{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:VRKKWP54YNFSPOE66CNOOQ5D2C","short_pith_number":"pith:VRKKWP54","canonical_record":{"source":{"id":"2405.12888","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-05-21T15:59:55Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"ca1c6e42320eafde56bd349d93091a65b8d41d4de6121dbd8bb4176923499fda","abstract_canon_sha256":"ce4f31852dccb3ad5022a3927cce93eaa23c90636ae1a57af4570263287d7808"},"schema_version":"1.0"},"canonical_sha256":"ac54ab3fbcc34b27b89ef09ae743a3d0aa568d46a30c29689e8049c56ad79ab7","source":{"kind":"arxiv","id":"2405.12888","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.12888","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"arxiv_version","alias_value":"2405.12888v1","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.12888","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"pith_short_12","alias_value":"VRKKWP54YNFS","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"pith_short_16","alias_value":"VRKKWP54YNFSPOE6","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"pith_short_8","alias_value":"VRKKWP54","created_at":"2026-07-05T08:21:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:VRKKWP54YNFSPOE66CNOOQ5D2C","target":"record","payload":{"canonical_record":{"source":{"id":"2405.12888","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-05-21T15:59:55Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"ca1c6e42320eafde56bd349d93091a65b8d41d4de6121dbd8bb4176923499fda","abstract_canon_sha256":"ce4f31852dccb3ad5022a3927cce93eaa23c90636ae1a57af4570263287d7808"},"schema_version":"1.0"},"canonical_sha256":"ac54ab3fbcc34b27b89ef09ae743a3d0aa568d46a30c29689e8049c56ad79ab7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:21:27.910721Z","signature_b64":"0qoR6OLTgZ4gne7TKcNqk9Gr7R/qC+iCjO9QEOKjEzxw7dcl68LC9IFEbNrziDlLsq8LYjFvWlxNvEihYILLBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac54ab3fbcc34b27b89ef09ae743a3d0aa568d46a30c29689e8049c56ad79ab7","last_reissued_at":"2026-07-05T08:21:27.910120Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:21:27.910120Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2405.12888","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":"2026-07-05T08:21:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fz5UsHm5X5SXzKIeDJQUN/8PBaH7o2kPASJh0Q3deJhOp7x8v2lmVCO4eh43Eo0MWdeuLiylA0mXetlv+TYCDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T02:23:16.625602Z"},"content_sha256":"ea4a11b9ef1ac56b501ec8eefa13502c2024ea46f2d6223694832247770693ac","schema_version":"1.0","event_id":"sha256:ea4a11b9ef1ac56b501ec8eefa13502c2024ea46f2d6223694832247770693ac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:VRKKWP54YNFSPOE66CNOOQ5D2C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Keep the Momentum: Conservation Laws beyond Euclidean Gradient Flows","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Gabriel Peyr\\'e, R\\'emi Gribonval, Sibylle Marcotte","submitted_at":"2024-05-21T15:59:55Z","abstract_excerpt":"Conservation laws are well-established in the context of Euclidean gradient flow dynamics, notably for linear or ReLU neural network training. Yet, their existence and principles for non-Euclidean geometries and momentum-based dynamics remain largely unknown. In this paper, we characterize \"all\" conservation laws in this general setting. In stark contrast to the case of gradient flows, we prove that the conservation laws for momentum-based dynamics exhibit temporal dependence. Additionally, we often observe a \"conservation loss\" when transitioning from gradient flow to momentum dynamics. Speci"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.12888","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.12888/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":"2026-07-05T08:21:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"krx+591hIL59/yNa+eSdQwKFcCiJVqlS14cozS3nEjc929cpbPfvXiEFS4AtL1AlrB3J7VcEMBv8A0SOYXOdCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T02:23:16.626111Z"},"content_sha256":"69b5626e46a6becc4f64fafd51118b251b927d3b30ab4726856bf20d8ea7185f","schema_version":"1.0","event_id":"sha256:69b5626e46a6becc4f64fafd51118b251b927d3b30ab4726856bf20d8ea7185f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VRKKWP54YNFSPOE66CNOOQ5D2C/bundle.json","state_url":"https://pith.science/