A CLT in Stein's distance for generalized Wishart matrices and higher order tensors

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. Specifi...

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Bibliographic Details
Published in:arXiv.org 2020-11
Main Author: Mikulincer, Dan
Format: Article
Language:English
Subjects:
Convergence
Mathematical analysis
Tensors
Online Access:Get full text
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Summary: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 \(X_i^{\otimes p}\).
ISSN:2331-8422