Gaussian discrepancy：向量平衡的概率松弛
Gaussian discrepancy: a probabilistic relaxation of vector balancing
摘要：
我们介绍了一种新的放松组合差异称为高斯差异，从而二进制签名取代相关的标准高斯随机变量。这种松弛有效地将布尔超立方体上的优化问题转化为相关矩阵空间上的优化问题。我们发现，高斯差异是一个更紧密的放松比以前研究的向量和球形差异的问题，我们构建了一个快速的在线算法，实现了一个版本的Banaszczyk约束高斯差异。这项工作也提出了新的问题，如高斯差异的Komlós猜想，这可能会揭示经典的差异问题。
英文摘要：
We introduce a novel relaxation of combinatorial discrepancy called Gaussian discrepancy, whereby binary signings are replaced with correlated standard Gaussian random variables. This relaxation effectively reformulates an optimization problem over the Boolean hypercube into one over the space of correlation matrices. We show that Gaussian discrepancy is a tighter relaxation than the previously studied vector and spherical discrepancy problems, and we construct a fast online algorithm that achieves a version of the Banaszczyk bound for Gaussian discrepancy. This work also raises new questions such as the Komlós conjecture for Gaussian discrepancy, which may shed light on classical discrepancy problems.
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