On the Estimation of Gaussian Moment Tensors

TL;DR

This paper compares two estimators for Gaussian moment tensors, showing that the Isserlis-based plug-in estimator outperforms the standard sample estimator in high-dimensional, non-asymptotic settings.

Contribution

The paper provides the first non-asymptotic, dimension-free error bounds for Gaussian moment tensor estimators, highlighting the advantages of Isserlis's theorem-based estimator.

Findings

Isserlis's estimator has lower error bounds than the sample estimator for even-order tensors.

Bounds are valid in operator and entrywise maximum norms.

Results apply to both symmetric and asymmetric tensors.

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.