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

Omar Al-Ghattas, Jiaheng Chen, Daniel Sanz-Alonso, Nathan Waniorek

TL;DR

It is proved 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.

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.

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

TL;DR

It is proved 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.

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.
Paper Structure (7 sections, 5 theorems, 27 equations, 2 figures)

This paper contains 7 sections, 5 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Thresholded Estimation of Covariance Operators
  4. Small Lengthscale Regime
  5. Application in Ensemble Kalman Filters
  6. Thresholded Estimation of Covariance Operators
  7. Covariance Function Estimation

Key Result

Theorem 2.2

Suppose that Assumption assumption:main1 holds. Let $1\le c_0\le \sqrt{N}$ and set Then, for any $p\ge 1$, where $c$ is a universal constant.

Figures (2)

  • Figure 1: Plots of the average relative errors and 95% confidence intervals achieved by the sample ($\varepsilon$, dashed blue) and thresholded ($\varepsilon_{\widehat{\rho}_N}$, solid red) covariance estimators based on sample size ($N$, dotted green) for the squared exponential kernel (left) and Matérn kernel (right) in $d=1$ over 100 trials.
  • Figure 2: Plots of the average relative errors and 95% confidence intervals achieved by the sample ($\varepsilon$, dashed blue) and thresholded ($\varepsilon_{\widehat{\rho}_N}$, solid red) covariance estimators based on sample size ($N$, dotted green) for the squared exponential kernel (left) and Matérn kernel (right) in $d=2$ over 30 trials.

Theorems & Definitions (10)

  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10: Approximation of Mean-Field EnKF
  • Proposition 3.1
  • proof