{"paper":{"title":"Computational Concentration of Measure: Optimal Bounds, Reductions, and More","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.CG","cs.CR","cs.LG"],"primary_cat":"cs.DS","authors_text":"Mohammad Mahmoody, Omid Etesami, Saeed Mahloujifar","submitted_at":"2019-07-11T17:33:03Z","abstract_excerpt":"Product measures of dimension $n$ are known to be concentrated in Hamming distance: for any set $S$ in the product space of probability $\\epsilon$, a random point in the space, with probability $1-\\delta$, has a neighbor in $S$ that is different from the original point in only $O(\\sqrt{n\\ln(1/(\\epsilon\\delta))})$ coordinates. We obtain the tight computational version of this result, showing how given a random point and access to an $S$-membership oracle, we can find such a close point in polynomial time. This resolves an open question of [Mahloujifar and Mahmoody, ALT 2019]. As corollaries, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05401","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"}