Approximation theorems in bilipschitz invariant theory
This work addresses a theoretical problem in bilipschitz invariant theory, providing near-optimal embeddings for specific cases, but it is incremental as it builds on known results without a unified treatment.
The paper tackles the problem of finding low-distortion embeddings of orbit spaces into Euclidean space for three specific cases (planar rotations, real phase retrieval, and finite reflection groups), proving that the smallest possible distortion is nearly achieved by a composition of a 'max filter bank' with a linear transformation.
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.