","proceeding":"NeurIPS 2018 12","authors":["Tam Le","Makoto Yamada"],"abstract":"Algebraic topology methods have recently played an important role for\nstatistical analysis with complicated geometric structured data such as shapes,\nlinked twist maps, and material data. Among them, \\textit{persistent homology}\nis a well-known tool to extract robust topological features, and outputs as\n\\textit{persistence diagrams} (PDs). However, PDs are point multi-sets which\ncan not be used in machine learning algorithms for vector data. To deal with\nit, an emerged approach is to use kernel methods, and an appropriate geometry\nfor PDs is an important factor to measure the similarity of PDs. A popular\ngeometry for PDs is the \\textit{Wasserstein metric}. However, Wasserstein\ndistance is not \\textit{negative definite}. Thus, it is limited to build\npositive definite kernels upon the Wasserstein distance \\textit{without\napproximation}. In this work, we rely upon the alternative \\textit{Fisher\ninformation geometry} to propose a positive definite kernel for PDs\n\\textit{without approximation}, namely the Persistence Fisher (PF) kernel.\nThen, we analyze eigensystem of the integral operator induced by the proposed\nkernel for kernel machines. Based on that, we derive generalization error\nbounds via covering numbers and Rademacher averages for kernel machines with\nthe PF kernel. Additionally, we show some nice properties such as stability and\ninfinite divisibility for the proposed kernel. Furthermore, we also propose a\nlinear time complexity over the number of points in PDs for an approximation of\nour proposed kernel with a bounded error. Throughout experiments with many\ndifferent tasks on various benchmark datasets, we illustrate that the PF kernel\ncompares favorably with other baseline kernels for PDs.","url_abs":"http://arxiv.org/abs/1802.03569v5","url_pdf":"http://arxiv.org/pdf/1802.03569v5.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":"persistence-fisher-kernel-a-riemannian","repo_url":"https://github.com/lttam/PersistenceFisher","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[],"methods":[{"method_slug":"affine-coupling","method_name":"Affine Coupling"},{"method_slug":"normalizing-flows","method_name":"Normalizing Flows"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1802.03569","atlas_url":"https://app.syntology.ai/?focus=1802.03569","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}