Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold.

Abstract

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of $R^d$. For restrictions to the Euclidean ball in odd dimensions, to the rotation group $\text{SO}\left(3\right)$, and to the Grassmannian manifold $G_{2,4}$, we compute the kernels' Fourier coefficients and determine their asymptotics. The $L_2$-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For $\text{SO}\left(3\right)$, the nonequispaced fast Fourier transform is publicly available, and, for $G_{2,4}$, the transform is derived here. We also provide numerical experiments for $\text{SO}\left(3\right)$ and $G_{2,4}$.