Stability of sorting based embeddings
<|reference_start|>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 $\alpha : \mathbb{R}^{M \times N} \to \mathbb{R}^D$ 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.<|reference_end|>
@article{balan2024stability,
    title={Stability of sorting based embeddings},
    author={Radu Balan, Efstratios Tsoukanis and Matthias Wellershoff},
    journal={arXiv preprint arXiv:2410.05446},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.05446},
    primaryClass={math.FA cs.LG}
}