{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:GWADZORIXCG36NA6UTERDWWWJ4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e31c8ca1e3343b822cf04e48540fa74a1e03c416d0a557529b8affbeeb4a7b55","cross_cats_sorted":["cs.LG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-05-26T13:22:14Z","title_canon_sha256":"453d9f7799509908249c374dd035812cac0a7bba3873f731ba094506cf9dda35"},"schema_version":"1.0","source":{"id":"2305.16910","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2305.16910","created_at":"2026-07-05T09:40:14Z"},{"alias_kind":"arxiv_version","alias_value":"2305.16910v3","created_at":"2026-07-05T09:40:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.16910","created_at":"2026-07-05T09:40:14Z"},{"alias_kind":"pith_short_12","alias_value":"GWADZORIXCG3","created_at":"2026-07-05T09:40:14Z"},{"alias_kind":"pith_short_16","alias_value":"GWADZORIXCG36NA6","created_at":"2026-07-05T09:40:14Z"},{"alias_kind":"pith_short_8","alias_value":"GWADZORI","created_at":"2026-07-05T09:40:14Z"}],"graph_snapshots":[{"event_id":"sha256:d7372a4973426587279a424f6a8b0bf1137d80fcb5b6ece99204f1729d209986","target":"graph","created_at":"2026-07-05T09:40:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2305.16910/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions $\\varrho:\\mathbb{C}\\to \\mathbb{C}$ that have the property that their associated networks are universal, i.e., are capable of approximating continuous functions to arbitrary accuracy on compact domains. Precisely, we show that deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor $\\mathbb{R}$-affine. This is a much larger class of","authors_text":"Hannes Matt, Paul Geuchen, Thomas Jahn","cross_cats":["cs.LG","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-05-26T13:22:14Z","title":"Universal approximation with complex-valued deep narrow neural networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.16910","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:007d93c9e72b28f03ecd8f765c8078063ff2114009adcc6f029532c72cea34dc","target":"record","created_at":"2026-07-05T09:40:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e31c8ca1e3343b822cf04e48540fa74a1e03c416d0a557529b8affbeeb4a7b55","cross_cats_sorted":["cs.LG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-05-26T13:22:14Z","title_canon_sha256":"453d9f7799509908249c374dd035812cac0a7bba3873f731ba094506cf9dda35"},"schema_version":"1.0","source":{"id":"2305.16910","kind":"arxiv","version":3}},"canonical_sha256":"35803cba28b88dbf341ea4c911dad64f3fe978b4813c5857243030dfca3e5152","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35803cba28b88dbf341ea4c911dad64f3fe978b4813c5857243030dfca3e5152","first_computed_at":"2026-07-05T09:40:14.148698Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:40:14.148698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PsE4Uy0CuXrm3KW0mRzH/IcBNdgviALdqJdSK35zW9aEsab8UgWE7PQKZGIo08c9cMqHi0Ove6oP90+U5ciGBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:40:14.149183Z","signed_message":"canonical_sha256_bytes"},"source_id":"2305.16910","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:007d93c9e72b28f03ecd8f765c8078063ff2114009adcc6f029532c72cea34dc","sha256:d7372a4973426587279a424f6a8b0bf1137d80fcb5b6ece99204f1729d209986"],"state_sha256":"f67801dd476cffef3ba9c342b0d0e40da87bc0f4d16ebd72e5a2218c94418c12"}