G-Invariant Representations using Coorbits: Injectivity Properties

Consider a finite-dimensional real vector space equipped with a finite group acting unitarily on it. We address the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our approach is based on subsets of sorted coorbits with respect to selected window vectors. We derive conditions under which such embeddings are injective, using techniques from semi-algebraic analysis. These results are then applied to the specific case of planar rotation invariance.