FFT-Based Fast Computation of Multivariate Kernel Density Estimators With Unconstrained Bandwidth Matrices

The problem of fast computation of multivariate kernel density estimation (KDE) is still an open research problem. In our view, the existing solutions do not resolve this matter in a satisfactory way. One of the most elegant and efficient approach uses the fast Fourier transform. Unfortunately, the existing FFT-based solution suffers from a serious limitation, as it can accurately operate only with the constrained (i.e., diagonal) multivariate bandwidth matrices. In this article, we describe the problem and give a satisfactory solution. The proposed solution may be successfully used also in other research problems, for example, for the fast computation of the optimal bandwidth for KDE. Supplementary materials for this article are available online.