{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:HPOEVWO3DRACURUBCMXJ5W5GMJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"14e341c2a21e5a8deeab46233b4cb888bca58e686fd721fb68cb4b03fa66529b","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"math.PR","submitted_at":"2006-12-27T10:32:59Z","title_canon_sha256":"51cb9988f5d06b372630c64134952dc148bf4742c0519ff70942f07cd8c6b519"},"schema_version":"1.0","source":{"id":"math/0612777","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0612777","created_at":"2026-05-18T01:08:49Z"},{"alias_kind":"arxiv_version","alias_value":"math/0612777v1","created_at":"2026-05-18T01:08:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612777","created_at":"2026-05-18T01:08:49Z"},{"alias_kind":"pith_short_12","alias_value":"HPOEVWO3DRAC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"HPOEVWO3DRACURUB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"HPOEVWO3","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:8ae7c84da55758ef251c97c5fd76f58b02944b6c255b3737c27dd23a7556f172","target":"graph","created_at":"2026-05-18T01:08:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Evarist Gin\\'e, Vladimir Koltchinskii","cross_cats":["math.DG"],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2006-12-27T10:32:59Z","title":"Empirical graph Laplacian approximation of Laplace--Beltrami operators: Large sample results"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612777","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6077eea471b8674908eb069461018cce13b79020fb493145221b783c2bd179c8","target":"record","created_at":"2026-05-18T01:08:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"14e341c2a21e5a8deeab46233b4cb888bca58e686fd721fb68cb4b03fa66529b","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"math.PR","submitted_at":"2006-12-27T10:32:59Z","title_canon_sha256":"51cb9988f5d06b372630c64134952dc148bf4742c0519ff70942f07cd8c6b519"},"schema_version":"1.0","source":{"id":"math/0612777","kind":"arxiv","version":1}},"canonical_sha256":"3bdc4ad9db1c402a4681132e9edba6626369bfa0408e1bfd6d26516cba4421bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3bdc4ad9db1c402a4681132e9edba6626369bfa0408e1bfd6d26516cba4421bc","first_computed_at":"2026-05-18T01:08:49.299172Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:49.299172Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F0BpmSTOiK7BVKTkMDRTSAmaW0VbfZqdd5/7uluBayOoEpOiLxQPBW8JFiDUijDKJkfbLw7r8t6Gt8tpuXPcCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:49.299698Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0612777","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6077eea471b8674908eb069461018cce13b79020fb493145221b783c2bd179c8","sha256:8ae7c84da55758ef251c97c5fd76f58b02944b6c255b3737c27dd23a7556f172"],"state_sha256":"19937ce1c15619882ebb19093329fcd2e56a4ecff955e28379b148ce6eacae3b"}