{"paper":{"title":"Representing smooth functions as compositions of near-identity functions with implications for deep network optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","cs.NE","math.ST","stat.ML","stat.TH"],"primary_cat":"cs.LG","authors_text":"Peter L. Bartlett, Philip M. Long, Steven N. Evans","submitted_at":"2018-04-13T16:24:17Z","abstract_excerpt":"We show that any smooth bi-Lipschitz $h$ can be represented exactly as a composition $h_m \\circ ... \\circ h_1$ of functions $h_1,...,h_m$ that are close to the identity in the sense that each $\\left(h_i-\\mathrm{Id}\\right)$ is Lipschitz, and the Lipschitz constant decreases inversely with the number $m$ of functions composed. This implies that $h$ can be represented to any accuracy by a deep residual network whose nonlinear layers compute functions with a small Lipschitz constant. Next, we consider nonlinear regression with a composition of near-identity nonlinear maps. We show that, regarding "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.05012","kind":"arxiv","version":2},"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"}