Covariance and Precision Operator Estimation for Gaussian Processes

Music City Center 

Gaussian processes (GPs) are popular models for random functions in statistics, machine learning and data science. This talk focuses on the estimation of covariance and precision operator of GPs. The first part of the talk will give minimax rates for structured covariance operator estimation, including banded operators with kernels that decay rapidly off-the-diagonal and $L^q$-sparse operators with unordered sparsity pattern. We identify the fundamental dimension-free quantities that determine the sample complexity and show that tapering and thresholding estimators attain the optimal rate. In the second part of the talk, I will discuss the estimation of large ill-conditioned precision matrices and Cholesky factors obtained by observing a GP at many locations. Under general assumptions, we show that the sample complexity scales poly-logarithmically with the size of the matrices. For precision estimation, our theory hinges on a local regression technique on the lattice graph exploiting the sparsity implied by the screening effect. For Cholesky factor estimation, we leverage a block-Cholesky decomposition recently used to establish complexity bounds for sparse Cholesky factorization.

Gaussian processes

Minimax theory

Dimension-free analysis

Covariance estimation

Precision estimation

Cholesky factor estimation 

Section on Nonparametric Statistics