Covariance Operator Estimation: Sparsity, Lengthscale, and Ensemble Kalman Filters

TL;DR

This paper develops thresholding methods for estimating covariance operators in Gaussian fields, showing they outperform standard estimators especially for fields with small correlation lengthscales, with applications to ensemble Kalman filters.

Contribution

It provides non-asymptotic bounds for thresholded covariance estimators and demonstrates their exponential sample complexity improvement over traditional methods for certain fields.

Findings

Thresholded estimators significantly reduce sample complexity.

Small correlation lengthscales lead to exponential improvements.

Application to ensemble Kalman filters enhances covariance estimation.

Abstract

This paper investigates covariance operator estimation via thresholding. For Gaussian random fields with approximately sparse covariance operators, we establish non-asymptotic bounds on the estimation error in terms of the sparsity level of the covariance and the expected supremum of the field. We prove that thresholded estimators enjoy an exponential improvement in sample complexity compared with the standard sample covariance estimator if the field has a small correlation lengthscale. As an application of the theory, we study thresholded estimation of covariance operators within ensemble Kalman filters.