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 $\alpha : \mathbb{ R}^{ M \times N} \to \mathbb{ R}^D$ to the invariant map $\beta_\Phi : V \to \mathbb{ R}^{ M \times N}$ given by sorting the coorbits $(\langle v, g \phi_i \rangle_V)_{ g \in G}$, where $(\phi_i)_{ i=1}^N \in V$, 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.
Categories: math.FA cs.LG