G-Invariant Representations using Coorbits: Bi-Lipschitz Properties

TL;DR

This paper investigates the construction of stable, bi-Lipschitz embeddings of orbit quotient spaces under finite group actions, using coorbit-based methods, and shows their optimality and universality.

Contribution

It introduces a method for Euclidean stable embeddings of orbit spaces using coorbits, proving their bi-Lipschitz nature and universality for invariant maps.

Findings

Injective embeddings are automatically bi-Lipschitz.

Stable embeddings can be achieved with lower dimensions.

Any Lipschitz G-invariant map factors through these embeddings.

Abstract

Consider a finite dimensional real vector space and a finite group acting unitarily on it. We study the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding is based on subsets of sorted coorbits. Our main result shows that, whenever such embeddings are injective, they are automatically bi-Lipschitz. Additionally, we demonstrate that stable embeddings can be achieved with reduced dimensionality, and that any continuous or Lipschitz $G$-invariant map can be factorized through these embeddings.