{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:R3YH4Z5FZDZ62Y6IAHEBQVHSD3","short_pith_number":"pith:R3YH4Z5F","canonical_record":{"source":{"id":"2504.04847","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-04-07T09:00:22Z","cross_cats_sorted":["cs.CC","cs.NA","math.NA"],"title_canon_sha256":"ac10904e067e4263d65eaf619272047115ffdf9a818c0e10d1e41b70cfa7714f","abstract_canon_sha256":"d99ae382bcb9625ac973ff53d588ee709cc9464b003bddd401ec708e0be8c6ec"},"schema_version":"1.0"},"canonical_sha256":"8ef07e67a5c8f3ed63c801c81854f21efab96fe2b545fc2e23988e11b221df35","source":{"kind":"arxiv","id":"2504.04847","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.04847","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"arxiv_version","alias_value":"2504.04847v1","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.04847","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"pith_short_12","alias_value":"R3YH4Z5FZDZ6","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"pith_short_16","alias_value":"R3YH4Z5FZDZ62Y6I","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"pith_short_8","alias_value":"R3YH4Z5F","created_at":"2026-07-05T10:45:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:R3YH4Z5FZDZ62Y6IAHEBQVHSD3","target":"record","payload":{"canonical_record":{"source":{"id":"2504.04847","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-04-07T09:00:22Z","cross_cats_sorted":["cs.CC","cs.NA","math.NA"],"title_canon_sha256":"ac10904e067e4263d65eaf619272047115ffdf9a818c0e10d1e41b70cfa7714f","abstract_canon_sha256":"d99ae382bcb9625ac973ff53d588ee709cc9464b003bddd401ec708e0be8c6ec"},"schema_version":"1.0"},"canonical_sha256":"8ef07e67a5c8f3ed63c801c81854f21efab96fe2b545fc2e23988e11b221df35","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:45:30.640264Z","signature_b64":"xghPD25S3pbW+eorxVoBxe96ktTafoR52LtFRFzZpjj03iYEh7UPX5NdeH66PgrUaYSxmiCPvhxzb+gtSGjqCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ef07e67a5c8f3ed63c801c81854f21efab96fe2b545fc2e23988e11b221df35","last_reissued_at":"2026-07-05T10:45:30.639786Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:45:30.639786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2504.04847","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:45:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MeeXg8VPMIAWUJBk4kH/Ig/VJjipNZ0kE6BTUw1VZFBe+BiZn+p+CCVuNIXidnF/kc4ptC/eRXyciRcuZlmsDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T04:39:26.995908Z"},"content_sha256":"2b70668167042fbb79a9558ac8aa65ce308e390023b5e1bfeaea7d6c6bd28255","schema_version":"1.0","event_id":"sha256:2b70668167042fbb79a9558ac8aa65ce308e390023b5e1bfeaea7d6c6bd28255"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:R3YH4Z5FZDZ62Y6IAHEBQVHSD3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlocal techniques for the analysis of deep ReLU neural network approximations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.NA","math.NA"],"primary_cat":"cs.LG","authors_text":"Cornelia Schneider, Jan Vybiral, Mario Ullrich","submitted_at":"2025-04-07T09:00:22Z","abstract_excerpt":"Recently, Daubechies, DeVore, Foucart, Hanin, and Petrova introduced a system of piece-wise linear functions, which can be easily reproduced by artificial neural networks with the ReLU activation function and which form a Riesz basis of $L_2([0,1])$. This work was generalized by two of the authors to the multivariate setting. We show that this system serves as a Riesz basis also for Sobolev spaces $W^s([0,1]^d)$ and Barron classes ${\\mathbb B}^s([0,1]^d)$ with smoothness $0<s<1$. We apply this fact to re-prove some recent results on the approximation of functions from these classes by deep neu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.04847","kind":"arxiv","version":1},"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/2504.04847/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":"2026-07-05T10:45:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uXeiaQUX5Di/9S7tj/z1LNd0i+gd4RiipWh9SMUcuB7DvMJK7QiuwSkCw+/pfK0Wr5f18C8UuX4KvXUyMvivDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T04:39:26.996766Z"},"content_sha256":"bc1ce043962403ff57c1886a043e3558b0880c5cbf0289f5000a83587d81700a","schema_version":"1.0","event_id":"sha256:bc1ce043962403ff57c1886a043e3558b0880c5cbf0289f5000a83587d81700a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R3YH4Z5FZDZ62Y6IAHEBQVHSD3/bundle.json","state_url":"https://pith.science