{"paper":{"title":"Remarks on the Most Informative Function Conjecture at fixed mean","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.IT"],"primary_cat":"cs.IT","authors_text":"David Witmer, Guy Kindler, Ryan O'Donnell","submitted_at":"2015-06-10T05:09:58Z","abstract_excerpt":"In 2013, Courtade and Kumar posed the following problem: Let $\\boldsymbol{x} \\sim \\{\\pm 1\\}^n$ be uniformly random, and form $\\boldsymbol{y} \\sim \\{\\pm 1\\}^n$ by negating each bit of $\\boldsymbol{x}$ independently with probability $\\alpha$. Is it true that the mutual information $I(f(\\boldsymbol{x}) \\mathbin{;} \\boldsymbol{y})$ is maximized among $f:\\{\\pm 1\\}^n \\to \\{\\pm 1\\}$ by $f(x) = x_1$? We do not resolve this problem. Instead, we make a couple of observations about the fixed-mean version of the conjecture. We show that Courtade and Kumar's stronger Lex Conjecture fails for small noise ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03167","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"}