{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:UBIOEGEH7P6OU5WSET3XJHTNNA","short_pith_number":"pith:UBIOEGEH","canonical_record":{"source":{"id":"1111.3120","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-14T07:36:52Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"9499668577898528035cd4d2ed8983e4e3f725419a3fa2b14a0ae61cabca55a1","abstract_canon_sha256":"8473ed4251607a21b7ce6bb32ee7d3eca7a0f3faff838086b1cf006d6c412c97"},"schema_version":"1.0"},"canonical_sha256":"a050e21887fbfcea76d224f7749e6d68195bb136cff38182802d8499de7c5af0","source":{"kind":"arxiv","id":"1111.3120","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3120","created_at":"2026-05-18T04:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3120v1","created_at":"2026-05-18T04:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3120","created_at":"2026-05-18T04:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"UBIOEGEH7P6O","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"UBIOEGEH7P6OU5WS","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"UBIOEGEH","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:UBIOEGEH7P6OU5WSET3XJHTNNA","target":"record","payload":{"canonical_record":{"source":{"id":"1111.3120","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-14T07:36:52Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"9499668577898528035cd4d2ed8983e4e3f725419a3fa2b14a0ae61cabca55a1","abstract_canon_sha256":"8473ed4251607a21b7ce6bb32ee7d3eca7a0f3faff838086b1cf006d6c412c97"},"schema_version":"1.0"},"canonical_sha256":"a050e21887fbfcea76d224f7749e6d68195bb136cff38182802d8499de7c5af0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:14.235412Z","signature_b64":"uz8QFb4zBt+bu8Iw6xQ4MmJOwX/5YeynjMImtF3uXH2StF3LfjdPG9idRJChk9ckp+VKN0w3480G8eLYEXMsAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a050e21887fbfcea76d224f7749e6d68195bb136cff38182802d8499de7c5af0","last_reissued_at":"2026-05-18T04:08:14.234891Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:14.234891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.3120","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ukHAJUDhygKMib0+o9h/wlw1b06c21dhXrg4Qt8+zMlUJXW93BJ1LSJLbaJeE+aSvTuxkY6wpE9YDBnv7oU8Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T00:32:06.837641Z"},"content_sha256":"e9238fe5288bc077c8b8b785b9195d5c1e2270289414b020cfa2333d6afa27a6","schema_version":"1.0","event_id":"sha256:e9238fe5288bc077c8b8b785b9195d5c1e2270289414b020cfa2333d6afa27a6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:UBIOEGEH7P6OU5WSET3XJHTNNA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Medians and means in Riemannian geometry: existence, uniqueness and computation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.DG","authors_text":"Fr\\'ed\\'eric Barbaresco, Le Yang (LMA), Marc Arnaudon (LMA)","submitted_at":"2011-11-14T07:36:52Z","abstract_excerpt":"This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Fr\\'echet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Fr\\'echet medians of generic data points are always u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3120","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k0XmICVhGvT2d1gAaaNjw2c6bNnXhoz9W0JRFg56QnGwYTbrNpwgAXHdCNEB+2NcyXBI6nAul/8WjHmTg5s0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T00:32:06.837982Z"},"content_sha256":"abd39cf6f095913b2f02349751804414ed27683af7cba519ed0cc029156c594e","schema_version":"1.0","event_id":"sha256:abd39cf6f095913b2f02349751804414ed27683af7cba519ed0cc029156c594e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UBIOEGEH7P6OU5WSET3XJHTNNA/bundle.json","state_url":"https://pith.science/pith/UBIOEGEH7P6OU5WSET3XJHTNNA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UBIOEGEH7P6OU5WSET3XJHTNNA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-29T00:32:06Z","links":{"resolver":"https://pith.science/pith/UBIOEGEH7P6OU5WSET3XJHTNNA","bundle":"https://pith.science/pith/UBIOEGEH7P6OU5WSET3XJHTNNA/bundle.json","state":"https://pith.science/pith/UBIOEGEH7P6OU5WSET3XJHTNNA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UBIOEGEH7P6OU5WSET3XJHTNNA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:UBIOEGEH7P6OU5WSET3XJHTNNA","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":"8473ed4251607a21b7ce6bb32ee7d3eca7a0f3faff838086b1cf006d6c412c97","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-14T07:36:52Z","title_canon_sha256":"9499668577898528035cd4d2ed8983e4e3f725419a3fa2b14a0ae61cabca55a1"},"schema_version":"1.0","source":{"id":"1111.3120","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3120","created_at":"2026-05-18T04:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3120v1","created_at":"2026-05-18T04:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3120","created_at":"2026-05-18T04:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"UBIOEGEH7P6O","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"UBIOEGEH7P6OU5WS","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"UBIOEGEH","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:abd39cf6f095913b2f02349751804414ed27683af7cba519ed0cc029156c594e","target":"graph","created_at":"2026-05-18T04:08:14Z","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":"This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Fr\\'echet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Fr\\'echet medians of generic data points are always u","authors_text":"Fr\\'ed\\'eric Barbaresco, Le Yang (LMA), Marc Arnaudon (LMA)","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-14T07:36:52Z","title":"Medians and means in Riemannian geometry: existence, uniqueness and computation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3120","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:e9238fe5288bc077c8b8b785b9195d5c1e2270289414b020cfa2333d6afa27a6","target":"record","created_at":"2026-05-18T04:08:14Z","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":"8473ed4251607a21b7ce6bb32ee7d3eca7a0f3faff838086b1cf006d6c412c97","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-14T07:36:52Z","title_canon_sha256":"9499668577898528035cd4d2ed8983e4e3f725419a3fa2b14a0ae61cabca55a1"},"schema_version":"1.0","source":{"id":"1111.3120","kind":"arxiv","version":1}},"canonical_sha256":"a050e21887fbfcea76d224f7749e6d68195bb136cff38182802d8499de7c5af0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a050e21887fbfcea76d224f7749e6d68195bb136cff38182802d8499de7c5af0","first_computed_at":"2026-05-18T04:08:14.234891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:14.234891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uz8QFb4zBt+bu8Iw6xQ4MmJOwX/5YeynjMImtF3uXH2StF3LfjdPG9idRJChk9ckp+VKN0w3480G8eLYEXMsAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:14.235412Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.3120","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9238fe5288bc077c8b8b785b9195d5c1e2270289414b020cfa2333d6afa27a6","sha256:abd39cf6f095913b2f02349751804414ed27683af7cba519ed0cc029156c594e"],"state_sha256":"3e82b6088b5b79818acad8185753312d167735fc8f0024e71f42dbb4c33296df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ia6vH73PK/BHihPo1cx2z88HPu1oIxd7NvGs7p7R3Ql7i9ee+oOtKKji4e3OGuXdD5SMCsMCsTqkS1RJ5BPtDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T00:32:06.840267Z","bundle_sha256":"ec51f77ef11bc43ec4c224082cad2a9eaf2e1fc42bf1d31b6c7a65224bb3bee3"}}