Samplet compression for conditionally positive definite kernels and universal Kriging

Abstract

We present a samplet-based framework for the efficient numerical solution of saddle-point systems arising from conditionally positive definite (CPD) kernel approximation in general and universal Kriging in particular. The vanishing moment property of samplets as well as the particular structure of the associated scaling distributions, which correspond to discrete orthogonal polynomials, allow for a numerically favorable representation of these saddle-point systems. Concretely, they enable a natural null-space reduction of the indefinite saddle-point system to a (large) linear system for the detail coefficients and a small triangular system for the polynomial coefficients. We derive error bounds for the approximation by polyharmonic splines in Beppo-Levi spaces and show that the detail coefficients span precisely the subspace on which the CPD kernel is positive definite, rendering the reduced block symmetric positive definite. In view of the quasi-sparsity of the samplet-transformed kernel matrix for asymptotically smooth kernels, the resulting method achieves O(N log N) cost for the assembly and the storage of the saddle-point system. The reduced system can efficiently be solved by a sparse Cholesky factorization. We illustrate the framework with three applications, namely Gaussian process regression with generalized covariances, landmark-based image registration via samplet-compressed thin plate splines, and three-dimensional mesh deformation.