{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:OCYV5ZSTYU3EB46GAB6ANNRSPW","short_pith_number":"pith:OCYV5ZST","schema_version":"1.0","canonical_sha256":"70b15ee653c53640f3c6007c06b6327da3e7e135daa655500876338d9e2209da","source":{"kind":"arxiv","id":"2307.02378","version":2},"attestation_state":"computed","paper":{"title":"Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.AP","stat.ML"],"primary_cat":"math.DG","authors_text":"Melanie Weber, Nicolas Garcia Trillos","submitted_at":"2023-07-05T15:45:53Z","abstract_excerpt":"Let $M$ denote a low-dimensional manifold embedded in Euclidean space and let ${X}= \\{ x_1, \\dots, x_n \\}$ be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph built from ${X}$ and the curvature of the manifold $M$ via continuum limits of Ollivier's discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if $M$ has Ricci curvature bounded from below by a positive constant, then the random geometric graph will inherit this global structural property with high probability. We d"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2307.02378","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2023-07-05T15:45:53Z","cross_cats_sorted":["cs.LG","math.AP","stat.ML"],"title_canon_sha256":"f65a0641fa2c79aa5ea0021b6c56a7c56bad1e2edbfe6953eab9962c32866898","abstract_canon_sha256":"bff7073f2b8122846543a9caeeeed82f09960c75b3d0d972bd9a80c34ab263e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:59:03.019349Z","signature_b64":"T6HlEeKGXwgsA7WN6qy0VgvK0g8XcPzBSfdaCY+BkGxUKcs0Ua7ndIa2x2p1EQcDgAQHPUNiutl5n+zD+BBUCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70b15ee653c53640f3c6007c06b6327da3e7e135daa655500876338d9e2209da","last_reissued_at":"2026-07-05T08:59:03.018876Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:59:03.018876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.AP","stat.ML"],"primary_cat":"math.DG","authors_text":"Melanie Weber, Nicolas Garcia Trillos","submitted_at":"2023-07-05T15:45:53Z","abstract_excerpt":"Let $M$ denote a low-dimensional manifold embedded in Euclidean space and let ${X}= \\{ x_1, \\dots, x_n \\}$ be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph built from ${X}$ and the curvature of the manifold $M$ via continuum limits of Ollivier's discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if $M$ has Ricci curvature bounded from below by a positive constant, then the random geometric graph will inherit this global structural property with high probability. We d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.02378","kind":"arxiv","version":2},"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/2307.02378/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2307.02378","created_at":"2026-07-05T08:59:03.018929+00:00"},{"alias_kind":"arxiv_version","alias_value":"2307.02378v2","created_at":"2026-07-05T08:59:03.018929+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.02378","created_at":"2026-07-05T08:59:03.018929+00:00"},{"alias_kind":"pith_short_12","alias_value":"OCYV5ZSTYU3E","created_at":"2026-07-05T08:59:03.018929+00:00"},{"alias_kind":"pith_short_16","alias_value":"OCYV5ZSTYU3EB46G","created_at":"2026-07-05T08:59:03.018929+00:00"},{"alias_kind":"pith_short_8","alias_value":"OCYV5ZST","created_at":"2026-07-05T08:59:03.018929+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.01107","citing_title":"Diffusion Operator Geometry of Feedforward Representations","ref_index":14,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW","json":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW.json","graph_json":"https://pith.science/api/pith-number/OCYV5ZSTYU3EB46GAB6ANNRSPW/graph.json","events_json":"https://pith.science/api/pith-number/OCYV5ZSTYU3EB46GAB6ANNRSPW/events.json","paper":"https://pith.science/paper/OCYV5ZST"},"agent_actions":{"view_html":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW","download_json":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW.json","view_paper":"https://pith.science/paper/OCYV5ZST","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2307.02378&json=true","fetch_graph":"https://pith.science/api/pith-number/OCYV5ZSTYU3EB46GAB6ANNRSPW/graph.json","fetch_events":"https://pith.science/api/pith-number/OCYV5ZSTYU3EB46GAB6ANNRSPW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW/action/storage_attestation","attest_author":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW/action/author_attestation","sign_citation":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW/action/citation_signature","submit_replication":"https://pith.science/pith/OCYV5ZSTYU3EB46GAB6ANNRSPW/action/replication_record"}},"created_at":"2026-07-05T08:59:03.018929+00:00","updated_at":"2026-07-05T08:59:03.018929+00:00"}