{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:RZ6Q44YHTMHC6DEX3KZCCKWDSW","short_pith_number":"pith:RZ6Q44YH","canonical_record":{"source":{"id":"1907.12998","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-07-30T15:04:01Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"a35b811f7d4c7026b4d260db5fcb006dc932f3ea91c18058276fd9f2b83f245d","abstract_canon_sha256":"814addccc2f3c67b261c26fc20ec1d0287cdd2d1968f815caafd7f597a9e2f1e"},"schema_version":"1.0"},"canonical_sha256":"8e7d0e73079b0e2f0c97dab2212ac395ab3df90f97b85114afb06ba265ed8879","source":{"kind":"arxiv","id":"1907.12998","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.12998","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"arxiv_version","alias_value":"1907.12998v2","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.12998","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"pith_short_12","alias_value":"RZ6Q44YHTMHC","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"pith_short_16","alias_value":"RZ6Q44YHTMHC6DEX","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"pith_short_8","alias_value":"RZ6Q44YH","created_at":"2026-07-05T00:44:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:RZ6Q44YHTMHC6DEX3KZCCKWDSW","target":"record","payload":{"canonical_record":{"source":{"id":"1907.12998","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-07-30T15:04:01Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"a35b811f7d4c7026b4d260db5fcb006dc932f3ea91c18058276fd9f2b83f245d","abstract_canon_sha256":"814addccc2f3c67b261c26fc20ec1d0287cdd2d1968f815caafd7f597a9e2f1e"},"schema_version":"1.0"},"canonical_sha256":"8e7d0e73079b0e2f0c97dab2212ac395ab3df90f97b85114afb06ba265ed8879","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:44:39.359453Z","signature_b64":"8HbzruZVEQ8S/GMGd9A+0bbAhKuaUB2atxEGRUruWieGlavWFeQC26O0jA5joyKhUezUbf9B2+OEk/RAefh9Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e7d0e73079b0e2f0c97dab2212ac395ab3df90f97b85114afb06ba265ed8879","last_reissued_at":"2026-07-05T00:44:39.359010Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:44:39.359010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.12998","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":"2026-07-05T00:44:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7RCQbUcE9ilfuy8B+ng5oH8dBVbPi1ijBpw4jEZCJhY4GH3qp5mtyorCy/82Fn7Hc8aeOT8xgX++6c/6BDAjBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:48:17.614095Z"},"content_sha256":"73a3991b352f1195ed561bb1261951adfb8555194ba814c01e3f2a0e9a6e2cb3","schema_version":"1.0","event_id":"sha256:73a3991b352f1195ed561bb1261951adfb8555194ba814c01e3f2a0e9a6e2cb3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:RZ6Q44YHTMHC6DEX3KZCCKWDSW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximation Capabilities of Neural ODEs and Invertible Residual Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Han Zhang, Jacob Unterman, Tom Arodz, Xi Gao","submitted_at":"2019-07-30T15:04:01Z","abstract_excerpt":"Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far the question of whether Neural ODEs and i-ResNets can model any continuous invertible function remained unresolved. Here, we show that both of these models are limited in their approximation capabilities. We then prove that any homeomorphism on a $p$-dimensional Euclidean space can be approximated by a Neural ODE operating on a $2p$-dimensional Euclidean space"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.12998","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/1907.12998/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":"2026-07-05T00:44:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"okUBW2cGrh1kwjIzWqjJs401ZM9xJ1PFJyCTtiJhvFRNDAodGW5BvM6n9Z5+5KLM5TPDMiO1GifbjdMsc6amDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:48:17.614581Z"},"content_sha256":"62b72ce39f3298a451a4d1608f74cfd8744b7079b5ab9e639ae5203f4f67b557","schema_version":"1.0","event_id":"sha256:62b72ce39f3298a451a4d1608f74cfd8744b7079b5ab9e639ae5203f4f67b557"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RZ6Q44YHTMHC6DEX3KZCCKWDSW/bundle.json","state