{"paper":{"title":"Conditioning and covariance on caterpillars","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DS","math.IT","math.PR"],"primary_cat":"cs.IT","authors_text":"Ryan O'Donnell, Sarah R. Allen","submitted_at":"2014-07-16T18:51:12Z","abstract_excerpt":"Let $X_1, \\dots, X_n$ be joint $\\{ \\pm 1\\}$-valued random variables. It is known that conditioning on a random subset of $O(1/\\epsilon^2)$ of them reduces their average pairwise covariance to below $\\epsilon$ (in expectation). We conjecture that $O(1/\\epsilon^2)$ can be improved to $O(1/\\epsilon)$. The motivation for the problem and our conjectured improvement comes from the theory of global correlation rounding for convex relaxation hierarchies. We suggest attempting the conjecture in the case that $X_1, \\dots, X_n$ are the leaves of an information flow tree. We prove the conjecture in the ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.4423","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"}