{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:3OSMFS2DUHLKQJYGNXKV7QXC3J","short_pith_number":"pith:3OSMFS2D","canonical_record":{"source":{"id":"2403.11730","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-03-18T12:39:58Z","cross_cats_sorted":["cs.NA","math.DG"],"title_canon_sha256":"7facf5b54cab8877d011ca2d29ce2e83f02dd272191a3b3e1dfc86ed7b1031cf","abstract_canon_sha256":"2ce9c09efb2cb14bbdb9accd829fb90691a26bf2ad2c4ab3b1c46a2330825449"},"schema_version":"1.0"},"canonical_sha256":"dba4c2cb43a1d6a827066dd55fc2e2da79ab8079b9b2b9e04b216997721f65d7","source":{"kind":"arxiv","id":"2403.11730","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.11730","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"arxiv_version","alias_value":"2403.11730v2","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.11730","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"pith_short_12","alias_value":"3OSMFS2DUHLK","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"pith_short_16","alias_value":"3OSMFS2DUHLKQJYG","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"pith_short_8","alias_value":"3OSMFS2D","created_at":"2026-07-05T09:54:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:3OSMFS2DUHLKQJYGNXKV7QXC3J","target":"record","payload":{"canonical_record":{"source":{"id":"2403.11730","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-03-18T12:39:58Z","cross_cats_sorted":["cs.NA","math.DG"],"title_canon_sha256":"7facf5b54cab8877d011ca2d29ce2e83f02dd272191a3b3e1dfc86ed7b1031cf","abstract_canon_sha256":"2ce9c09efb2cb14bbdb9accd829fb90691a26bf2ad2c4ab3b1c46a2330825449"},"schema_version":"1.0"},"canonical_sha256":"dba4c2cb43a1d6a827066dd55fc2e2da79ab8079b9b2b9e04b216997721f65d7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:54:25.405397Z","signature_b64":"KmrWhhDe0S80AqvjrvYG4O29RK2+tpc1cKBPcRqRhQX3Aa5bVHan+PQv0t8QdnnYDv8OIdvB+hFZGyGyuK78AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dba4c2cb43a1d6a827066dd55fc2e2da79ab8079b9b2b9e04b216997721f65d7","last_reissued_at":"2026-07-05T09:54:25.404953Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:54:25.404953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2403.11730","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:54:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lyu+OtDCOPqVbrjRjf0EsuWgmw3DlR1oVQofliECgvT9kUxlxYh2z0qnZEXlvlGdlJlUF/EEZ8cfjGtrQpxDAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T09:47:14.026520Z"},"content_sha256":"66e11d22373e56129fc9c7ee06bd6acfc446cf29897ff0943b8022479c0aa5a3","schema_version":"1.0","event_id":"sha256:66e11d22373e56129fc9c7ee06bd6acfc446cf29897ff0943b8022479c0aa5a3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:3OSMFS2DUHLKQJYGNXKV7QXC3J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An efficient algorithm for the Riemannian logarithm on the Stiefel manifold for a family of Riemannian metrics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.DG"],"primary_cat":"math.NA","authors_text":"Nina Miolane, Ralf Zimmermann, Simon Mataigne","submitted_at":"2024-03-18T12:39:58Z","abstract_excerpt":"Since the popularization of the Stiefel manifold for numerical applications in 1998 in a seminal paper from Edelman et al., it has been exhibited to be a key to solve many problems from optimization, statistics and machine learning. In 2021, H\\\"uper et al. proposed a one-parameter family of Riemannian metrics on the Stiefel manifold, subsuming the well-known Euclidean and canonical metrics. Since then, several methods have been proposed to obtain a candidate for the Riemannian logarithm given any metric from the family. Most of these methods are based on the shooting method or rely on optimiza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.11730","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/2403.11730/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:54:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tQLDFd+cz+fEZ8kGXle7Xb00OYV2ZaIWVbh5aYsP2K8Xldy0W8jmAXax/1VqveDOjOy7F85plY5sfTk44wqWBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T09:47:14.027113Z"},"content_sha256":"468ec83b42f44ffbe851ec18b17228f547f664cc15e75d1b91eeb0e5766a0ea6","schema_version":"1.0","event_id":"sha256:468ec83b42f44ffbe851ec18b17228f547f664cc15e75d1b91eeb0e5766a0ea6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3OSMFS2DUHLKQJYGNXKV7QXC3J/bundle.json","state_url":"https://pith.science/pith/3OSMFS2DUHLKQJYGNXKV7QXC3J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3OSMFS2DUHLKQJYGNXKV7QXC3J/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":"2026-08-18T09:47:14Z","links":{"resolver":"https://pith.science/pith/3OSMFS2DUHLKQJYGNXKV7QXC3J","bundle":"https://pith.science/pith/3OSMFS2DUHLKQJYGNXKV7QXC3J/bundle.json","state":"https://pith.science/pith/3OSMFS2DUHLKQJYGNXKV7QXC3J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3OSMFS2DUHLKQJYGNXKV7QXC3J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:3OSMFS2DUHLKQJYGNXKV7QXC3J","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":"2ce9c09efb2cb14bbdb9accd829fb90691a26bf2ad2c4ab3b1c46a2330825449","cross_cats_sorted":["cs.NA","math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-03-18T12:39:58Z","title_canon_sha256":"7facf5b54cab8877d011ca2d29ce2e83f02dd272191a3b3e1dfc86ed7b1031cf"},"schema_version":"1.0","source":{"id":"2403.11730","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.11730","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"arxiv_version","alias_value":"2403.11730v2","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.11730","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"pith_short_12","alias_value":"3OSMFS2DUHLK","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"pith_short_16","alias_value":"3OSMFS2DUHLKQJYG","created_at":"2026-07-05T09:54:25Z"},{"alias_kind":"pith_short_8","alias_value":"3OSMFS2D","created_at":"2026-07-05T09:54:25Z"}],"graph_snapshots":[{"event_id":"sha256:468ec83b42f44ffbe851ec18b17228f547f664cc15e75d1b91eeb0e5766a0ea6","target":"graph","created_at":"2026-07-05T09:54:25Z","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/2403.11730/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Since the popularization of the Stiefel manifold for numerical applications in 1998 in a seminal paper from Edelman et al., it has been exhibited to be a key to solve many problems from optimization, statistics and machine learning. In 2021, H\\\"uper et al. proposed a one-parameter family of Riemannian metrics on the Stiefel manifold, subsuming the well-known Euclidean and canonical metrics. Since then, several methods have been proposed to obtain a candidate for the Riemannian logarithm given any metric from the family. Most of these methods are based on the shooting method or rely on optimiza","authors_text":"Nina Miolane, Ralf Zimmermann, Simon Mataigne","cross_cats":["cs.NA","math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-03-18T12:39:58Z","title":"An efficient algorithm for the Riemannian logarithm on the Stiefel manifold for a family of Riemannian metrics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.11730","kind":"arxiv","version":2},"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":"sha256:66e11d22373e56129fc9c7ee06bd6acfc446cf29897ff0943b8022479c0aa5a3","target":"record","created_at":"2026-07-05T09:54:25Z","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":"2ce9c09efb2cb14bbdb9accd829fb90691a26bf2ad2c4ab3b1c46a2330825449","cross_cats_sorted":["cs.NA","math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-03-18T12:39:58Z","title_canon_sha256":"7facf5b54cab8877d011ca2d29ce2e83f02dd272191a3b3e1dfc86ed7b1031cf"},"schema_version":"1.0","source":{"id":"2403.11730","kind":"arxiv","version":2}},"canonical_sha256":"dba4c2cb43a1d6a827066dd55fc2e2da79ab8079b9b2b9e04b216997721f65d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dba4c2cb43a1d6a827066dd55fc2e2da79ab8079b9b2b9e04b216997721f65d7","first_computed_at":"2026-07-05T09:54:25.404953Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:54:25.404953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KmrWhhDe0S80AqvjrvYG4O29RK2+tpc1cKBPcRqRhQX3Aa5bVHan+PQv0t8QdnnYDv8OIdvB+hFZGyGyuK78AA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:54:25.405397Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.11730","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66e11d22373e56129fc9c7ee06bd6acfc446cf29897ff0943b8022479c0aa5a3","sha256:468ec83b42f44ffbe851ec18b17228f547f664cc15e75d1b91eeb0e5766a0ea6"],"state_sha256":"7b90c0ddbeb1551320673d0b316644a83bf81a11e5390007015e62547ec8bafa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nnS+JXtEmZMD/HLVHj26L7ZwASGQc4WIjcyHBzwPexx2edqF1uas9ZlXeG0dww1IBThfttvBwgyZ1+2z34ZLDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T09:47:14.032735Z","bundle_sha256":"8a9895850053959f2c302cd2ce366b7105e11899b38801fc953806098c70b869"}}