{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:TJEZ3DCLRVGNIR5RHRDTGGI7Y2","short_pith_number":"pith:TJEZ3DCL","schema_version":"1.0","canonical_sha256":"9a499d8c4b8d4cd447b13c4733191fc68c66f20d7e879fdd3e90f799c93948bd","source":{"kind":"arxiv","id":"1709.02540","version":3},"attestation_state":"computed","paper":{"title":"The Expressive Power of Neural Networks: A View from the Width","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Feicheng Wang, Hongming Pu, Liwei Wang, Zhiqiang Hu, Zhou Lu","submitted_at":"2017-09-08T05:00:20Z","abstract_excerpt":"The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that depth-bounded (e.g. depth-$2$) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for width-bounded ReLU networks: width-$(n+4)$ ReLU networks, where $n$ is the input dimension, are universal approximators. Moreover, except for a measure zero set, all funct"},"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":"1709.02540","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-08T05:00:20Z","cross_cats_sorted":[],"title_canon_sha256":"beeebe47f8510900afb5780d20ae25c1199b8a28a61df55ac0931f39c1fc1c5c","abstract_canon_sha256":"260da47cd201022e01e734f2e749605581516cb337569287bb043a61877b8ef8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:35.627855Z","signature_b64":"1zkzKDA+MUSDCYJDaXdOfpgvvShgoYqWeK0BVesfpbXGYf9v4wh8WLfttvXR6bGehPvVejuQ/1rzhfRa/SpcBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a499d8c4b8d4cd447b13c4733191fc68c66f20d7e879fdd3e90f799c93948bd","last_reissued_at":"2026-05-18T00:31:35.627233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:35.627233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Expressive Power of Neural Networks: A View from the Width","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Feicheng Wang, Hongming Pu, Liwei Wang, Zhiqiang Hu, Zhou Lu","submitted_at":"2017-09-08T05:00:20Z","abstract_excerpt":"The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that depth-bounded (e.g. depth-$2$) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for width-bounded ReLU networks: width-$(n+4)$ ReLU networks, where $n$ is the input dimension, are universal approximators. Moreover, except for a measure zero set, all funct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02540","kind":"arxiv","version":3},"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":"1709.02540","created_at":"2026-05-18T00:31:35.627327+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.02540v3","created_at":"2026-05-18T00:31:35.627327+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.02540","created_at":"2026-05-18T00:31:35.627327+00:00"},{"alias_kind":"pith_short_12","alias_value":"TJEZ3DCLRVGN","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_16","alias_value":"TJEZ3DCLRVGNIR5R","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_8","alias_value":"TJEZ3DCL","created_at":"2026-05-18T12:31:46.661854+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2405.08319","citing_title":"Measurement-based quantum machine learning","ref_index":73,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2","json":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2.json","graph_json":"https://pith.science/api/pith-number/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/graph.json","events_json":"https://pith.science/api/pith-number/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/events.json","paper":"https://pith.science/paper/TJEZ3DCL"},"agent_actions":{"view_html":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2","download_json":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2.json","view_paper":"https://pith.science/paper/TJEZ3DCL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.02540&json=true","fetch_graph":"https://pith.science/api/pith-number/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/graph.json","fetch_events":"https://pith.science/api/pith-number/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/action/storage_attestation","attest_author":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/action/author_attestation","sign_citation":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/action/citation_signature","submit_replication":"https://pith.science/pith/TJEZ3DCLRVGNIR5RHRDTGGI7Y2/action/replication_record"}},"created_at":"2026-05-18T00:31:35.627327+00:00","updated_at":"2026-05-18T00:31:35.627327+00:00"}