/pith/R3YH4Z5FZDZ62Y6IAHEBQVHSD3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R3YH4Z5FZDZ62Y6IAHEBQVHSD3/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":"2026-08-10T04:39:27Z","links":{"resolver":"https://pith.science/pith/R3YH4Z5FZDZ62Y6IAHEBQVHSD3","bundle":"https://pith.science/pith/R3YH4Z5FZDZ62Y6IAHEBQVHSD3/bundle.json","state":"https://pith.science/pith/R3YH4Z5FZDZ62Y6IAHEBQVHSD3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R3YH4Z5FZDZ62Y6IAHEBQVHSD3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:R3YH4Z5FZDZ62Y6IAHEBQVHSD3","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":"d99ae382bcb9625ac973ff53d588ee709cc9464b003bddd401ec708e0be8c6ec","cross_cats_sorted":["cs.CC","cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-04-07T09:00:22Z","title_canon_sha256":"ac10904e067e4263d65eaf619272047115ffdf9a818c0e10d1e41b70cfa7714f"},"schema_version":"1.0","source":{"id":"2504.04847","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.04847","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"arxiv_version","alias_value":"2504.04847v1","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.04847","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"pith_short_12","alias_value":"R3YH4Z5FZDZ6","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"pith_short_16","alias_value":"R3YH4Z5FZDZ62Y6I","created_at":"2026-07-05T10:45:30Z"},{"alias_kind":"pith_short_8","alias_value":"R3YH4Z5F","created_at":"2026-07-05T10:45:30Z"}],"graph_snapshots":[{"event_id":"sha256:bc1ce043962403ff57c1886a043e3558b0880c5cbf0289f5000a83587d81700a","target":"graph","created_at":"2026-07-05T10:45:30Z","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/2504.04847/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Recently, Daubechies, DeVore, Foucart, Hanin, and Petrova introduced a system of piece-wise linear functions, which can be easily reproduced by artificial neural networks with the ReLU activation function and which form a Riesz basis of $L_2([0,1])$. This work was generalized by two of the authors to the multivariate setting. We show that this system serves as a Riesz basis also for Sobolev spaces $W^s([0,1]^d)$ and Barron classes ${\\mathbb B}^s([0,1]^d)$ with smoothness $0<s<1$. We apply this fact to re-prove some recent results on the approximation of functions from these classes by deep neu","authors_text":"Cornelia Schneider, Jan Vybiral, Mario Ullrich","cross_cats":["cs.CC","cs.NA","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-04-07T09:00:22Z","title":"Nonlocal techniques for the analysis of deep ReLU neural network approximations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.04847","kind":"arxiv","version":1},"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":"sha256:2b70668167042fbb79a9558ac8aa65ce308e390023b5e1bfeaea7d6c6bd28255","target":"record","created_at":"2026-07-05T10:45:30Z","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":"d99ae382bcb9625ac973ff53d588ee709cc9464b003bddd401ec708e0be8c6ec","cross_cats_sorted":["cs.CC","cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-04-07T09:00:22Z","title_canon_sha256":"ac10904e067e4263d65eaf619272047115ffdf9a818c0e10d1e41b70cfa7714f"},"schema_version":"1.0","source":{"id":"2504.04847","kind":"arxiv","version":1}},"canonical_sha256":"8ef07e67a5c8f3ed63c801c81854f21efab96fe2b545fc2e23988e11b221df35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ef07e67a5c8f3ed63c801c81854f21efab96fe2b545fc2e23988e11b221df35","first_computed_at":"2026-07-05T10:45:30.639786Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:45:30.639786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xghPD25S3pbW+eorxVoBxe96ktTafoR52LtFRFzZpjj03iYEh7UPX5NdeH66PgrUaYSxmiCPvhxzb+gtSGjqCw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:45:30.640264Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.04847","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b70668167042fbb79a9558ac8aa65ce308e390023b5e1bfeaea7d6c6bd28255","sha256:bc1ce043962403ff57c1886a043e3558b0880c5cbf0289f5000a83587d81700a"],"state_sha256":"4ae7119bfbe9914c9791817e078566fd7186d45ee83c1d0495e613c63723bfa6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+kGG+sVbdLEdkEWvzOxnqvJyTDCRNhLgur2jY7+A/kzzV1u0RQPXEA2sws/iVM+XdZIXkBlAHepTxWtrSkIgCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T04:39:27.002938Z","bundle_sha256":"892628f0e4c98114e8235cc772b7502df9488b64043535c6981295c3dadbc153"}}