{"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-fractal-dimension-for-measures-via","title":"A fractal dimension for measures via persistent homology","arxiv_id":"1808.01079","date":"2018-08-03","proceeding":null,"authors":["Henry Adams","Manuchehr Aminian","Elin Farnell","Michael Kirby","Chris Peterson","Joshua Mirth","Rachel Neville","Patrick Shipman","Clayton Shonkwiler"],"abstract":"We use persistent homology in order to define a family of fractal dimensions, denoted $\\mathrm{dim}_{\\mathrm{PH}}^i(\\mu)$ for each homological dimension $i\\ge 0$, assigned to a probability measure $\\mu$ on a metric space. The case of $0$-dimensional homology ($i=0$) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if $\\mu$ is supported on a compact subset of Euclidean space $\\mathbb{R}^m$ for $m\\ge2$, then Steele's work implies that $\\mathrm{dim}_{\\mathrm{PH}}^0(\\mu)=m$ if the absolutely continuous part of $\\mu$ has positive mass, and otherwise $\\mathrm{dim}_{\\mathrm{PH}}^0(\\mu)<m$. Experiments suggest that similar results may be true for higher-dimensional homology $0<i<m$, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points $n$ goes to infinity, of the total sum of the $i$-dimensional persistent homology interval lengths for $n$ random points selected from $\\mu$ in an i.i.d. fashion. To some measures $\\mu,$ we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of $0$-dimensional homology when $\\mu$ is the uniform distribution over the unit interval, and conjecture that it exists when $\\mu$ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.","url_abs":"http://arxiv.org/abs/1808.01079v4","url_pdf":"http://arxiv.org/pdf/1808.01079v4.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-fractal-dimension-for-measures-via","repo_url":"https://github.com/CSU-PHdimension/PHdimension","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1808.01079","atlas_url":"https://app.syntology.ai/?focus=1808.01079","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}