{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:HJN3TP7F65KVKTRJJILXIPDNDR","short_pith_number":"pith:HJN3TP7F","schema_version":"1.0","canonical_sha256":"3a5bb9bfe5f755554e294a17743c6d1c5eb5f2c463d9e978f28f942469b0cd62","source":{"kind":"arxiv","id":"1706.03301","version":1},"attestation_state":"computed","paper":{"title":"Neural networks and rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","stat.ML"],"primary_cat":"cs.LG","authors_text":"Matus Telgarsky","submitted_at":"2017-06-11T03:07:42Z","abstract_excerpt":"Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree $O(\\text{polylog}(1/\\epsilon))$ which is $\\epsilon$-close, and similarly for any rational function there exists a ReLU network of size $O(\\text{polylog}(1/\\epsilon))$ which is $\\epsilon$-close. By contrast, polynomials need degree $\\Omega(\\text{poly}(1/\\epsilon))$ to approximate even a single ReLU. When converting a ReLU network to a rational function as above, the hidden constants depend exponentially on the number of"},"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":"1706.03301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-06-11T03:07:42Z","cross_cats_sorted":["cs.NE","stat.ML"],"title_canon_sha256":"132d117ae67f945fa6b9f712a0a64b17a6fc2885afefaacef5c0a472c370ada9","abstract_canon_sha256":"4e4868c8e06e374088513ffcacedef4b71c0fa9dd3306115a7d61af09fbbf23b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:37.851037Z","signature_b64":"qkHIEQ1RzSKg5mAWQU4xZ8dbMLHAwmIauNgcseDSTfLZwHb4j/rrXLZzNDQSUfrN4rRm4xKMNJHDCZSqiLVgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a5bb9bfe5f755554e294a17743c6d1c5eb5f2c463d9e978f28f942469b0cd62","last_reissued_at":"2026-05-18T00:42:37.850517Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:37.850517Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Neural networks and rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","stat.ML"],"primary_cat":"cs.LG","authors_text":"Matus Telgarsky","submitted_at":"2017-06-11T03:07:42Z","abstract_excerpt":"Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree $O(\\text{polylog}(1/\\epsilon))$ which is $\\epsilon$-close, and similarly for any rational function there exists a ReLU network of size $O(\\text{polylog}(1/\\epsilon))$ which is $\\epsilon$-close. By contrast, polynomials need degree $\\Omega(\\text{poly}(1/\\epsilon))$ to approximate even a single ReLU. When converting a ReLU network to a rational function as above, the hidden constants depend exponentially on the number of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03301","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":""},"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":"1706.03301","created_at":"2026-05-18T00:42:37.850598+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.03301v1","created_at":"2026-05-18T00:42:37.850598+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03301","created_at":"2026-05-18T00:42:37.850598+00:00"},{"alias_kind":"pith_short_12","alias_value":"HJN3TP7F65KV","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_16","alias_value":"HJN3TP7F65KVKTRJ","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_8","alias_value":"HJN3TP7F","created_at":"2026-05-18T12:31:18.294218+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/HJN3TP7F65KVKTRJJILXIPDNDR","json":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR.json","graph_json":"https://pith.science/api/pith-number/HJN3TP7F65KVKTRJJILXIPDNDR/graph.json","events_json":"https://pith.science/api/pith-number/HJN3TP7F65KVKTRJJILXIPDNDR/events.json","paper":"https://pith.science/paper/HJN3TP7F"},"agent_actions":{"view_html":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR","download_json":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR.json","view_paper":"https://pith.science/paper/HJN3TP7F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.03301&json=true","fetch_graph":"https://pith.science/api/pith-number/HJN3TP7F65KVKTRJJILXIPDNDR/graph.json","fetch_events":"https://pith.science/api/pith-number/HJN3TP7F65KVKTRJJILXIPDNDR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR/action/storage_attestation","attest_author":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR/action/author_attestation","sign_citation":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR/action/citation_signature","submit_replication":"https://pith.science/pith/HJN3TP7F65KVKTRJJILXIPDNDR/action/replication_record"}},"created_at":"2026-05-18T00:42:37.850598+00:00","updated_at":"2026-05-18T00:42:37.850598+00:00"}