{"paper":{"title":"The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Daniel Dadush, Nikhil Bansal, Shachar Lovett, Shashwat Garg","submitted_at":"2017-08-03T09:49:42Z","abstract_excerpt":"An important result in discrepancy due to Banaszczyk states that for any set of $n$ vectors in $\\mathbb{R}^m$ of $\\ell_2$ norm at most $1$ and any convex body $K$ in $\\mathbb{R}^m$ of Gaussian measure at least half, there exists a $\\pm 1$ combination of these vectors which lies in $5K$. This result implies the best known bounds for several problems in discrepancy. Banaszczyk's proof of this result is non-constructive and a major open problem has been to give an efficient algorithm to find such a $\\pm 1$ combination of the vectors.\n  In this paper, we resolve this question and give an efficient"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01079","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"}