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

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 SO(3) , 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 SO(3) , the nonequispaced fast Fourier transform is publicly available, and, for 𝒢_2,4 , the transform is derived here. We also provide numerical experiments for SO(3) and 𝒢_2,4 .