Approximation to uniform distribution in SO(3)
Carlos Beltrán, Damir Ferizović
TL;DR
This work addresses how to assess and achieve near-uniform point distributions on the rotation group $SO(3)$. It uses a determinantal point process (DPP) built from the first eigenspaces of the Laplace–Beltrami operator to generate $N$-point configurations and derives sharp bounds for both Riesz and Green energies, including an explicit Green function $\mathcal{G}(\alpha,\beta)=(\pi-\omega(\alpha^{-1}\beta))\cot(\omega(\alpha^{-1}\beta)/2)-1$ and $N=\binom{2L+3}{3}$; these results enable principled comparisons to Haar-uniform distributions. The authors prove a lower bound $\mathcal{E}_{\mathcal{G}}(N)\ge -3\sqrt[3]{\pi}\,N^{4/3}+O(N)$ and an upper bound $\mathcal{E}_{\mathcal{G}}(N)\le -4(3/4)^{4/3}\,N^{4/3}+O(N)$, and they obtain a variance bound $\mathrm{Var}(\eta_A)=O\left(\frac{\varepsilon^2}{\cos\varepsilon}N^{2/3}\log N\right)$ for Haar-ball counts, ensuring concentration toward uniformity. In addition, the paper provides a practical, simple Halton-based sampling scheme (HArDiSh) for $SO(3)$ as a fast comparator to DPP samples. These results enable principled comparison of uniform point constructions on $SO(3)$ and contribute explicit Gegenbauer-based identities relevant beyond the present application.
Abstract
Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO(3), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO(3). The variance of rotation matrices sampled by the determinantal point process is estimated, and formulas for the L2 -norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest. Also a simple but effective algorithm to sample points in SO(3) is given.