G-invariant representations using coorbits: bi-Lipschitz properties

Let $${\mathcal {V}}$$ V be a finite-dimensional real vector space with a finite group G acting orthogonally on it. We address the general problem of constructing Euclidean-stable embeddings of the orbit space $${\mathcal {V}}/G$$ V / G . Our approach builds on embeddings derived from subsets of sorted coorbits. The main result establishes that whenever such embeddings are injective, they are automatically bi-Lipschitz. Furthermore, we show that stable embeddings can be achieved in reduced dimensions, and that any continuous or Lipschitz G -invariant map can be factored through these embeddings.