pith/VRKKWP54YNFSPOE66CNOOQ5D2C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VRKKWP54YNFSPOE66CNOOQ5D2C/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":"2026-08-21T02:23:16Z","links":{"resolver":"https://pith.science/pith/VRKKWP54YNFSPOE66CNOOQ5D2C","bundle":"https://pith.science/pith/VRKKWP54YNFSPOE66CNOOQ5D2C/bundle.json","state":"https://pith.science/pith/VRKKWP54YNFSPOE66CNOOQ5D2C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VRKKWP54YNFSPOE66CNOOQ5D2C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:VRKKWP54YNFSPOE66CNOOQ5D2C","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":"ce4f31852dccb3ad5022a3927cce93eaa23c90636ae1a57af4570263287d7808","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-05-21T15:59:55Z","title_canon_sha256":"ca1c6e42320eafde56bd349d93091a65b8d41d4de6121dbd8bb4176923499fda"},"schema_version":"1.0","source":{"id":"2405.12888","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.12888","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"arxiv_version","alias_value":"2405.12888v1","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.12888","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"pith_short_12","alias_value":"VRKKWP54YNFS","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"pith_short_16","alias_value":"VRKKWP54YNFSPOE6","created_at":"2026-07-05T08:21:27Z"},{"alias_kind":"pith_short_8","alias_value":"VRKKWP54","created_at":"2026-07-05T08:21:27Z"}],"graph_snapshots":[{"event_id":"sha256:69b5626e46a6becc4f64fafd51118b251b927d3b30ab4726856bf20d8ea7185f","target":"graph","created_at":"2026-07-05T08:21:27Z","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/2405.12888/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Conservation laws are well-established in the context of Euclidean gradient flow dynamics, notably for linear or ReLU neural network training. Yet, their existence and principles for non-Euclidean geometries and momentum-based dynamics remain largely unknown. In this paper, we characterize \"all\" conservation laws in this general setting. In stark contrast to the case of gradient flows, we prove that the conservation laws for momentum-based dynamics exhibit temporal dependence. Additionally, we often observe a \"conservation loss\" when transitioning from gradient flow to momentum dynamics. Speci","authors_text":"Gabriel Peyr\\'e, R\\'emi Gribonval, Sibylle Marcotte","cross_cats":["math.OC"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-05-21T15:59:55Z","title":"Keep the Momentum: Conservation Laws beyond Euclidean Gradient Flows"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.12888","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:ea4a11b9ef1ac56b501ec8eefa13502c2024ea46f2d6223694832247770693ac","target":"record","created_at":"2026-07-05T08:21:27Z","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":"ce4f31852dccb3ad5022a3927cce93eaa23c90636ae1a57af4570263287d7808","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-05-21T15:59:55Z","title_canon_sha256":"ca1c6e42320eafde56bd349d93091a65b8d41d4de6121dbd8bb4176923499fda"},"schema_version":"1.0","source":{"id":"2405.12888","kind":"arxiv","version":1}},"canonical_sha256":"ac54ab3fbcc34b27b89ef09ae743a3d0aa568d46a30c29689e8049c56ad79ab7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac54ab3fbcc34b27b89ef09ae743a3d0aa568d46a30c29689e8049c56ad79ab7","first_computed_at":"2026-07-05T08:21:27.910120Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:21:27.910120Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0qoR6OLTgZ4gne7TKcNqk9Gr7R/qC+iCjO9QEOKjEzxw7dcl68LC9IFEbNrziDlLsq8LYjFvWlxNvEihYILLBA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:21:27.910721Z","signed_message":"canonical_sha256_bytes"},"source_id":"2405.12888","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea4a11b9ef1ac56b501ec8eefa13502c2024ea46f2d6223694832247770693ac","sha256:69b5626e46a6becc4f64fafd51118b251b927d3b30ab4726856bf20d8ea7185f"],"state_sha256":"200c52ad9e677cae6efd477f4e4a1bdc8e4f1c96b2ec85cef2fba90df517dc92"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yl39YqBIJuvymts51k4aeocDbdi05Pxe0aCX2T77ZEhNHXbIjZoXyI3JRz55Fy80ZKBWy+A0R6xCDoxKMaksDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T02:23:16.631196Z","bundle_sha256":"6eca4174de952066010409bd8a3dee498494281a8c0d8bbf92ca3d05e08ff0ec"}}