Fast Computation of Sums of Gaussians in High Dimensions

TitleFast Computation of Sums of Gaussians in High Dimensions
Publication TypeJournal Articles
Year of Publication2005
AuthorsRaykar V C, Yang C, Duraiswami R, Gumerov NA
JournalTechnical Reports from UMIACS UMIACS-TR-2005-69
Date Published2005/11/15/T19:2
KeywordsTechnical Report

Evaluating sums of multivariate Gaussian kernels is a key computational task in many problems in computational statistics andmachine learning. The computational cost of the direct evaluation of such sums scales as the product of the number of kernel
functions and the evaluation points. The fast Gauss transform proposed by Greengard and Strain (1991) is a $\epsilon$-exact
approximation algorithm that reduces the computational complexity of the evaluation of the sum of $N$ Gaussians at $M$ points
in $d$ dimensions from $\mathcal{O}(MN)$ to $\mathcal{O}(M+N)$. However, the constant factor in $\mathcal{O}(M+N)$ grows
exponentially with increasing dimensionality $d$, which makes the algorithm impractical for dimensions greater than three. In
this paper we present a new algorithm where the constant factor is reduced to asymptotically polynomial order. The reduction
is based on a new multivariate Taylor's series expansion (which can act both as a local as well as a far field expansion)
scheme combined with the efficient space subdivision using the $k$-center algorithm. The proposed method differs from the
original fast Gauss transform in terms of a different factorization, efficient space subdivision, and the use of point-wise
error bounds. Algorithm details, error bounds, procedure to choose the parameters and numerical experiments are presented.
As an example we shows how the proposed method can be used for very fast $\epsilon$-exact multivariate kernel density