This paper derives a new strong Gaussian approximation bound for the sum of independent random vectors. The approach relies on the optimal transport theory and yields \textit{explicit} dependence on the dimension size $p$ and the sample size $n$. This dependence establishes a new fundamental limit for all practical applications of statistical learning theory. Particularly, based on this bound, we prove approximation in distribution for the maximum norm in a high-dimensional setting ($p >n$).
翻译:本文为独立随机矢量的总和引出一个新的强大的高斯近似值。 这种方法依赖于最佳的运输理论, 并产生\ textit{ exclit} 依赖维度大小 $p$ 和样本大小 $n$。 这种依赖性为统计学习理论的所有实际应用设定了新的基本限制。 特别是, 基于此约束性, 我们证明在高维环境中分配最大规范的近似值 ($p>n$ ) 。