Precision and Cholesky Factor Estimation for Gaussian Processes

TL;DR

This paper investigates efficient estimation methods for large precision matrices and Cholesky factors in Gaussian processes, demonstrating poly-logarithmic sample complexity under certain assumptions.

Contribution

It introduces novel estimation techniques leveraging local regression and block-Cholesky decomposition, improving understanding of sample complexity for high-dimensional Gaussian process models.

Findings

Sample complexity scales poly-logarithmically with matrix size.

Local regression exploits sparsity from the screening effect.

Block-Cholesky decomposition aids in efficient Cholesky factor estimation.

Abstract

This paper studies the estimation of large precision matrices and Cholesky factors obtained by observing a Gaussian process at many locations. Under general assumptions on the precision and the observations, we show that the sample complexity scales poly-logarithmically with the size of the precision matrix and its Cholesky factor. The key challenge in these estimation tasks is the polynomial growth of the condition number of the target matrices with their size. For precision estimation, our theory hinges on an intuitive local regression technique on the lattice graph which exploits the approximate 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.