_url":"https://pith.science/pith/RZ6Q44YHTMHC6DEX3KZCCKWDSW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RZ6Q44YHTMHC6DEX3KZCCKWDSW/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":"2026-08-16T13:48:17Z","links":{"resolver":"https://pith.science/pith/RZ6Q44YHTMHC6DEX3KZCCKWDSW","bundle":"https://pith.science/pith/RZ6Q44YHTMHC6DEX3KZCCKWDSW/bundle.json","state":"https://pith.science/pith/RZ6Q44YHTMHC6DEX3KZCCKWDSW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RZ6Q44YHTMHC6DEX3KZCCKWDSW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:RZ6Q44YHTMHC6DEX3KZCCKWDSW","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":"814addccc2f3c67b261c26fc20ec1d0287cdd2d1968f815caafd7f597a9e2f1e","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-07-30T15:04:01Z","title_canon_sha256":"a35b811f7d4c7026b4d260db5fcb006dc932f3ea91c18058276fd9f2b83f245d"},"schema_version":"1.0","source":{"id":"1907.12998","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.12998","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"arxiv_version","alias_value":"1907.12998v2","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.12998","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"pith_short_12","alias_value":"RZ6Q44YHTMHC","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"pith_short_16","alias_value":"RZ6Q44YHTMHC6DEX","created_at":"2026-07-05T00:44:39Z"},{"alias_kind":"pith_short_8","alias_value":"RZ6Q44YH","created_at":"2026-07-05T00:44:39Z"}],"graph_snapshots":[{"event_id":"sha256:62b72ce39f3298a451a4d1608f74cfd8744b7079b5ab9e639ae5203f4f67b557","target":"graph","created_at":"2026-07-05T00:44:39Z","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/1907.12998/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far the question of whether Neural ODEs and i-ResNets can model any continuous invertible function remained unresolved. Here, we show that both of these models are limited in their approximation capabilities. We then prove that any homeomorphism on a $p$-dimensional Euclidean space can be approximated by a Neural ODE operating on a $2p$-dimensional Euclidean space","authors_text":"Han Zhang, Jacob Unterman, Tom Arodz, Xi Gao","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-07-30T15:04:01Z","title":"Approximation Capabilities of Neural ODEs and Invertible Residual Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.12998","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":"sha256:73a3991b352f1195ed561bb1261951adfb8555194ba814c01e3f2a0e9a6e2cb3","target":"record","created_at":"2026-07-05T00:44:39Z","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":"814addccc2f3c67b261c26fc20ec1d0287cdd2d1968f815caafd7f597a9e2f1e","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-07-30T15:04:01Z","title_canon_sha256":"a35b811f7d4c7026b4d260db5fcb006dc932f3ea91c18058276fd9f2b83f245d"},"schema_version":"1.0","source":{"id":"1907.12998","kind":"arxiv","version":2}},"canonical_sha256":"8e7d0e73079b0e2f0c97dab2212ac395ab3df90f97b85114afb06ba265ed8879","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e7d0e73079b0e2f0c97dab2212ac395ab3df90f97b85114afb06ba265ed8879","first_computed_at":"2026-07-05T00:44:39.359010Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:44:39.359010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8HbzruZVEQ8S/GMGd9A+0bbAhKuaUB2atxEGRUruWieGlavWFeQC26O0jA5joyKhUezUbf9B2+OEk/RAefh9Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T00:44:39.359453Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.12998","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73a3991b352f1195ed561bb1261951adfb8555194ba814c01e3f2a0e9a6e2cb3","sha256:62b72ce39f3298a451a4d1608f74cfd8744b7079b5ab9e639ae5203f4f67b557"],"state_sha256":"cd170b2c7200fa98e08f5ea787c76f04726b151f59800bbef23a5b964e54b3e1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b/AStkGXMxRKUyVwtMZEb1K8ufi7shVuBTa03n2I1NQn1Frnu666U+EdHZJt7CS2iuT/ATMuON9z7Q1YtkzjCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T13:48:17.618158Z","bundle_sha256":"3f372a0aa20f5b887378b3a958cbbf8aa5933666c0872d9260220753606c18ae"}}