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Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold.
- Source :
-
Constructive Approximation . Jun2023, Vol. 57 Issue 3, p983-1026. 44p. - Publication Year :
- 2023
-
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 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 . [ABSTRACT FROM AUTHOR]
- Subjects :
- *KERNEL (Mathematics)
*BOREL sets
*PROBABILITY measures
*FAST Fourier transforms
Subjects
Details
- Language :
- English
- ISSN :
- 01764276
- Volume :
- 57
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Constructive Approximation
- Publication Type :
- Academic Journal
- Accession number :
- 164275724
- Full Text :
- https://doi.org/10.1007/s00365-023-09638-0