{"paper":{"title":"A fractal dimension for measures via persistent homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.PR"],"primary_cat":"math.DS","authors_text":"Chris Peterson, Clayton Shonkwiler, Elin Farnell, Henry Adams, Joshua Mirth, Manuchehr Aminian, Michael Kirby, Patrick Shipman, Rachel Neville","submitted_at":"2018-08-03T03:45:22Z","abstract_excerpt":"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 $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01079","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1808.01079/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}