On the Estimation of Gaussian Moment Tensors

TL;DR

The paper develops a dimension-free, non-asymptotic theory for estimating Gaussian moment tensors of arbitrary order by comparing the standard sample moment estimator with a plug-in Isserlis-based estimator. It provides sharp operator- and entrywise-norm bounds showing that Isserlis's estimator requires fewer samples to achieve consistency for even $p>2$, in both symmetric and asymmetric tensor settings, and it derives matching lower bounds to establish tightness. Additionally, the work delivers an explicit entrywise maximum-norm bound for the sample moment tensor and proves perturbation-based lower and upper bounds that quantify the stability and advantages of Isserlis-based estimation under covariance perturbations. Together, these results enhance understanding of high-order tensor estimation in Gaussian models and inform practical sample-complexity improvements for tensor methods in statistics and ML.

Abstract

This paper studies two estimators for Gaussian moment tensors: the standard sample moment estimator and a plug-in estimator based on Isserlis's theorem. We establish dimension-free, non-asymptotic error bounds that demonstrate and quantify the advantage of Isserlis's estimator for tensors of even order $p>2$. Our bounds hold in operator and entrywise maximum norms, and apply to symmetric and asymmetric tensors.

Theorems & Definitions (14)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3
• Theorem 3.4
• proof : Proof of Theorem \ref{['thm:IsserlissBest_asymmetry']}
• Proposition 3.5
• Corollary 3.6
• proof : Proof of Corollary \ref{['coro:perturbation']}