{"paper":{"title":"An Algorithm for Computing Lipschitz Inner Functions in Kolmogorov's Superposition Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Jonas Actor, Matthew G. Knepley","submitted_at":"2017-12-22T02:39:20Z","abstract_excerpt":"Kolmogorov famously proved that multivariate continuous functions can be represented as a superposition of a small number of univariate continuous functions, $$ f(x_1,\\dots,x_n) = \\sum_{q=0}^{2n+1} \\chi^q \\left( \\sum_{p=1}^n \\psi^{pq}(x_p) \\right).$$ Fridman \\cite{fridman} posed the best smoothness bound for the functions $\\psi^{pq}$, that such functions can be constructed to be Lipschitz continuous with constant 1. Previous algorithms to describe these inner functions have only been H\\\"older continuous, such as those proposed by K\\\"oppen and Braun and Griebel. This is problematic, as pointed "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08286","kind":"arxiv","version":1},"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"}