Stability of sorting based embeddings

TL;DR

This paper investigates the stability of sorting-based embeddings for group actions on inner product spaces, establishing conditions for bi-Lipschitz properties and universal factorization of invariant maps.

Contribution

It provides a characterization of when sorting-based embeddings are bi-Lipschitz and shows their universality in factoring invariant Lipschitz and continuous maps.

Findings

Sorting-based embeddings are bi-Lipschitz if and only if they separate orbits.

Invariant Lipschitz maps into Hilbert spaces factor through these embeddings.

Invariant continuous maps into locally convex spaces also factor through these embeddings.

Abstract

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.