{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:ZZCKXIX62JVKF2ODB75KEGJVIS","short_pith_number":"pith:ZZCKXIX6","schema_version":"1.0","canonical_sha256":"ce44aba2fed26aa2e9c30ffaa219354483ef74d4b1c1a699146cd2d9bd73417d","source":{"kind":"arxiv","id":"2202.03841","version":2},"attestation_state":"computed","paper":{"title":"Width is Less Important than Depth in ReLU Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","stat.ML"],"primary_cat":"cs.LG","authors_text":"Gal Vardi, Gilad Yehudai, Ohad Shamir","submitted_at":"2022-02-08T13:07:22Z","abstract_excerpt":"We solve an open question from Lu et al. (2017), by showing that any target network with inputs in $\\mathbb{R}^d$ can be approximated by a width $O(d)$ network (independent of the target network's architecture), whose number of parameters is essentially larger only by a linear factor. In light of previous depth separation theorems, which imply that a similar result cannot hold when the roles of width and depth are interchanged, it follows that depth plays a more significant role than width in the expressive power of neural networks.\n  We extend our results to constructing networks with bounded"},"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":"2202.03841","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2022-02-08T13:07:22Z","cross_cats_sorted":["cs.NE","stat.ML"],"title_canon_sha256":"2bd444659e2f846f51da41905bb57a7c0e666e676b3a61fda1571128c5e0dc50","abstract_canon_sha256":"6f7f897ec63d6fb8d1aea44dde42486f981bf7afcc40730150a036a2e9e07b02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:28:09.984777Z","signature_b64":"FLrR/7xrGmluIUJ5HBqcwN7fPE5W+layan6hAeu82zaQjDTpirLsOPqIDW5SCN2lv2H4imavZeAjRKHYhBMHBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce44aba2fed26aa2e9c30ffaa219354483ef74d4b1c1a699146cd2d9bd73417d","last_reissued_at":"2026-07-05T04:28:09.984323Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:28:09.984323Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Width is Less Important than Depth in ReLU Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","stat.ML"],"primary_cat":"cs.LG","authors_text":"Gal Vardi, Gilad Yehudai, Ohad Shamir","submitted_at":"2022-02-08T13:07:22Z","abstract_excerpt":"We solve an open question from Lu et al. (2017), by showing that any target network with inputs in $\\mathbb{R}^d$ can be approximated by a width $O(d)$ network (independent of the target network's architecture), whose number of parameters is essentially larger only by a linear factor. In light of previous depth separation theorems, which imply that a similar result cannot hold when the roles of width and depth are interchanged, it follows that depth plays a more significant role than width in the expressive power of neural networks.\n  We extend our results to constructing networks with bounded"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.03841","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/2202.03841/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":"2202.03841","created_at":"2026-07-05T04:28:09.984380+00:00"},{"alias_kind":"arxiv_version","alias_value":"2202.03841v2","created_at":"2026-07-05T04:28:09.984380+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.03841","created_at":"2026-07-05T04:28:09.984380+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZZCKXIX62JVK","created_at":"2026-07-05T04:28:09.984380+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZZCKXIX62JVKF2OD","created_at":"2026-07-05T04:28:09.984380+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZZCKXIX6","created_at":"2026-07-05T04:28:09.984380+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/ZZCKXIX62JVKF2ODB75KEGJVIS","json":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS.json","graph_json":"https://pith.science/api/pith-number/ZZCKXIX62JVKF2ODB75KEGJVIS/graph.json","events_json":"https://pith.science/api/pith-number/ZZCKXIX62JVKF2ODB75KEGJVIS/events.json","paper":"https://pith.science/paper/ZZCKXIX6"},"agent_actions":{"view_html":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS","download_json":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS.json","view_paper":"https://pith.science/paper/ZZCKXIX6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2202.03841&json=true","fetch_graph":"https://pith.science/api/pith-number/ZZCKXIX62JVKF2ODB75KEGJVIS/graph.json","fetch_events":"https://pith.science/api/pith-number/ZZCKXIX62JVKF2ODB75KEGJVIS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS/action/storage_attestation","attest_author":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS/action/author_attestation","sign_citation":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS/action/citation_signature","submit_replication":"https://pith.science/pith/ZZCKXIX62JVKF2ODB75KEGJVIS/action/replication_record"}},"created_at":"2026-07-05T04:28:09.984380+00:00","updated_at":"2026-07-05T04:28:09.984380+00:00"}