Spectral decomposition of discrepancy kernels on the Euclidean ball, the special orthogonal group, and the Grassmannian manifold
(32 pages)
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 Rd.
For restrictions to the
Euclidean ball in odd dimensions, to the rotation group SO(3), and
to the Grassmannian manifold G2,4, we compute the kernels'
Fourier coefficients and determine their asymptotics. The
L2-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 G2,4, the transform is
derived here. We also provide numerical experiments for SO(3) and
G2,4.
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