{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:4AWBJ5JCGWIFVERDGZHEFCPQGO","short_pith_number":"pith:4AWBJ5JC","canonical_record":{"source":{"id":"2310.10766","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-10-16T19:00:28Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"6ee4bdad6fc529a47c8cb305b7044da2b40e15c7ebe85fa30790d642d314b7e4","abstract_canon_sha256":"92adb80d4f9d82a797e103dc250e4a63d72c877bac820a7fa0a04a5f86d812a9"},"schema_version":"1.0"},"canonical_sha256":"e02c14f52235905a9223364e4289f03381564894a73fd7b0f833226cf7e27341","source":{"kind":"arxiv","id":"2310.10766","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2310.10766","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"arxiv_version","alias_value":"2310.10766v5","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2310.10766","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"pith_short_12","alias_value":"4AWBJ5JCGWIF","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"pith_short_16","alias_value":"4AWBJ5JCGWIFVERD","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"pith_short_8","alias_value":"4AWBJ5JC","created_at":"2026-07-05T12:02:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:4AWBJ5JCGWIFVERDGZHEFCPQGO","target":"record","payload":{"canonical_record":{"source":{"id":"2310.10766","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-10-16T19:00:28Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"6ee4bdad6fc529a47c8cb305b7044da2b40e15c7ebe85fa30790d642d314b7e4","abstract_canon_sha256":"92adb80d4f9d82a797e103dc250e4a63d72c877bac820a7fa0a04a5f86d812a9"},"schema_version":"1.0"},"canonical_sha256":"e02c14f52235905a9223364e4289f03381564894a73fd7b0f833226cf7e27341","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:02:32.465454Z","signature_b64":"Pz2U1QnRIEsotNY6H47TgkYfoFCghGVAqx/8l6afOghCZ38/9aTKrQnDGCvIP5uVnO6AWIIOBPuEJnJK0a0aDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e02c14f52235905a9223364e4289f03381564894a73fd7b0f833226cf7e27341","last_reissued_at":"2026-07-05T12:02:32.464958Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:02:32.464958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2310.10766","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:02:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NyT0858z8TTKm7D4hrGWM8jV+V/32M15fl6dPIy9w6gxUZUJycyNEvuZmtYWxfmD2i8uXWN3P3qC6xlSgfMoDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T01:06:48.356653Z"},"content_sha256":"478cb2a5f1e9265066f67163f27841fa61f588fc9efe89a50479074f5cd06333","schema_version":"1.0","event_id":"sha256:478cb2a5f1e9265066f67163f27841fa61f588fc9efe89a50479074f5cd06333"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:4AWBJ5JCGWIFVERDGZHEFCPQGO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Haizhao Yang, Yahong Yang, Yang Xiang, Yue Wu","submitted_at":"2023-10-16T19:00:28Z","abstract_excerpt":"This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\\in\\mathbb{N}$ with $m\\ge 2$ and $1\\le p\\le +\\infty$. Standard ReLU deep neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily of ReLU DNNs, and several layers of the square of ReLU added at the end to smooth the networks output. This approach retains the advantages of ReLU DNNs, leading to the straightforward training. The paper also proves the optimality of DSRNs by estimating the VC-dimension of higher-order deriva"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.10766","kind":"arxiv","version":5},"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/2310.10766/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":"2026-07-05T12:02:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rT762HDpHvtXlcrH55H0Y3o7i4gTXGIgsYgW3LxhCgBRzz7LMPdUFszyBXWti/d2HlV+WFlbdCqr9+y9jhGzBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T01:06:48.357145Z"},"content_sha256":"e517d8ed648a99014ed81b8e7dd4f9f5dac9b74d95bb3511458412c92b5a9c45","schema_version":"1.0","event_id":"sha256:e517d8ed648a99014ed81b8e7dd4f9f5dac9b74d95bb3511458412c92b5a9c45"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4AWBJ5JCGWIFVERDGZHEFCPQGO/bundle.json","state_url":"https://pith.science/pith/4AWBJ5JCGWIFVERDGZHEFCPQGO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4AWBJ5JCGWIFVERDGZHEFCPQGO/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":"2026-08-06T01:06:48Z","links":{"resolver":"https://pith.science/pith/4AWBJ5JCGWIFVERDGZHEFCPQGO","bundle":"https://pith.science/pith/4AWBJ5JCGWIFVERDGZHEFCPQGO/bundle.json","state":"https://pith.science/pith/4AWBJ5JCGWIFVERDGZHEFCPQGO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4AWBJ5JCGWIFVERDGZHEFCPQGO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:4AWBJ5JCGWIFVERDGZHEFCPQGO","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":"92adb80d4f9d82a797e103dc250e4a63d72c877bac820a7fa0a04a5f86d812a9","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-10-16T19:00:28Z","title_canon_sha256":"6ee4bdad6fc529a47c8cb305b7044da2b40e15c7ebe85fa30790d642d314b7e4"},"schema_version":"1.0","source":{"id":"2310.10766","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2310.10766","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"arxiv_version","alias_value":"2310.10766v5","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2310.10766","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"pith_short_12","alias_value":"4AWBJ5JCGWIF","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"pith_short_16","alias_value":"4AWBJ5JCGWIFVERD","created_at":"2026-07-05T12:02:32Z"},{"alias_kind":"pith_short_8","alias_value":"4AWBJ5JC","created_at":"2026-07-05T12:02:32Z"}],"graph_snapshots":[{"event_id":"sha256:e517d8ed648a99014ed81b8e7dd4f9f5dac9b74d95bb3511458412c92b5a9c45","target":"graph","created_at":"2026-07-05T12:02:32Z","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/2310.10766/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\\in\\mathbb{N}$ with $m\\ge 2$ and $1\\le p\\le +\\infty$. Standard ReLU deep neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily of ReLU DNNs, and several layers of the square of ReLU added at the end to smooth the networks output. This approach retains the advantages of ReLU DNNs, leading to the straightforward training. The paper also proves the optimality of DSRNs by estimating the VC-dimension of higher-order deriva","authors_text":"Haizhao Yang, Yahong Yang, Yang Xiang, Yue Wu","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-10-16T19:00:28Z","title":"Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.10766","kind":"arxiv","version":5},"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":"sha256:478cb2a5f1e9265066f67163f27841fa61f588fc9efe89a50479074f5cd06333","target":"record","created_at":"2026-07-05T12:02:32Z","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":"92adb80d4f9d82a797e103dc250e4a63d72c877bac820a7fa0a04a5f86d812a9","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-10-16T19:00:28Z","title_canon_sha256":"6ee4bdad6fc529a47c8cb305b7044da2b40e15c7ebe85fa30790d642d314b7e4"},"schema_version":"1.0","source":{"id":"2310.10766","kind":"arxiv","version":5}},"canonical_sha256":"e02c14f52235905a9223364e4289f03381564894a73fd7b0f833226cf7e27341","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e02c14f52235905a9223364e4289f03381564894a73fd7b0f833226cf7e27341","first_computed_at":"2026-07-05T12:02:32.464958Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:02:32.464958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Pz2U1QnRIEsotNY6H47TgkYfoFCghGVAqx/8l6afOghCZ38/9aTKrQnDGCvIP5uVnO6AWIIOBPuEJnJK0a0aDg==","signature_status":"signed_v1","signed_at":"2026-07-05T12:02:32.465454Z","signed_message":"canonical_sha256_bytes"},"source_id":"2310.10766","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:478cb2a5f1e9265066f67163f27841fa61f588fc9efe89a50479074f5cd06333","sha256:e517d8ed648a99014ed81b8e7dd4f9f5dac9b74d95bb3511458412c92b5a9c45"],"state_sha256":"e1f9115fe8ac78fd706b3f401047858b8d7406efd8763fc0673e9d916869be3d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QjnxglcY1duX95L6oAAxOLIUYKKIpGAXOdCAoT6GR9/8yxksPtygYAGbnhTHDQ6BpaypOoY88NgAnOKwDaecDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T01:06:48.360484Z","bundle_sha256":"710ac5e87f1357e49285e3a5322ba0d3879b8079192b03570367b11000da20b5"}}