{"paper":{"title":"Central limit theorems for the eigenvalues of graph Laplacians on data clouds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.AP","math.DG","math.PR"],"primary_cat":"stat.ML","authors_text":"Chenghui Li, Housen Li, Leo Suchan, Nicol\\'as Garc\\'ia Trillos","submitted_at":"2025-07-24T21:03:20Z","abstract_excerpt":"Given i.i.d.\\ samples $X_n =\\{ x_1, \\dots, x_n \\}$ from a distribution supported on a low dimensional manifold ${M}$ embedded in Eucliden space, we consider the graph Laplacian operator $\\Delta_n$ associated to an $\\varepsilon$-proximity graph over $X_n$ and study the asymptotic fluctuations of its eigenvalues around their means. In particular, letting $\\hat{\\lambda}_l^\\varepsilon$ denote the $l$-th eigenvalue of $\\Delta_n$, and under suitable assumptions on the data generating model and on the rate of decay of $\\varepsilon$, we prove that $\\sqrt{n } (\\hat{\\lambda}_{l}^\\varepsilon - \\mathbb{E}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.18803","kind":"arxiv","version":1},"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/2507.18803/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"}