Points on $\operatorname{SO}(3)$ with low logarithmic energy

TL;DR

This paper introduces a novel randomized construction of rotation matrices on SO(3) that achieves improved upper bounds on minimal logarithmic energy by combining spherical point distributions and evenly spaced rotations.

Contribution

It presents a new randomized method for constructing rotation matrices with low logarithmic energy, improving bounds on minimal energy configurations on SO(3).

Findings

New upper bound on minimal logarithmic energy for rotation matrices.

Construction uses zeros of random polynomials and evenly spaced rotations.

Provides insights into energy minimization on the rotation group.

Abstract

We describe several randomized collections of $3\times 3$ rotation matrices and analyze their associated logarithmic energy. The best one (i.e. the one attaining the lowest expected logarithmic energy) is constructed by choosing $r$ spherical points, which come from the zeros of a randomly chosen degree $r$ polynomial, and considering at each of these points a set of $s$ evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of $n=rs$ rotation matrices.