Approximation theorems in bilipschitz invariant theory
Abstract
Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase retrieval, and (c) finite reflection groups. Here, we prove that for all three of these cases, the smallest possible distortion is nearly achieved by a composition of a "max filter bank" with a linear transformation. Our proof amounts to a two-step process: first, we show it suffices to demonstrate a certain inclusion of Lipschitz function spaces, and second, we prove that inclusion, using fundamentally different approaches for the three cases. We also show that these cases interact differently with a few related function spaces, which suggests that a unified treatment would be nontrivial.
Cite
@article{arxiv.2603.23643,
title = {Approximation theorems in bilipschitz invariant theory},
author = {Jameson Cahill and Joseph W. Iverson and Dustin G. Mixon and Nathan Willey},
journal= {arXiv preprint arXiv:2603.23643},
year = {2026}
}