{"paper":{"title":"Four Deviations Suffice for Rank 1 Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.FA"],"primary_cat":"math.CO","authors_text":"Kyle Luh, Rasmus Kyng, Zhao Song","submitted_at":"2019-01-20T21:05:04Z","abstract_excerpt":"We prove a matrix discrepancy bound that strengthens the famous Kadison-Singer result of Marcus, Spielman, and Srivastava. Consider any independent scalar random variables $\\xi_1, \\ldots, \\xi_n$ with finite support, e.g.\n  $\\{ \\pm 1 \\}$ or $\\{ 0,1 \\}$-valued random variables, or some combination thereof. Let $u_1, \\dots, u_n \\in \\mathbb{C}^m$ and $$ \\sigma^2 = \\left\\| \\sum_{i=1}^n \\text{Var}[ \\xi_i ] (u_i u_i^{*})^2 \\right\\|. $$ Then there exists a choice of outcomes $\\varepsilon_1,\\ldots,\\varepsilon_n$ in the support of $\\xi_1, \\ldots, \\xi_n$ s.t. $$ \\left \\|\\sum_{i=1}^n \\mathbb{E} [ \\xi_i] u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.06731","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1901.06731/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}