{"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/the-persistence-landscape-and-some-of-its","title":"The persistence landscape and some of its properties","arxiv_id":"1810.04963","date":"2018-10-11","proceeding":null,"authors":["Peter Bubenik"],"abstract":"Persistence landscapes map persistence diagrams into a function space, which\nmay often be taken to be a Banach space or even a Hilbert space. In the latter\ncase, it is a feature map and there is an associated kernel. The main advantage\nof this summary is that it allows one to apply tools from statistics and\nmachine learning. Furthermore, the mapping from persistence diagrams to\npersistence landscapes is stable and invertible. We introduce a weighted\nversion of the persistence landscape and define a one-parameter family of\nPoisson-weighted persistence landscape kernels that may be useful for learning.\nWe also demonstrate some additional properties of the persistence landscape.\nFirst, the persistence landscape may be viewed as a tropical rational function.\nSecond, in many cases it is possible to exactly reconstruct all of the\ncomponent persistence diagrams from an average persistence landscape. It\nfollows that the persistence landscape kernel is characteristic for certain\ngeneric empirical measures. Finally, the persistence landscape distance may be\narbitrarily small compared to the interleaving distance.","url_abs":"http://arxiv.org/abs/1810.04963v2","url_pdf":"http://arxiv.org/pdf/1810.04963v2.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":"the-persistence-landscape-and-some-of-its","repo_url":"https://github.com/gabbyangeloro/pyscapes","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}