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.

翻译：本文研究通过阈值化方法进行协方差算子估计。对于具有近似稀疏协方差算子的高斯随机场，我们根据协方差的稀疏程度与场的期望上确界，建立了估计误差的非渐近上界。我们证明，若随机场具有较小的相关尺度，与标准样本协方差估计器相比，阈值化估计器在样本复杂度上可实现指数级提升。作为该理论的应用，我们研究了集合卡尔曼滤波器中协方差算子的阈值化估计方法。