论文标题
Stein距离的CLT，用于广义的WishArt矩阵和高阶张量
A CLT in Stein's distance for generalized Wishart matrices and higher order tensors
论文作者
论文摘要
我们研究了独立张量幂的总和，$ \ frac {1} {\ sqrt {d}} \ sum \ limits_ {i = 1}^d x_i^{\ otimes p} $。我们专注于高维度，其中$ x_i \ in \ mathbb {r}^n $和$ n $可以用$ d $扩展。我们的主要结果是提出的收敛阈值。具体来说，我们表明，在某些规律性的假设下，如果$ n^{2p-1} \ gg d $，则标准化的总和会收敛到高斯。结果除其他外，该结果适用于对称统一的对数符号测量和产品度量。这概括了文献中发现的几个结果。 我们的主要技术是最佳传输到斯坦因的方法的新颖应用，该应用是$ x_i^{\ otimes p} $固有的低维结构。
We study the central limit theorem for sums of independent tensor powers, $\frac{1}{\sqrt{d}}\sum\limits_{i=1}^d X_i^{\otimes p}$. We focus on the high-dimensional regime where $X_i \in \mathbb{R}^n$ and $n$ may scale with $d$. Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if $n^{2p-1}\gg d$, then the normalized sum converges to a Gaussian. The results apply, among others, to symmetric uniform log-concave measures and to product measures. This generalizes several results found in the literature. Our main technique is a novel application of optimal transport to Stein's method which accounts for the low dimensional structure which is inherent in $X_i^{\otimes p}$.
