Scalable Gaussian Process Computations Using Hierarchical Matrices

Abstract

We present a kernel-independent method that applies hierarchical matrices to the problem of maximum likelihood estimation for Gaussian processes. The proposed approximation provides natural and scalable stochastic estimators for its gradient and Hessian, as well as the expected Fisher information matrix, that are computable in quasilinear $O(n{\hspace{0.17em}\text{log}\hspace{0.17em}}^{2}n)$ complexity for a large range of models. To accomplish this, we (i) choose a specific hierarchical approximation for covariance matrices that enables the computation of their exact derivatives and (ii) use a stabilized form of the Hutchinson stochastic trace estimator. Since both the observed and expected information matrices can be computed in quasilinear complexity, covariance matrices for maximum likelihood estimators (MLEs) can also be estimated efficiently. In this study, we demonstrate the scalability of the method, show how details of its implementation effect numerical accuracy and computational effort, and validate that the resulting MLEs and confidence intervals based on the inverse Fisher information matrix faithfully approach those obtained by the exact likelihood. Supplementary materials for this article are available online.

KEYWORDS:

• Algorithms
• Numerical linear algebra
• Spatial analysis
• Statistical computing

Additional information

Funding

This material was based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR) under contracts DE-AC02-06CH11347 and DE-AC02-06CH11357. We acknowledge partial NSF funding through awards FP061151-01-PR and CNS-1545046 to MA.