Approximation theorems in bilipschitz invariant theory

TL;DR

This paper investigates low-distortion embeddings of orbit spaces in Euclidean space within bilipschitz invariant theory, providing near-optimal solutions for specific cases and highlighting the complexity of unifying these approaches.

Contribution

It demonstrates that for planar rotations, real phase retrieval, and finite reflection groups, near-optimal embeddings are achieved via max filter banks combined with linear transformations, using novel proof techniques.

Findings

Near-optimal embeddings for three key cases

Different approaches needed for each case

Insights into the complexity of unified treatment

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.