{"paper":{"title":"Noise Stability and Correlation with Half Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.PR","authors_text":"Elchanan Mossel, Joe Neeman","submitted_at":"2016-03-06T07:17:55Z","abstract_excerpt":"Benjamini, Kalai and Schramm showed that a monotone function $f : \\{-1,1\\}^n \\to \\{-1,1\\}$ is noise stable if and only if it is correlated with a half-space (a set of the form $\\{x: \\langle x, a\\rangle \\le b\\}$).\n  We study noise stability in terms of correlation with half-spaces for general (not necessarily monotone) functions. We show that a function $f: \\{-1, 1\\}^n \\to \\{-1, 1\\}$ is noise stable if and only if it becomes correlated with a half-space when we modify $f$ by randomly restricting a constant fraction of its coordinates.\n  Looking at random restrictions is necessary: we construct "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01799","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"}