Fast Spatial Gaussian Process Maximum Likelihood Estimation Via Skeletonization Factorizations
In many computational science applications, the problem of interest exhibits a strong underlying spatial structure that can be exploited for algorithmic efficiency. For example, the nested dissection method (George, 1973) takes the underlying two-dimensional or three-dimensional nature of a sparse finite-difference stencil or finite-element mesh and hierarchically subdivides the domain to exploit spatial locality. More recently, algorithms based on similar hierarchical matrix structure have come in to the limelight for dealing with dense matrices that are morally the same as these sparse matrices, exhibiting low-rank blocks where before there were zeros (see work by Hackbusch et al.; Xia et al.; Chandrasekaran et al.; Greengard et al., among others). In the context of a framework for maximum likelihood estimation for estimating parameters of 2-D Gaussian processes, I will introduce at a high level skeletonization (Martinsson & Rokhlin, 2005), the recursive skeletonization factorization (Ho & Ying, 2015), and matrix peeling (Lin, Lu & Ying, 2011) and show how hierarchical rank structure can be exploited for this problem.