Spectral decomposition of discrepancy kernels on the Euclidean ball, the special orthogonal group, and the Grassmannian manifold

TL;DR

This paper analyzes the spectral decomposition of discrepancy kernels on specific geometric spaces to enable efficient numerical approximation of probability measures, with explicit computations and numerical experiments on the Euclidean ball, SO(3), and Grassmannian manifolds.

Contribution

It computes Fourier coefficients and asymptotics of discrepancy kernels on these spaces, deriving transforms for efficient numerical minimization and providing new numerical methods and experiments.

Findings

Fourier coefficients and asymptotics for discrepancy kernels are computed.

Efficient numerical minimization of discrepancy is enabled via Fourier domain analysis.

Numerical experiments demonstrate the practical applicability of the methods.

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 $\mathbb{R}^d$. For restrictions to the Euclidean ball in odd dimensions, to the rotation group $SO(3)$, and to the Grassmannian manifold $\mathcal{G}_{2,4}$, 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 $\mathcal{G}_{2,4}$, the transform is derived here. We also provide numerical experiments for $SO(3)$ and $\mathcal{G}_{2,4}$.