{"paper":{"title":"Neural networks and rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","stat.ML"],"primary_cat":"cs.LG","authors_text":"Matus Telgarsky","submitted_at":"2017-06-11T03:07:42Z","abstract_excerpt":"Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree $O(\\text{polylog}(1/\\epsilon))$ which is $\\epsilon$-close, and similarly for any rational function there exists a ReLU network of size $O(\\text{polylog}(1/\\epsilon))$ which is $\\epsilon$-close. By contrast, polynomials need degree $\\Omega(\\text{poly}(1/\\epsilon))$ to approximate even a single ReLU. When converting a ReLU network to a rational function as above, the hidden constants depend exponentially on the number of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03301","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"}