{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-gram-schmidt-walk-a-cure-for-the","title":"The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues","arxiv_id":"1708.01079","date":"2017-08-03","proceeding":null,"authors":["Nikhil Bansal","Daniel Dadush","Shashwat Garg","Shachar Lovett"],"abstract":"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. In this paper, we resolve this question and give an efficient randomized algorithm to find a $\\pm 1$ combination of the vectors which lies in $cK$ for $c>0$ an absolute constant. This leads to new efficient algorithms for several problems in discrepancy theory.","url_abs":"https://arxiv.org/abs/1708.01079v1","url_pdf":"https://arxiv.org/pdf/1708.01079v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-gram-schmidt-walk-a-cure-for-the","repo_url":"https://github.com/crharshaw/GSWDesign.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1708.01079","atlas_url":"https://app.syntology.ai/?focus=1708.01079","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}