{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:QH5L3TCHMNPQT6TED2Y6FFN5HE","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":"51a784119604e5e3716bab83f5783cd65bec1433c346e0ac0bafb09b5be456cd","cross_cats_sorted":["stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-10-08T16:08:38Z","title_canon_sha256":"040d94d3b5bf32352ca155f286273ff1027d02b2db1c9dfaf6819946106d1d8b"},"schema_version":"1.0","source":{"id":"2410.06164","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.06164","created_at":"2026-07-05T09:17:56Z"},{"alias_kind":"arxiv_version","alias_value":"2410.06164v1","created_at":"2026-07-05T09:17:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.06164","created_at":"2026-07-05T09:17:56Z"},{"alias_kind":"pith_short_12","alias_value":"QH5L3TCHMNPQ","created_at":"2026-07-05T09:17:56Z"},{"alias_kind":"pith_short_16","alias_value":"QH5L3TCHMNPQT6TE","created_at":"2026-07-05T09:17:56Z"},{"alias_kind":"pith_short_8","alias_value":"QH5L3TCH","created_at":"2026-07-05T09:17:56Z"}],"graph_snapshots":[{"event_id":"sha256:fad652c97f305a48f905f2f1c39e0914d5c6535a24cb51293867e664b52a852d","target":"graph","created_at":"2026-07-05T09:17:56Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2410.06164/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The extension of bivariate measures of dependence to non-Euclidean spaces is a challenging problem. The non-linear nature of these spaces makes the generalisation of classical measures of linear dependence (such as the covariance) not trivial. In this paper, we propose a novel approach to measure stochastic dependence between two random variables taking values in a Riemannian manifold, with the aim of both generalising the classical concepts of covariance and correlation and building a connection to Fr\\'echet moments of random variables on manifolds. We introduce generalised local measures of ","authors_text":"Davide Pigoli, Meshal Abuqrais","cross_cats":["stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-10-08T16:08:38Z","title":"A Riemannian covariance for manifold-valued data"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06164","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:bafc4cc01053151d35c97ba8b580ddface18f360a2400c7c68a4e6ae11d73be3","target":"record","created_at":"2026-07-05T09:17:56Z","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":"51a784119604e5e3716bab83f5783cd65bec1433c346e0ac0bafb09b5be456cd","cross_cats_sorted":["stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-10-08T16:08:38Z","title_canon_sha256":"040d94d3b5bf32352ca155f286273ff1027d02b2db1c9dfaf6819946106d1d8b"},"schema_version":"1.0","source":{"id":"2410.06164","kind":"arxiv","version":1}},"canonical_sha256":"81fabdcc47635f09fa641eb1e295bd391b8baad50672e001803772afd9763749","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81fabdcc47635f09fa641eb1e295bd391b8baad50672e001803772afd9763749","first_computed_at":"2026-07-05T09:17:56.873678Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:17:56.873678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L9a77zOHshc5TBTmgGCFUDzdzyDw2JpFOk5DUVSpKUhghaFsRDvy5dR7qi8TQ+orU+OIEvzrMXX89HYyK0rDCg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:17:56.874069Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.06164","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bafc4cc01053151d35c97ba8b580ddface18f360a2400c7c68a4e6ae11d73be3","sha256:fad652c97f305a48f905f2f1c39e0914d5c6535a24cb51293867e664b52a852d"],"state_sha256":"dde04f0a6c10cf7477b62acb673a3fa45bdd389b1e3eae4c8044159f99697de2"}