{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:7DZ4SGXM44VNQAOW54LC4EINTO","short_pith_number":"pith:7DZ4SGXM","schema_version":"1.0","canonical_sha256":"f8f3c91aece72ad801d6ef162e110d9ba556f9e4a3dbf2dc6e2b271d19e56a4e","source":{"kind":"arxiv","id":"2607.01266","version":1},"attestation_state":"computed","paper":{"title":"Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.FA"],"primary_cat":"math.LO","authors_text":"Clemens Kinn, Philipp Petersen","submitted_at":"2026-06-29T20:44:20Z","abstract_excerpt":"We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\\varepsilon>0$ by ReLU neural networks of si"},"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":"2607.01266","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-06-29T20:44:20Z","cross_cats_sorted":["cs.LG","math.FA"],"title_canon_sha256":"6daf02e5906b23515c8af2f5d673710be46660a95d370f2de2e5637f1efd322f","abstract_canon_sha256":"c2dc2edf0579083191d90093a034fa71632b4ab98364f5d1c65c26ee09e48eb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T00:16:55.890652Z","signature_b64":"9IV5JAJTYtxyWMpekMcsjx9e2j+yTaTewWj0jOa3y3MaI1irg3VIWkND6pjdRsvIxUo+7T6t/4/Krhid/KkHDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8f3c91aece72ad801d6ef162e110d9ba556f9e4a3dbf2dc6e2b271d19e56a4e","last_reissued_at":"2026-07-03T00:16:55.890296Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T00:16:55.890296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.FA"],"primary_cat":"math.LO","authors_text":"Clemens Kinn, Philipp Petersen","submitted_at":"2026-06-29T20:44:20Z","abstract_excerpt":"We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\\varepsilon>0$ by ReLU neural networks of si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01266","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/2607.01266/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":"2607.01266","created_at":"2026-07-03T00:16:55.890364+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.01266v1","created_at":"2026-07-03T00:16:55.890364+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01266","created_at":"2026-07-03T00:16:55.890364+00:00"},{"alias_kind":"pith_short_12","alias_value":"7DZ4SGXM44VN","created_at":"2026-07-03T00:16:55.890364+00:00"},{"alias_kind":"pith_short_16","alias_value":"7DZ4SGXM44VNQAOW","created_at":"2026-07-03T00:16:55.890364+00:00"},{"alias_kind":"pith_short_8","alias_value":"7DZ4SGXM","created_at":"2026-07-03T00:16:55.890364+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO","json":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO.json","graph_json":"https://pith.science/api/pith-number/7DZ4SGXM44VNQAOW54LC4EINTO/graph.json","events_json":"https://pith.science/api/pith-number/7DZ4SGXM44VNQAOW54LC4EINTO/events.json","paper":"https://pith.science/paper/7DZ4SGXM"},"agent_actions":{"view_html":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO","download_json":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO.json","view_paper":"https://pith.science/paper/7DZ4SGXM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.01266&json=true","fetch_graph":"https://pith.science/api/pith-number/7DZ4SGXM44VNQAOW54LC4EINTO/graph.json","fetch_events":"https://pith.science/api/pith-number/7DZ4SGXM44VNQAOW54LC4EINTO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/action/storage_attestation","attest_author":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/action/author_attestation","sign_citation":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/action/citation_signature","submit_replication":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/action/replication_record"}},"created_at":"2026-07-03T00:16:55.890364+00:00","updated_at":"2026-07-03T00:16:55.890364+00:00"}