# The Gram-Schmidt walk: a cure for the Banaszczyk blues

@article{Bansal2018TheGW, title={The Gram-Schmidt walk: a cure for the Banaszczyk blues}, author={Nikhil Bansal and Daniel Dadush and Shashwat Garg and Shachar Lovett}, journal={Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing}, year={2018} }

An important result in discrepancy due to Banaszczyk states that for any set of n vectors in ℝm of ℓ2 norm at most 1 and any convex body K in ℝm of Gaussian measure at least half, there exists a ± 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 an open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors. In this paper…

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