G-Invariant Representations using Coorbits: Bi-Lipschitz Properties

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.