{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:LKLLDU4DT3JQXQREPPFMNRR3XW","short_pith_number":"pith:LKLLDU4D","canonical_record":{"source":{"id":"2008.02491","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2020-08-06T07:33:17Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"e0eb2542e9137cd7680952411a2087d9b9bdfc6c30a959c3852216fe06015eb1","abstract_canon_sha256":"6857e2ea15e552b8912f8afe817f629fb25a582efe001094c4882e1f3e5ffe09"},"schema_version":"1.0"},"canonical_sha256":"5a96b1d3839ed30bc2247bcac6c63bbd8b5d49ce095a3c7237a7ff822b94ebec","source":{"kind":"arxiv","id":"2008.02491","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2008.02491","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"arxiv_version","alias_value":"2008.02491v2","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2008.02491","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"pith_short_12","alias_value":"LKLLDU4DT3JQ","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"pith_short_16","alias_value":"LKLLDU4DT3JQXQRE","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"pith_short_8","alias_value":"LKLLDU4D","created_at":"2026-07-05T02:27:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:LKLLDU4DT3JQXQREPPFMNRR3XW","target":"record","payload":{"canonical_record":{"source":{"id":"2008.02491","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2020-08-06T07:33:17Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"e0eb2542e9137cd7680952411a2087d9b9bdfc6c30a959c3852216fe06015eb1","abstract_canon_sha256":"6857e2ea15e552b8912f8afe817f629fb25a582efe001094c4882e1f3e5ffe09"},"schema_version":"1.0"},"canonical_sha256":"5a96b1d3839ed30bc2247bcac6c63bbd8b5d49ce095a3c7237a7ff822b94ebec","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:27:10.679464Z","signature_b64":"2wIH+Q/oDWrCwBfTtm3klk7/U8vbbXBfqxPwiXrnJUe2FR1X5v6cheYOHYSB61Kk8bRBTlEx0w5FkNBqOIhnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a96b1d3839ed30bc2247bcac6c63bbd8b5d49ce095a3c7237a7ff822b94ebec","last_reissued_at":"2026-07-05T02:27:10.678962Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:27:10.678962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2008.02491","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":"2026-07-05T02:27:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jpM88uhbjoYeMuNwvUFTa36yWGUgDUG+Oq2F0JDsLGFnd7VYJXAQIFfTqKqorJfTOcM5xizgBNPYaaxwJbnABQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T23:36:01.093346Z"},"content_sha256":"fc69e79e87fb6544e712f1742b6313f0e56800e19749ce8f2302b8eb1a3ba897","schema_version":"1.0","event_id":"sha256:fc69e79e87fb6544e712f1742b6313f0e56800e19749ce8f2302b8eb1a3ba897"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:LKLLDU4DT3JQXQREPPFMNRR3XW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Large-time asymptotics in deep learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"math.OC","authors_text":"Borjan Geshkovski, Carlos Esteve, Dario Pighin, Enrique Zuazua","submitted_at":"2020-08-06T07:33:17Z","abstract_excerpt":"We consider the neural ODE perspective of supervised learning and study the impact of the final time $T$ (which may indicate the depth of a corresponding ResNet) in training. For the classical $L^2$--regularized empirical risk minimization problem, whenever the neural ODE dynamics are homogeneous with respect to the parameters, we show that the training error is at most of the order $\\mathcal{O}\\left(\\frac{1}{T}\\right)$. Furthermore, if the loss inducing the empirical risk attains its minimum, the optimal parameters converge to minimal $L^2$--norm parameters which interpolate the dataset. By a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.02491","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/2008.02491/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":"2026-07-05T02:27:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zsrQE7KzrqdfL4MhNeNgRviJZUqvNId4xi9EWap1G/to7GX0pEzNb49+QN7C4nMxMy3rMAOU0ei9PDqNepKTBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T23:36:01.094206Z"},"content_sha256":"fc946bd5307f6eedb4fa74f17e9776bcc8f98cbb3701b9dd876c93876dfe43ef","schema_version":"1.0","event_id":"sha256:fc946bd5307f6eedb4fa74f17e9776bcc8f98cbb3701b9dd876c93876dfe43ef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LKLLDU4DT3JQXQREPPFMNRR3XW/bundle.json","state_url":"https