Stability of sorting based embeddings

Consider a group G of order M acting unitarily on a real inner product space V. We show that the sorting based embedding obtained by applying a general linear map α : ℝM × N → ℝD to the invariant map β_Phi : V → ℝM × N given by sorting the coorbits (langle v, g φi rangleV)g ∈ G, where (φi)i=1N ∈ V, satisfies a bi-Lipschitz condition if and only if it separates orbits. Additionally, we note that any invariant Lipschitz continuous map (into a Hilbert space) factors through the sorting based embedding, and that any invariant continuous map (into a locally convex space) factors through the sorting based embedding as well.