Precision and Cholesky Factor Estimation for Gaussian Processes
Jiaheng Chen, Daniel Sanz-Alonso
TL;DR
The paper develops nonparametric estimation methods for large precision matrices and their Cholesky factors arising from Gaussian processes observed at scattered locations. By exploiting the screening-induced approximate sparsity and leveraging a lattice-based local regression framework, it achieves poly-logarithmic sample complexity in the matrix size, despite ill-conditioning. It introduces a Hall’s marriage theorem-based reduction from scattered observations to a regular lattice and provides three main results: precision estimation with a high-probability spectral-norm bound, Cholesky-factor estimation under a maximin ordering, and precision-operator estimation on the lattice with explicit dependence on the condition number. The work connects operator-adapted wavelets, Vecchia-inspired local regressions, and hierarchical block-Cholesky decompositions to deliver scalable estimation procedures with provable guarantees. Practically, these results enable efficient uncertainty quantification and sampling for Gaussian-process-based models in high dimensions and large spatial domains.
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.