Fast spatial Gaussian process maximum likelihood estimation via skeletonization factorizations

Maximum likelihood estimation for parameter fitting given observations from a Gaussian process in space is a computationally demanding task that restricts the use of such methods to moderately sized datasets. We present a framework for unstructured observations in two spatial dimensions that allows for evaluation of the log-likelihood and its gradient (i.e., the score equations) in Õ(n3/2) time under certain assumptions, where n is the number of observations. Our method relies on the skeletonization procedure described by Martinsson and Rokhlin [J. Cornput. Phys., 205 (2005), pp. 1-23] in the form of the recursive skeletonization factorization of Ho and Ying [Cornrn. Pure Appl. Math., 69 (2015), pp. 1415-1451]. Combining this with an adaptation of the matrix peeling algorithm of Lin, Lu, and Ying [J. Cornput. Phys., 230 (2011), pp, 4071-4087] for constructing ℋ-matrix representations of black-box operators, we obtain a framework that can be used in the context of any first-order optimization routine to quickly and accurately compute maximum likelihood estimates. © 2017 Society for Industrial and Applied Mathematics.