{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:S24JTVYT4XVAF7UUPH3TXF2CMN","short_pith_number":"pith:S24JTVYT","canonical_record":{"source":{"id":"2406.06100","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-06-10T08:34:59Z","cross_cats_sorted":[],"title_canon_sha256":"76f2e0421237fbe1a236c94c2febc5759eb8187696693ee779494c48a93fccd9","abstract_canon_sha256":"741fdee1a78adeeece8df235d001f5c9471b95a0ac53ce43f2f1e95176932563"},"schema_version":"1.0"},"canonical_sha256":"96b899d713e5ea02fe9479f73b97426351fad539b0ee6c80e01127192499e852","source":{"kind":"arxiv","id":"2406.06100","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.06100","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"arxiv_version","alias_value":"2406.06100v2","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.06100","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"pith_short_12","alias_value":"S24JTVYT4XVA","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"pith_short_16","alias_value":"S24JTVYT4XVAF7UU","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"pith_short_8","alias_value":"S24JTVYT","created_at":"2026-07-05T10:56:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:S24JTVYT4XVAF7UUPH3TXF2CMN","target":"record","payload":{"canonical_record":{"source":{"id":"2406.06100","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-06-10T08:34:59Z","cross_cats_sorted":[],"title_canon_sha256":"76f2e0421237fbe1a236c94c2febc5759eb8187696693ee779494c48a93fccd9","abstract_canon_sha256":"741fdee1a78adeeece8df235d001f5c9471b95a0ac53ce43f2f1e95176932563"},"schema_version":"1.0"},"canonical_sha256":"96b899d713e5ea02fe9479f73b97426351fad539b0ee6c80e01127192499e852","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:56:33.409270Z","signature_b64":"oTfw1MPN1iziPFvHBgJQ7LeJahmBuc5yobd2hv0gN7ZAiLiuIV2rwJ66YJjmGsfjCNJppcBuf/ieEgq6600fBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"96b899d713e5ea02fe9479f73b97426351fad539b0ee6c80e01127192499e852","last_reissued_at":"2026-07-05T10:56:33.408783Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:56:33.408783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2406.06100","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":"2026-07-05T10:56:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+thbjzA2L5j5J1vqix3FusBHHnyA6yUIZ4FA3Cby+1wNniweiNl8bA6XQm/8nS0ufcXhAvbVidRs964/s7n/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T11:17:18.043816Z"},"content_sha256":"ff8e309e90d425c330e1a8be8288676c565a293435d4047dd66911bda5be1062","schema_version":"1.0","event_id":"sha256:ff8e309e90d425c330e1a8be8288676c565a293435d4047dd66911bda5be1062"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:S24JTVYT4XVAF7UUPH3TXF2CMN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Heavy-ball Differential Equation Achieves $O(\\varepsilon^{-7/4})$ Convergence for Nonconvex Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Akiko Takeda, Kaito Okamura, Naoki Marumo","submitted_at":"2024-06-10T08:34:59Z","abstract_excerpt":"First-order optimization methods for nonconvex functions with Lipschitz continuous gradient and Hessian have been extensively studied. State-of-the-art methods for finding an $\\varepsilon$-stationary point within $O(\\varepsilon^{-{7/4}})$ or $\\tilde{O}(\\varepsilon^{-{7/4}})$ gradient evaluations are based on Nesterov's accelerated gradient descent (AGD) or Polyak's heavy-ball (HB) method. However, these algorithms employ additional mechanisms, such as restart schemes and negative curvature exploitation, which complicate their behavior and make it challenging to apply them to more advanced sett"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.06100","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/2406.06100/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":"2026-07-05T10:56:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NdTR1Ym1/+C7M1Dmk4A6RIEiSfmbb7cD9lwlX03LfEHv/awDHoSwCurstSGPcYKh3fAmP7XBf5Jva9mh42OqCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T11:17:18.044338Z"},"content_sha256":"5cbd8314af8d5fd610b01bfcb6c9914e33f7c260d8c7d957f4cf0ba99f7477ba","schema_version":"1.0","event_id":"sha256:5cbd8314af8d5fd610b01bfcb6c9914e33f7c260d8c7d957f4cf0ba99f7477ba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S24JTVYT4XVAF7UUPH3TXF2CMN/bundle.json","state_url":"https://pith.science