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

Author:

Dick Josef,Ehler Martin,Gräf Manuel,Krattenthaler Christian

Abstract

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

Funder

Austrian Science Fund

Publisher

Springer Science and Business Media LLC

Subject

Computational Mathematics,General Mathematics,Analysis

Reference49 articles.

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