Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms

Abstract

Kernel-based methods are heavily used in machine learning. However, they suffer from $O(N^2)$ complexity in the number $N$ of considered data points. In this paper, we propose an approximation procedure, which reduces this complexity to $O(N)$. Our approach is based on two ideas. First, we prove that any radial kernel with analytic basis function can be represented as sliced version of some one-dimensional kernel and derive an analytic formula for the one-dimensional counterpart. It turns out that the relation between one- and $d$-dimensional kernels is given by a generalized Riemann-Liouville fractional integral. Hence, we can reduce the $d$-dimensional kernel summation to a one-dimensional setting. Second, for solving these one-dimensional problems efficiently, we apply fast Fourier summations on non-equispaced data, a sorting algorithm or a combination of both. Due to its practical importance we pay special attention to the Gaussian kernel, where we show a dimension-independent error bound and represent its one-dimensional counterpart via a closed-form Fourier transform. We provide a run time comparison and error estimate of our fast kernel summations.