{"paper":{"title":"Weighted Persistent Homology Sums of Random \\v{C}ech Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.PR","authors_text":"Benjamin Schweinhart","submitted_at":"2018-07-18T17:40:12Z","abstract_excerpt":"We study the asymptotic behavior of random variables of the form \\begin{equation*} E_{\\alpha}^i\\left(x_1,\\ldots,x_n\\right)=\\sum_{\\left(b,d\\right)\\in \\mathit{PH}_i\\left(x_1,\\ldots,x_n\\right)} \\left(d-b\\right)^{\\alpha} \\end{equation*} where $\\left\\{x_j\\right\\}_{j\\in\\mathbb{N}}$ are i.i.d. samples from a probability measure on a triangulable metric space, and $\\textit{PH}_i\\left(x_1,\\ldots,x_n\\right)$ denotes the $i$-dimensional reduced persistent homology of the \\v{C}ech complex of $\\left\\{x_1,\\ldots,x_n\\right\\}.$ These quantities are a higher-dimensional generalization of the $\\alpha$-weighted "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.07054","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}