://pith.science/pith/LKLLDU4DT3JQXQREPPFMNRR3XW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LKLLDU4DT3JQXQREPPFMNRR3XW/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":"2026-08-04T23:36:01Z","links":{"resolver":"https://pith.science/pith/LKLLDU4DT3JQXQREPPFMNRR3XW","bundle":"https://pith.science/pith/LKLLDU4DT3JQXQREPPFMNRR3XW/bundle.json","state":"https://pith.science/pith/LKLLDU4DT3JQXQREPPFMNRR3XW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LKLLDU4DT3JQXQREPPFMNRR3XW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:LKLLDU4DT3JQXQREPPFMNRR3XW","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":"6857e2ea15e552b8912f8afe817f629fb25a582efe001094c4882e1f3e5ffe09","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2020-08-06T07:33:17Z","title_canon_sha256":"e0eb2542e9137cd7680952411a2087d9b9bdfc6c30a959c3852216fe06015eb1"},"schema_version":"1.0","source":{"id":"2008.02491","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2008.02491","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"arxiv_version","alias_value":"2008.02491v2","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2008.02491","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"pith_short_12","alias_value":"LKLLDU4DT3JQ","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"pith_short_16","alias_value":"LKLLDU4DT3JQXQRE","created_at":"2026-07-05T02:27:10Z"},{"alias_kind":"pith_short_8","alias_value":"LKLLDU4D","created_at":"2026-07-05T02:27:10Z"}],"graph_snapshots":[{"event_id":"sha256:fc946bd5307f6eedb4fa74f17e9776bcc8f98cbb3701b9dd876c93876dfe43ef","target":"graph","created_at":"2026-07-05T02:27:10Z","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/2008.02491/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the neural ODE perspective of supervised learning and study the impact of the final time $T$ (which may indicate the depth of a corresponding ResNet) in training. For the classical $L^2$--regularized empirical risk minimization problem, whenever the neural ODE dynamics are homogeneous with respect to the parameters, we show that the training error is at most of the order $\\mathcal{O}\\left(\\frac{1}{T}\\right)$. Furthermore, if the loss inducing the empirical risk attains its minimum, the optimal parameters converge to minimal $L^2$--norm parameters which interpolate the dataset. By a","authors_text":"Borjan Geshkovski, Carlos Esteve, Dario Pighin, Enrique Zuazua","cross_cats":["cs.LG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2020-08-06T07:33:17Z","title":"Large-time asymptotics in deep learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.02491","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:fc69e79e87fb6544e712f1742b6313f0e56800e19749ce8f2302b8eb1a3ba897","target":"record","created_at":"2026-07-05T02:27:10Z","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":"6857e2ea15e552b8912f8afe817f629fb25a582efe001094c4882e1f3e5ffe09","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2020-08-06T07:33:17Z","title_canon_sha256":"e0eb2542e9137cd7680952411a2087d9b9bdfc6c30a959c3852216fe06015eb1"},"schema_version":"1.0","source":{"id":"2008.02491","kind":"arxiv","version":2}},"canonical_sha256":"5a96b1d3839ed30bc2247bcac6c63bbd8b5d49ce095a3c7237a7ff822b94ebec","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5a96b1d3839ed30bc2247bcac6c63bbd8b5d49ce095a3c7237a7ff822b94ebec","first_computed_at":"2026-07-05T02:27:10.678962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:27:10.678962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2wIH+Q/oDWrCwBfTtm3klk7/U8vbbXBfqxPwiXrnJUe2FR1X5v6cheYOHYSB61Kk8bRBTlEx0w5FkNBqOIhnCg==","signature_status":"signed_v1","signed_at":"2026-07-05T02:27:10.679464Z","signed_message":"canonical_sha256_bytes"},"source_id":"2008.02491","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fc69e79e87fb6544e712f1742b6313f0e56800e19749ce8f2302b8eb1a3ba897","sha256:fc946bd5307f6eedb4fa74f17e9776bcc8f98cbb3701b9dd876c93876dfe43ef"],"state_sha256":"9518369ee684574555a16a0e954aaf24934b1832e1ea89c68c195174363a93e1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ilA6znG3CrUrOGzElp+CwypfFxmz0Apmj7dqUIlsnwU1ow2Q/r4zfJTaB28A6OspNGq24YRnwpyYArcGcw9+Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T23:36:01.102210Z","bundle_sha256":"892ed1592cfd03b32a1c2d5ed56eebf4a859385dd302c825ea70a5ccfe06ec5b"}}