{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:CYJHYOVH5PF7TTVSWXO3KPNLQL","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":"b8ce54b006bb98edbfd0a8deb52d9945d5453fa7beda643287966e9477d8520f","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","title_canon_sha256":"fb4e396a9125c7c1eb46ff83fbe2c0394995360145bd545d6b14590df88f0023"},"schema_version":"1.0","source":{"id":"2307.04204","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2307.04204","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"arxiv_version","alias_value":"2307.04204v2","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.04204","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_12","alias_value":"CYJHYOVH5PF7","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_16","alias_value":"CYJHYOVH5PF7TTVS","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_8","alias_value":"CYJHYOVH","created_at":"2026-07-05T07:05:31Z"}],"graph_snapshots":[{"event_id":"sha256:fd6988d800c731d5f18b3596b942e273412c3000397bc4f929267c2e6259e24e","target":"graph","created_at":"2026-07-05T07:05:31Z","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/2307.04204/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Cohen et al. (2021) empirically study the evolution of the largest eigenvalue of the loss Hessian, also known as sharpness, along the gradient descent (GD) trajectory and observe the Edge of Stability (EoS) phenomenon. The sharpness increases at the early phase of training (referred to as progressive sharpening), and eventually saturates close to the threshold of $2 / \\text{(step size)}$. In this paper, we start by demonstrating through empirical studies that when the EoS phenomenon occurs, different GD trajectories (after a proper reparameterization) align on a specific bifurcation diagram in","authors_text":"Chulhee Yun, Minhak Song","cross_cats":["math.OC","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","title":"Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.04204","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:ef0bb8f4821889d2e8400a104f9332166f27ef475e87135e0f111a6069b4bf48","target":"record","created_at":"2026-07-05T07:05:31Z","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":"b8ce54b006bb98edbfd0a8deb52d9945d5453fa7beda643287966e9477d8520f","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","title_canon_sha256":"fb4e396a9125c7c1eb46ff83fbe2c0394995360145bd545d6b14590df88f0023"},"schema_version":"1.0","source":{"id":"2307.04204","kind":"arxiv","version":2}},"canonical_sha256":"16127c3aa7ebcbf9ceb2b5ddb53dab82cc7d8835de8af652c18b3eae69503322","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16127c3aa7ebcbf9ceb2b5ddb53dab82cc7d8835de8af652c18b3eae69503322","first_computed_at":"2026-07-05T07:05:31.556550Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:05:31.556550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3MgYxCdsYfnJRr5OCyoNhC+9ELJNS0aGMHiNL8VqbaxyF+koxKy5ELdemBOQhEgF+AypkrxfjYbC0rTB2VtbBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T07:05:31.557017Z","signed_message":"canonical_sha256_bytes"},"source_id":"2307.04204","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef0bb8f4821889d2e8400a104f9332166f27ef475e87135e0f111a6069b4bf48","sha256:fd6988d800c731d5f18b3596b942e273412c3000397bc4f929267c2e6259e24e"],"state_sha256":"bb7fb1a652fccee318288d7445ca436e197f87d8159e6b3aacca3d1e64a0ca41"}