{"paper":{"title":"Ricci Curvature and the Manifold Learning Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.MG","stat.ML"],"primary_cat":"math.DG","authors_text":"Antonio G. Ache, Micah W. Warren","submitted_at":"2014-10-13T15:37:20Z","abstract_excerpt":"Consider a sample of $n$ points taken i.i.d from a submanifold $\\Sigma$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $\\Sigma$ with respect to the induced metric from the sample. Our method is grounded in the notions of Carr\\'e du Champ for diffusion semi-groups, the theory of Empirical processes and local Principal Component Analysis."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3351","kind":"arxiv","version":5},"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"}