Gram-Schmidt Walk：治疗Banaszczyk Blues
The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues
摘要：
一个重要的结果，在差异，由于Banaszczyk指出，任何一组$n$向量在$\mathbb{R}^m$的$\ell_2$规范至多$1$和任何凸体$K$在$\mathbb{R}^m$的高斯措施至少一半，存在一个$\pm 1$组合这些向量在于$5K$。这一结果意味着最好的知名的界限，几个问题的差异。Banaszczyk对这一结果的证明是非建设性的，一个主要的开放问题是给出一个有效的算法来找到这样一个$\pm 1$向量组合。本文解决了这一问题，并给出了一个有效的随机算法来寻找位于$cK$中的向量的$\pm 1$组合，其中$c>0$是一个绝对常数。这导致新的高效算法的差异理论中的几个问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
