Fast spatial Gaussian process maximum likelihood estimation via skeletonization factorizations
For practitioners in spatial statistics and machine learning, this work reduces the computational bottleneck of Gaussian process maximum likelihood estimation, enabling application to larger datasets.
The paper presents a framework for maximum likelihood estimation of Gaussian process parameters from spatial observations, achieving $ ilde O(n^{3/2})$ time complexity for log-likelihood and gradient evaluation. This enables parameter fitting for larger datasets than previously feasible.
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 $\tilde O(n^{3/2})$ time under certain assumptions, where $n$ is the number of observations. Our method relies on the skeletonization procedure described by Martinsson & Rokhlin in the form of the recursive skeletonization factorization of Ho & Ying. Combining this with an adaptation of the matrix peeling algorithm of Lin et al. for constructing $\mathcal{H}$-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.