{"paper":{"title":"Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Deep KANs achieve universal approximation on compact sets precisely when they include one fixed non-affine continuous edge function.","cross_cats":["cs.NE","math.FA"],"primary_cat":"cs.LG","authors_text":"Vugar Ismailov","submitted_at":"2026-04-26T15:31:51Z","abstract_excerpt":"We analyze the universal approximation property of Kolmogorov-Arnold Networks (KANs) in terms of their edge functions. If these functions are all affine, then universality clearly fails. How many non-affine functions are needed, in addition to affine ones, to ensure universality? We show that a single one suffices. More precisely, we prove that deep KANs in which all edge functions are either affine or equal to a fixed continuous function $\\sigma$ are dense in $C(K)$ for every compact set $K\\subset\\mathbb{R}^n$ if and only if $\\sigma$ is non-affine. In contrast, for KANs with exactly two hidde"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that deep KANs in which all edge functions are either affine or equal to a fixed continuous function σ are dense in C(K) for every compact set K⊂R^n if and only if σ is non-affine.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The edge functions are continuous real-valued functions and the KAN architecture follows the standard Kolmogorov-Arnold representation with summation at nodes; the proofs rely on this continuity and the specific layered structure without additional restrictions on width or activation placement.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Deep KANs with edge functions restricted to affine maps plus one fixed non-affine continuous function σ are dense in C(K) for any compact K if and only if σ is non-affine.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Deep KANs achieve universal approximation on compact sets precisely when they include one fixed non-affine continuous edge function.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f5035a5fadedb87aeef7f5ac38311853f82a36d13bf6eaf0c7507a8489bdc962"},"source":{"id":"2604.23765","kind":"arxiv","version":2},"verdict":{"id":"9c284f1c-7076-4f5d-9344-51f1ca65a38e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T06:23:18.008354Z","strongest_claim":"We show that deep KANs in which all edge functions are either affine or equal to a fixed continuous function σ are dense in C(K) for every compact set K⊂R^n if and only if σ is non-affine.","one_line_summary":"Deep KANs with edge functions restricted to affine maps plus one fixed non-affine continuous function σ are dense in C(K) for any compact K if and only if σ is non-affine.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The edge functions are continuous real-valued functions and the KAN architecture follows the standard Kolmogorov-Arnold representation with summation at nodes; the proofs rely on this continuity and the specific layered structure without additional restrictions on width or activation placement.","pith_extraction_headline":"Deep KANs achieve universal approximation on compact sets precisely when they include one fixed non-affine continuous edge function."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.23765/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T22:46:58.820231Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"9fa96978f917cd1b15ce850e2f0c6ba4983686e5075c8936a04ef3d8a36ba9ad"},"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"}