{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:ETV7IEREXWDZY5ZTB7Y55WDEUS","short_pith_number":"pith:ETV7IERE","schema_version":"1.0","canonical_sha256":"24ebf41224bd879c77330ff1ded864a48cd62eaadebe6172d75978d527b333c1","source":{"kind":"arxiv","id":"2206.09527","version":2},"attestation_state":"computed","paper":{"title":"Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.ST","stat.ML","stat.TH"],"primary_cat":"math.NA","authors_text":"Alexey Naumov, Denis Belomestny, Nikita Puchkin, Sergey Samsonov","submitted_at":"2022-06-20T01:18:29Z","abstract_excerpt":"This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any H\\\"{o}lder smooth function up to a given approximation error in H\\\"{o}lder norms in such a way that all weights of this neural network are bounded by $1$. The latter feature is essential to control generalization errors in many statistical and machine learning applications."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2206.09527","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2022-06-20T01:18:29Z","cross_cats_sorted":["cs.NA","math.ST","stat.ML","stat.TH"],"title_canon_sha256":"aa24d386b5348a415f8adada8a5bfbb81827ae25c364fd9cff6a9d98584b55fb","abstract_canon_sha256":"c03816d49a76003adc39b2bca02ebdfe15084e6e491b0299b9e00f889be7057d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:22:08.823031Z","signature_b64":"R7Hijt5cTY2vqaU2PvShoWWNNCWX9ev/aGXtYhU98BlcYFHz7ldDauRZ63ahnVQm0zx1un0bkX1BcfPZ5WmUDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24ebf41224bd879c77330ff1ded864a48cd62eaadebe6172d75978d527b333c1","last_reissued_at":"2026-07-05T05:22:08.822577Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:22:08.822577Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.ST","stat.ML","stat.TH"],"primary_cat":"math.NA","authors_text":"Alexey Naumov, Denis Belomestny, Nikita Puchkin, Sergey Samsonov","submitted_at":"2022-06-20T01:18:29Z","abstract_excerpt":"This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any H\\\"{o}lder smooth function up to a given approximation error in H\\\"{o}lder norms in such a way that all weights of this neural network are bounded by $1$. The latter feature is essential to control generalization errors in many statistical and machine learning applications."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.09527","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/2206.09527/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2206.09527","created_at":"2026-07-05T05:22:08.822641+00:00"},{"alias_kind":"arxiv_version","alias_value":"2206.09527v2","created_at":"2026-07-05T05:22:08.822641+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.09527","created_at":"2026-07-05T05:22:08.822641+00:00"},{"alias_kind":"pith_short_12","alias_value":"ETV7IEREXWDZ","created_at":"2026-07-05T05:22:08.822641+00:00"},{"alias_kind":"pith_short_16","alias_value":"ETV7IEREXWDZY5ZT","created_at":"2026-07-05T05:22:08.822641+00:00"},{"alias_kind":"pith_short_8","alias_value":"ETV7IERE","created_at":"2026-07-05T05:22:08.822641+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.18386","citing_title":"Reconstructing Galactic Gravitational Potentials from Stellar Kinematics with Physics-Informed Neural Networks","ref_index":71,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS","json":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS.json","graph_json":"https://pith.science/api/pith-number/ETV7IEREXWDZY5ZTB7Y55WDEUS/graph.json","events_json":"https://pith.science/api/pith-number/ETV7IEREXWDZY5ZTB7Y55WDEUS/events.json","paper":"https://pith.science/paper/ETV7IERE"},"agent_actions":{"view_html":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS","download_json":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS.json","view_paper":"https://pith.science/paper/ETV7IERE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2206.09527&json=true","fetch_graph":"https://pith.science/api/pith-number/ETV7IEREXWDZY5ZTB7Y55WDEUS/graph.json","fetch_events":"https://pith.science/api/pith-number/ETV7IEREXWDZY5ZTB7Y55WDEUS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS/action/storage_attestation","attest_author":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS/action/author_attestation","sign_citation":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS/action/citation_signature","submit_replication":"https://pith.science/pith/ETV7IEREXWDZY5ZTB7Y55WDEUS/action/replication_record"}},"created_at":"2026-07-05T05:22:08.822641+00:00","updated_at":"2026-07-05T05:22:08.822641+00:00"}