{"paper":{"title":"Fr\\'echet Means for Distributions of Persistence diagrams","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN","math.MG","stat.TH"],"primary_cat":"math.ST","authors_text":"John Harer, Katharine Turner, Sayan Mukherjee, Yuriy Mileyko","submitted_at":"2012-06-13T13:17:02Z","abstract_excerpt":"Given a distribution $\\rho$ on persistence diagrams and observations $X_1,...X_n \\stackrel{iid}{\\sim} \\rho$ we introduce an algorithm in this paper that estimates a Fr\\'echet mean from the set of diagrams $X_1,...X_n$. If the underlying measure $\\rho$ is a combination of Dirac masses $\\rho = \\frac{1}{m} \\sum_{i=1}^m \\delta_{Z_i}$ then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fr\\'echet mean computed by the algorithm given observations drawn iid from $\\rho$. We illustrate the convergence of an empirical mean computed by the algorithm to a popula"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2790","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"}