Optimal Estimation of Structured Covariance Operators

This paper establishes optimal convergence rates for estimation of structured covariance operators of Gaussian processes. We study banded operators with kernels that decay rapidly off-the-diagonal and $L^q$-sparse operators with an unordered sparsity pattern. For these classes of operators, we find the minimax optimal rate of estimation in operator norm, identifying the fundamental dimension-free quantities that determine the sample complexity. In addition, we prove that tapering and thresholding estimators attain the optimal rate. The proof of the upper bound for tapering estimators requires novel techniques to circumvent the issue that discretization of a banded operator does not result, in general, in a banded covariance matrix. To derive lower bounds for banded and $L^q$-sparse classes, we introduce a general framework to lift theory from high-dimensional matrix estimation to the operator setting. Our work contributes to the growing literature on operator estimation and learning, building on ideas from high-dimensional statistics while also addressing new challenges that emerge in infinite dimension.