{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:TAP5RIDSYCFKL6DBMLMASSG45Z","short_pith_number":"pith:TAP5RIDS","canonical_record":{"source":{"id":"2102.09235","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-18T09:37:49Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"ba6ed1322b0e5283ef156c049cabd16dc1684bf5e1e7472258a397d7a7aeae24","abstract_canon_sha256":"17862507dac2ebc93b2055944416b973e02eef9d111549aaba567b65ebb16819"},"schema_version":"1.0"},"canonical_sha256":"981fd8a072c08aa5f86162d80948dcee6c41e2101976a0866a6603a8bf391891","source":{"kind":"arxiv","id":"2102.09235","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.09235","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"arxiv_version","alias_value":"2102.09235v3","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.09235","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"pith_short_12","alias_value":"TAP5RIDSYCFK","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"pith_short_16","alias_value":"TAP5RIDSYCFKL6DB","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"pith_short_8","alias_value":"TAP5RIDS","created_at":"2026-06-29T01:15:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:TAP5RIDSYCFKL6DBMLMASSG45Z","target":"record","payload":{"canonical_record":{"source":{"id":"2102.09235","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-18T09:37:49Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"ba6ed1322b0e5283ef156c049cabd16dc1684bf5e1e7472258a397d7a7aeae24","abstract_canon_sha256":"17862507dac2ebc93b2055944416b973e02eef9d111549aaba567b65ebb16819"},"schema_version":"1.0"},"canonical_sha256":"981fd8a072c08aa5f86162d80948dcee6c41e2101976a0866a6603a8bf391891","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-29T01:15:01.537351Z","signature_b64":"INrJ2nLsaNk/dWkQ1D+OYDoRLXBS5QC5VecmXRcWWO0+N8GHbJCUcigYKmFdcx+FWSVIkw12hU6H9zR7jwEbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"981fd8a072c08aa5f86162d80948dcee6c41e2101976a0866a6603a8bf391891","last_reissued_at":"2026-06-29T01:15:01.536932Z","signature_status":"signed_v1","first_computed_at":"2026-06-29T01:15:01.536932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2102.09235","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-29T01:15:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gKCBjP6KLsu+8QqTGEdtt9oGAVORq6FySIgX+Vq3TMadW00Y+5J6YdIQD0LYsRlg+mRorCjKcgl91UG9sY4sAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T20:20:01.917268Z"},"content_sha256":"e947cdf7a8441be52f848a1a0d92ec86c3cf46dbea5867fda4f715eb3cb6def9","schema_version":"1.0","event_id":"sha256:e947cdf7a8441be52f848a1a0d92ec86c3cf46dbea5867fda4f715eb3cb6def9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:TAP5RIDSYCFKL6DBMLMASSG45Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Deep Residual Networks Learn the Geodesic Curve in the Wasserstein Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Kuo Gai, Shihua Zhang","submitted_at":"2021-02-18T09:37:49Z","abstract_excerpt":"Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems. However, the specific dynamics that DNNs, especially deep residual networks (ResNets), tend to learn during training remain insufficiently characterized. To this end, we model the forward propagation of deep residual networks using continuity equations, in which the measure is conserved and infinite curves in the measure space connect the input distribution to the output one of a ResNet. We find ResNets with $L_2$ regularization attempt to learn the geodesic curve in the Wasserstein spa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.09235","kind":"arxiv","version":3},"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/2102.09235/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":"2026-06-29T01:15:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LP/Drv46Sb7/9edtCMrJ7g04YGtVum68cL7kA/NeLjkCLna5SUghgl0yqekCIiGgyrGAA3AjX6taqqbvXvLfCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T20:20:01.917680Z"},"content_sha256":"37d5a98fec3de3d5e1e49f1188975e5cf9978b8e2b227405540835b7725d5ccb","schema_version":"1.0","event_id":"sha256:37d5a98fec3de3d5e1e49f1188975e5cf9978b8e2b227405540835b7725d5ccb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TAP5RIDSYCFKL6DBMLMASSG45Z/bundle.json