{"paper":{"title":"Empirical graph Laplacian approximation of Laplace--Beltrami operators: Large sample results","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"math.PR","authors_text":"Evarist Gin\\'e, Vladimir Koltchinskii","submitted_at":"2006-12-27T10:32:59Z","abstract_excerpt":"Let ${M}$ be a compact Riemannian submanifold of ${{\\bf R}^m}$ of dimension $\\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators $$ \\Delta_{h_n,n}f(p):=\\frac{1}{nh_n^{d+2}}\\sum_{i=1}^n K(\\frac{p-X_i}{h_n})(f(X_i)-f(p)), p\\in M $$ where ${K(u):={\\frac{1}{(4\\pi)^{d/2}}}e^{-\\|u\\|^2/4}}$ is the Gaussian kernel and ${h_n\\to 0}$ as ${n\\to\\infty.}$ Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approxima"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612777","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"}