Scalable Gaussian Process Computations Using Hierarchical Matrices
Christopher J. Geoga, Mihai Anitescu, Michael L. Stein · Journal of Computational and Graphical Statistics · 2019
Christopher J. Geogaa* , Mihai Anitescuab & Michael L. Steinba Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL; b Department of Statistics, University of Chicago, Chicago, ILColor versions of one or more of the figures in the article can be found online at www.tandfonline.com/r/JCGS.Supplementary materials for this article are available online. Please go to www.tandfonline.com/r/JCGS.CONTACT Christopher J. Geoga [email protected] Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL 60439.AbstractWe 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 log 2n) 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.