","state_url":"https://pith.science/pith/TAP5RIDSYCFKL6DBMLMASSG45Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TAP5RIDSYCFKL6DBMLMASSG45Z/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":"2026-07-01T20:20:01Z","links":{"resolver":"https://pith.science/pith/TAP5RIDSYCFKL6DBMLMASSG45Z","bundle":"https://pith.science/pith/TAP5RIDSYCFKL6DBMLMASSG45Z/bundle.json","state":"https://pith.science/pith/TAP5RIDSYCFKL6DBMLMASSG45Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TAP5RIDSYCFKL6DBMLMASSG45Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:TAP5RIDSYCFKL6DBMLMASSG45Z","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":"17862507dac2ebc93b2055944416b973e02eef9d111549aaba567b65ebb16819","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-18T09:37:49Z","title_canon_sha256":"ba6ed1322b0e5283ef156c049cabd16dc1684bf5e1e7472258a397d7a7aeae24"},"schema_version":"1.0","source":{"id":"2102.09235","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.09235","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"arxiv_version","alias_value":"2102.09235v3","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.09235","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"pith_short_12","alias_value":"TAP5RIDSYCFK","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"pith_short_16","alias_value":"TAP5RIDSYCFKL6DB","created_at":"2026-06-29T01:15:01Z"},{"alias_kind":"pith_short_8","alias_value":"TAP5RIDS","created_at":"2026-06-29T01:15:01Z"}],"graph_snapshots":[{"event_id":"sha256:37d5a98fec3de3d5e1e49f1188975e5cf9978b8e2b227405540835b7725d5ccb","target":"graph","created_at":"2026-06-29T01:15:01Z","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/2102.09235/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems. However, the specific dynamics that DNNs, especially deep residual networks (ResNets), tend to learn during training remain insufficiently characterized. To this end, we model the forward propagation of deep residual networks using continuity equations, in which the measure is conserved and infinite curves in the measure space connect the input distribution to the output one of a ResNet. We find ResNets with $L_2$ regularization attempt to learn the geodesic curve in the Wasserstein spa","authors_text":"Kuo Gai, Shihua Zhang","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-18T09:37:49Z","title":"Deep Residual Networks Learn the Geodesic Curve in the Wasserstein Space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.09235","kind":"arxiv","version":3},"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:e947cdf7a8441be52f848a1a0d92ec86c3cf46dbea5867fda4f715eb3cb6def9","target":"record","created_at":"2026-06-29T01:15:01Z","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":"17862507dac2ebc93b2055944416b973e02eef9d111549aaba567b65ebb16819","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-18T09:37:49Z","title_canon_sha256":"ba6ed1322b0e5283ef156c049cabd16dc1684bf5e1e7472258a397d7a7aeae24"},"schema_version":"1.0","source":{"id":"2102.09235","kind":"arxiv","version":3}},"canonical_sha256":"981fd8a072c08aa5f86162d80948dcee6c41e2101976a0866a6603a8bf391891","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"981fd8a072c08aa5f86162d80948dcee6c41e2101976a0866a6603a8bf391891","first_computed_at":"2026-06-29T01:15:01.536932Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-29T01:15:01.536932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"INrJ2nLsaNk/dWkQ1D+OYDoRLXBS5QC5VecmXRcWWO0+N8GHbJCUcigYKmFdcx+FWSVIkw12hU6H9zR7jwEbDA==","signature_status":"signed_v1","signed_at":"2026-06-29T01:15:01.537351Z","signed_message":"canonical_sha256_bytes"},"source_id":"2102.09235","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e947cdf7a8441be52f848a1a0d92ec86c3cf46dbea5867fda4f715eb3cb6def9","sha256:37d5a98fec3de3d5e1e49f1188975e5cf9978b8e2b227405540835b7725d5ccb"],"state_sha256":"751aede26eea45c3cff780cf6a92e4a5c8e44626aede276c2e1e280f08ad3cb0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y4iystR9XcXqRP/72o6K1ZcDEozjun+zHAwAdfxIpqQFEol3SGEKUW5L8kH+1gCUDQg4m7O2aQI0QcbiHdZ/Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T20:20:01.921534Z","bundle_sha256":"73ec502d44c02b08ce795036f479e873bfdb158e471c96aa380a3b69b6893466"}}