{"paper":{"title":"Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG","math.PR"],"primary_cat":"stat.ML","authors_text":"Dejan Slepcev, Matthias Hein, Moritz Gerlach, Nicolas Garcia Trillos","submitted_at":"2018-01-30T17:23:27Z","abstract_excerpt":"We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of $O\\Big(\\big(\\frac{\\log n}{n}\\big)^\\frac{1}{2m}\\Big)$ to the eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of $M$.\n  No information on the submanifold $M$ is needed in the construction of the graph or the \"out-of-sample extension\" of the eigenvectors. O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.10108","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":""},"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"}