/pith/S24JTVYT4XVAF7UUPH3TXF2CMN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S24JTVYT4XVAF7UUPH3TXF2CMN/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":"2026-08-09T11:17:18Z","links":{"resolver":"https://pith.science/pith/S24JTVYT4XVAF7UUPH3TXF2CMN","bundle":"https://pith.science/pith/S24JTVYT4XVAF7UUPH3TXF2CMN/bundle.json","state":"https://pith.science/pith/S24JTVYT4XVAF7UUPH3TXF2CMN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S24JTVYT4XVAF7UUPH3TXF2CMN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:S24JTVYT4XVAF7UUPH3TXF2CMN","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":"741fdee1a78adeeece8df235d001f5c9471b95a0ac53ce43f2f1e95176932563","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-06-10T08:34:59Z","title_canon_sha256":"76f2e0421237fbe1a236c94c2febc5759eb8187696693ee779494c48a93fccd9"},"schema_version":"1.0","source":{"id":"2406.06100","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.06100","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"arxiv_version","alias_value":"2406.06100v2","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.06100","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"pith_short_12","alias_value":"S24JTVYT4XVA","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"pith_short_16","alias_value":"S24JTVYT4XVAF7UU","created_at":"2026-07-05T10:56:33Z"},{"alias_kind":"pith_short_8","alias_value":"S24JTVYT","created_at":"2026-07-05T10:56:33Z"}],"graph_snapshots":[{"event_id":"sha256:5cbd8314af8d5fd610b01bfcb6c9914e33f7c260d8c7d957f4cf0ba99f7477ba","target":"graph","created_at":"2026-07-05T10:56:33Z","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/2406.06100/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"First-order optimization methods for nonconvex functions with Lipschitz continuous gradient and Hessian have been extensively studied. State-of-the-art methods for finding an $\\varepsilon$-stationary point within $O(\\varepsilon^{-{7/4}})$ or $\\tilde{O}(\\varepsilon^{-{7/4}})$ gradient evaluations are based on Nesterov's accelerated gradient descent (AGD) or Polyak's heavy-ball (HB) method. However, these algorithms employ additional mechanisms, such as restart schemes and negative curvature exploitation, which complicate their behavior and make it challenging to apply them to more advanced sett","authors_text":"Akiko Takeda, Kaito Okamura, Naoki Marumo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-06-10T08:34:59Z","title":"Heavy-ball Differential Equation Achieves $O(\\varepsilon^{-7/4})$ Convergence for Nonconvex Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.06100","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:ff8e309e90d425c330e1a8be8288676c565a293435d4047dd66911bda5be1062","target":"record","created_at":"2026-07-05T10:56:33Z","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":"741fdee1a78adeeece8df235d001f5c9471b95a0ac53ce43f2f1e95176932563","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-06-10T08:34:59Z","title_canon_sha256":"76f2e0421237fbe1a236c94c2febc5759eb8187696693ee779494c48a93fccd9"},"schema_version":"1.0","source":{"id":"2406.06100","kind":"arxiv","version":2}},"canonical_sha256":"96b899d713e5ea02fe9479f73b97426351fad539b0ee6c80e01127192499e852","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"96b899d713e5ea02fe9479f73b97426351fad539b0ee6c80e01127192499e852","first_computed_at":"2026-07-05T10:56:33.408783Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:56:33.408783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oTfw1MPN1iziPFvHBgJQ7LeJahmBuc5yobd2hv0gN7ZAiLiuIV2rwJ66YJjmGsfjCNJppcBuf/ieEgq6600fBA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:56:33.409270Z","signed_message":"canonical_sha256_bytes"},"source_id":"2406.06100","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ff8e309e90d425c330e1a8be8288676c565a293435d4047dd66911bda5be1062","sha256:5cbd8314af8d5fd610b01bfcb6c9914e33f7c260d8c7d957f4cf0ba99f7477ba"],"state_sha256":"f86d830d908d463b78b437b3f3d744350d43157c7b413e683c38bbd9255a5309"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2/JuHIeI0xblkqVPMG1IiSbxLn4ZEXZWOYLevvRdArh1cd/FvRzhPuRCRKi8ttS6HEYxzku713a4xQx9gOP3DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T11:17:18.048362Z","bundle_sha256":"058cca9e89dac1042ec73d745b8d01be176c52242a756b89e83335be90e9e203"}}