{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:F7RUVTRPD74UILQYMSC3KS7SXB","short_pith_number":"pith:F7RUVTRP","schema_version":"1.0","canonical_sha256":"2fe34ace2f1ff9442e186485b54bf2b855efb27ff1046f2c85b6292b526616f0","source":{"kind":"arxiv","id":"1710.11278","version":2},"attestation_state":"computed","paper":{"title":"Approximating Continuous Functions by ReLU Nets of Minimal Width","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.LG","math.CO","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Boris Hanin, Mark Sellke","submitted_at":"2017-10-31T00:26:56Z","abstract_excerpt":"This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\\geq 1,$ what is the minimal width $w$ so that neural nets with ReLU activations, input dimension $d_{in}$, hidden layer widths at most $w,$ and arbitrary depth can approximate any continuous, real-valued function of $d_{in}$ variables arbitrarily well? It turns out that this minimal width is exactly equal to $d_{in}+1.$ That is, if all the hidden layer widths are bounded by $d_{in}$, then even in the infinite depth limit"},"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":"1710.11278","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2017-10-31T00:26:56Z","cross_cats_sorted":["cs.CC","cs.LG","math.CO","math.ST","stat.TH"],"title_canon_sha256":"bc2b9772aa1a27ab4ba4230e024cad1a6be2373db87a6ee2ae604b1f2846b618","abstract_canon_sha256":"4f98bb4dc492ff73584fdcd62bb3461006bea5b764088cf0e1d064638a2319b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:34.566561Z","signature_b64":"W6/BE+Bkc0ve4+mglbLVFGPLNziEG/pJZv7neq5ykpMWy7yZM5yVTR0YAXEnuCsR31X3d8/LfJyu+Wj2okI5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fe34ace2f1ff9442e186485b54bf2b855efb27ff1046f2c85b6292b526616f0","last_reissued_at":"2026-05-18T00:21:34.565831Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:34.565831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximating Continuous Functions by ReLU Nets of Minimal Width","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.LG","math.CO","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Boris Hanin, Mark Sellke","submitted_at":"2017-10-31T00:26:56Z","abstract_excerpt":"This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\\geq 1,$ what is the minimal width $w$ so that neural nets with ReLU activations, input dimension $d_{in}$, hidden layer widths at most $w,$ and arbitrary depth can approximate any continuous, real-valued function of $d_{in}$ variables arbitrarily well? It turns out that this minimal width is exactly equal to $d_{in}+1.$ That is, if all the hidden layer widths are bounded by $d_{in}$, then even in the infinite depth limit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.11278","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":""},"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":"1710.11278","created_at":"2026-05-18T00:21:34.565961+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.11278v2","created_at":"2026-05-18T00:21:34.565961+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.11278","created_at":"2026-05-18T00:21:34.565961+00:00"},{"alias_kind":"pith_short_12","alias_value":"F7RUVTRPD74U","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_16","alias_value":"F7RUVTRPD74UILQY","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_8","alias_value":"F7RUVTRP","created_at":"2026-05-18T12:31:15.632608+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":8,"internal_anchor_count":4,"sample":[{"citing_arxiv_id":"2503.15696","citing_title":"Approximation properties of neural ODEs","ref_index":15,"is_internal_anchor":true},{"citing_arxiv_id":"2605.21451","citing_title":"Approximation Theory for Neural Networks: Old and New","ref_index":16,"is_internal_anchor":true},{"citing_arxiv_id":"2509.21280","citing_title":"Model reduction of parametric ordinary differential equations via autoencoders: representation properties and convergence analysis","ref_index":28,"is_internal_anchor":true},{"citing_arxiv_id":"2512.01015","citing_title":"Upper Approximation Bounds for Neural Oscillators","ref_index":18,"is_internal_anchor":true},{"citing_arxiv_id":"2605.12301","citing_title":"Approximation of Maximally Monotone Operators : A Graph Convergence Perspective","ref_index":24,"is_internal_anchor":false},{"citing_arxiv_id":"2604.27052","citing_title":"Man, Machine, and Mathematics","ref_index":39,"is_internal_anchor":false},{"citing_arxiv_id":"2605.05659","citing_title":"Structural Correspondence and Universal Approximation in Diagonal plus Low-Rank Neural Networks","ref_index":22,"is_internal_anchor":false},{"citing_arxiv_id":"2604.21393","citing_title":"Relocation of compact sets in $\\mathbb{R}^n$ by diffeomorphisms and linear separability of datasets in $\\mathbb{R}^n$","ref_index":4,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB","json":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB.json","graph_json":"https://pith.science/api/pith-number/F7RUVTRPD74UILQYMSC3KS7SXB/graph.json","events_json":"https://pith.science/api/pith-number/F7RUVTRPD74UILQYMSC3KS7SXB/events.json","paper":"https://pith.science/paper/F7RUVTRP"},"agent_actions":{"view_html":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB","download_json":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB.json","view_paper":"https://pith.science/paper/F7RUVTRP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.11278&json=true","fetch_graph":"https://pith.science/api/pith-number/F7RUVTRPD74UILQYMSC3KS7SXB/graph.json","fetch_events":"https://pith.science/api/pith-number/F7RUVTRPD74UILQYMSC3KS7SXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB/action/storage_attestation","attest_author":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB/action/author_attestation","sign_citation":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB/action/citation_signature","submit_replication":"https://pith.science/pith/F7RUVTRPD74UILQYMSC3KS7SXB/action/replication_record"}},"created_at":"2026-05-18T00:21:34.565961+00:00","updated_at":"2026-05-18T00:21:34.565961+00:00"}