Scalable Gaussian Process Computations Using Hierarchical Matrices

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 ma...

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Bibliographic Details
Published in:Journal of computational and graphical statistics Vol. 29; no. 2; pp. 227 - 237
Main Authors: Geoga, Christopher J., Anitescu, Mihai, Stein, Michael L.
Format: Journal Article
Language:English
Published: Alexandria Taylor & Francis 02.04.2020
Taylor & Francis Ltd
Subjects:
Algorithms
Approximation
Complexity
Confidence intervals
Covariance matrix
Fisher information
Gaussian process
Mathematical analysis
Maximum likelihood estimation
Maximum likelihood estimators
Numerical linear algebra
Spatial analysis
Statistical computing
ISSN:1061-8600, 1537-2715
Online Access:Get full text
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Summary: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 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.
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ISSN:1061-8600
1537-2715
DOI:10.1080/10618600.2019.1652616