{"paper":{"title":"Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with $ \\ell^1 $ and $ \\ell^0 $ Controls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Andrew R. Barron, Jason M. Klusowski","submitted_at":"2016-07-26T17:52:00Z","abstract_excerpt":"We establish $ L^{\\infty} $ and $ L^2 $ error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with $ \\ell^1 $ and $ \\ell^0 $ controls on their inner and outer parameters. With the squared ReLU ridge function, we show that the $ L^2 $ approximation error is inversely proportional to the inner layer $ \\ell^0 $ sparsity and it need only be sublinear in the outer layer $ \\ell^0 $ sparsity. Our constructions are obtained using a variant of the Jones-Barron probabilistic method, which can be interpre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07819","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":""},"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"}