{"paper":{"title":"On best approximation by multivariate ridge functions with applications to generalized translation networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","stat.ML"],"primary_cat":"math.FA","authors_text":"Felix Voigtlaender, Olov Schavemaker, Palina Salanevich, Paul Geuchen","submitted_at":"2024-12-11T15:16:16Z","abstract_excerpt":"In this paper, we prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., for approximation by functions of the form $\\mathbb{R}^d \\ni x \\mapsto \\sum_{k=1}^n \\varrho_k(A_k x) \\in \\mathbb{R}$ with $\\varrho_k : \\mathbb{R}^\\ell \\to \\mathbb{R}$ and $A_k \\in \\mathbb{R}^{\\ell \\times d}$. We show that the order of approximation asymptotically behaves as $n^{-r/(d-\\ell)}$, where $r$ is the regularity (order of differentiability) of the Sobolev functions to be approximated. Our lower bound even holds when approximating $L^\\infty$-Sobo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.08453","kind":"arxiv","version":3},"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/2412.08453/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"}