Spectral-norm risk rates for multi-taper estimation of Gaussian processes

TL;DR

This paper analyzes the spectral-norm risk rates of multi-taper estimators for Gaussian process covariance estimation, demonstrating their optimality in one dimension and near-optimality in multi-dimensional settings.

Contribution

It derives spectral-norm risk rates for multi-taper estimators and extends lower bounds to multi-dimensional grids, establishing near-optimal performance in complex domains.

Findings

Thomson's multi-taper achieves optimal risk rates in 1D.

Multi-taper estimators have near-optimal risk in 2D acquisition domains.

Spectral-norm risk rates match known benchmarks in specific cases.

Abstract

We consider the estimation of the covariance of a stationary Gaussian process on a multi-dimensional grid from observations taken on a general acquisition domain. We derive spectral-norm risk rates for multi-taper estimators. When applied to one dimensional acquisition intervals, these show that Thomson's classical multi-taper has optimal risk rates, as they match known benchmarks. We also extend existing lower risk bounds to multi-dimensional grids and conclude that multi-taper estimators associated with certain two-dimensional acquisition domains also have almost optimal risk rates.