Orbit Computation for Atomically Generated Subgroups of Isometries of $\mathbb{Z}^n$

Abstract

Isometries are ubiquitous in nature; isometries of discrete (quantized) objects---abstracted as the group of isometries of $\mathbb{Z}^n$ denoted by $\mathsf{ISO}(\mathbb{Z}^n)$---are important concepts in the computational world. In this paper, we compute various isometric invariances which mathematically are orbit-computation problems under various isometry-subgroup actions $H \curvearrowright \mathbb{Z}^n, H \leq \mathsf{ISO}(\mathbb{Z}^n)$. One computational challenge here is about the \emph{infinite}: in general, we can have an infinite subgroup acting on $\mathbb{Z}^n$, resulting in possibly an infinite number of orbits of possibly infinite size. In practice, we restrict the set of orbits (a partition of $\mathbb{Z}^n$) to a finite subset $Z \subseteq \mathbb{Z}^n$ (a partition of $Z$), where $Z$ is specified a priori by an application domain or a data set. Our main contribution is an efficient algorithm to solve this \emph{restricted} orbit-computation problem in the special case of \emph{atomically generated subgroups}---a new notion partially motivated from interpretable AI. The atomic property is key to preserving the \emph{semidirect-product structure}---the core structure we leverage to make our algorithm outperform generic approaches. Besides algorithmic merit, our approach enables \emph{parallel-computing} implementations in many subroutines, which can further benefit from hardware boosts. Moreover, our algorithm works efficiently for \emph{any} finite subset ($Z$) regardless of the shape (continuous/discrete, (non)convex) or location; so it is application-independent.