On Low Discrepancy Samplings in Product Spaces of Motion Groups

TL;DR

This paper introduces a novel deterministic method for constructing low discrepancy point samplings in product spaces of motion groups like SO(3) and SE(3), significantly improving sampling quality for applications in computational sciences.

Contribution

It presents the first nontrivial constructions of low discrepancy samplings for SO(3)^n, SE(3)^n, and T^n, with an innovative two-step discrepancy construction strategy.

Findings

Achieved almost exponential reduction in sampling size compared to trivial methods.

Constructed explicit low discrepancy points in S^2 with optimal bounds.

Extended discrepancy concepts to local Cartesian product sets.

Abstract

Deterministically generating near-uniform point samplings of the motion groups like SO(3), SE(3) and their n-wise products SO(3)^n, SE(3)^n is fundamental to numerous applications in computational and data sciences. The natural measure of sampling quality is discrepancy. In this work, our main goal is construct low discrepancy deterministic samplings in product spaces of the motion groups. To this end, we develop a novel strategy (using a two-step discrepancy construction) that leads to an almost exponential improvement in size (from the trivial direct product). To the best of our knowledge, this is the first nontrivial construction for SO(3)^n, SE(3)^n and the hypertorus T^n. We also construct new low discrepancy samplings of S^2 and SO(3). The central component in our construction for SO(3) is an explicit construction of N points in S^2 with discrepancy \tilde{\O}(1/\sqrt{N}) with respect to convex sets, matching the bound achieved for the special case of spherical caps in \cite{ABD_12}. We also generalize the discrepancy of Cartesian product sets \cite{Chazelle04thediscrepancy} to the discrepancy of local Cartesian product sets. The tools we develop should be useful in generating low discrepancy samplings of other complicated geometric spaces.