Askit: approximate skeletonization kernel-independent treecode in high dimensions ∗

TLDR: In this article, a fast algorithm for kernel summation in high dimensions is presented, based on a kernel function that is a pair potential defined on a dataset of points in a high-dimensional Euclidean space.

Abstract: We present a fast algorithm for kernel summation problems in high dimensions. Such problems appear in computational physics, numerical approximation, nonparametric statistics, and machine learning. In our context, the sums depend on a kernel function that is a pair potential defined on a dataset of points in a high-dimensional Euclidean space. A direct evaluation of the sum scales quadratically with the number of points. Fast kernel summation methods can reduce this cost to linear complexity, but the constants involved do not scale well with the dimensionality of the dataset. The main algorithmic components of fast kernel summation algorithms are the separation of the kernel sum between near and far field (which is the basis for pruning) and the efficient and accurate approximation of the far field. We introduce novel methods for pruning and for approximating the far field. Our far field approximation requires only kernel evaluations and does not use analytic expansions. Pruning is not done using bounding...