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Approximation theorems in bilipschitz invariant theory

Jameson Cahill, Joseph W. Iverson, Dustin G. Mixon, Nathan Willey

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.

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.
Paper Structure (15 sections, 14 theorems, 113 equations, 4 figures, 7 tables)

This paper contains 15 sections, 14 theorems, 113 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Bilipschitz invariant theory
  3. Max filtering and main result
  4. Lipschitz function classes and proof of main result
  5. Roadmap
  6. Continuity of distortion and proof of Lemma \ref{['lem.lipschitz closures suffice']}
  7. Integral combinations of max filters
  8. Primary technology
  9. Application 1: Planar rotations
  10. Application 2: Real phase retrieval
  11. Containment of function spaces and proof of Theorem \ref{['thm.main result ii']}
  12. Proof of Theorem \ref{['thm.main result ii']}(a): Planar rotations
  13. Proof of Theorem \ref{['thm.main result ii']}(b): Real phase retrieval
  14. Proof of Theorem \ref{['thm.main result ii']}(c): Reflection groups
  15. Numerical results

Key Result

Theorem 3

Fix a group $G\leq\operatorname{O}(d)$ and a target dimension $n$ such that For each $\epsilon > 0$, there exists $m=m(\epsilon)$, along with a max filter bank $\Phi\colon \mathbb{R}^d/G\to\mathbb{R}^m$ and a linear map $L\colon\mathbb{R}^m\to\mathbb{R}^n$ such that the composition $L\circ\Phi$ has distortion at most $c_2(\mathbb{R}^d/G) + \epsilon$. (In fact, in the case o

Figures (4)

  • Figure 1: Illustration of Example \ref{['ex.ksl']} in the special case where $d=3$. Here, we display the value of $f(x)$ when $x$ is in the first octant of the unit sphere. For $x$ residing in most other octants, we have $f(x)=0$, though the function values in the octant opposite the first are determined by the fact that $f$ is even. The set of $x\in\mathbb{R}^3$ at which $f(x)$ is not differentiable consists of all scalar multiples of the points along the solid black curves.
  • Figure 2: The embedding of $\mathbb{R}^2/\{\pm I\}$ into increasing embedding dimensions, visualized in $\mathbb{R}^3$ with the help of principal component analysis. The MF embeddings are above, and the LMF embeddings are below. Both embeddings use the optimal max filter bank, namely, roots of unity.
  • Figure 3: Connected U.S. congressional districts embedded using LMF, and then visualized using principal component analysis.
  • Figure 4: 2D Shape Structure Dataset embedded using LMF, and then visualized using principal component analysis.

Theorems & Definitions (31)

  • Example 1
  • Example 2
  • Theorem 3: Main result
  • Lemma 4
  • Theorem 5
  • proof : Proof of Theorem \ref{['thm.main result']}
  • Lemma 8: Weyl's inequality for Lipschitz maps
  • proof
  • Lemma 9
  • proof
  • ...and 21 more