Estimating the Euclidean distortion of an orbit space

TL;DR

This paper investigates how to embed orbit spaces formed by finite-dimensional inner product spaces and isometry groups into Hilbert spaces while preserving the metric, with applications in invariant machine learning.

Contribution

It introduces new theoretical tools and addresses fundamental instances of embedding orbit spaces into Hilbert spaces with minimal distortion.

Findings

Developed novel theoretical tools for embedding orbit spaces.

Analyzed fundamental cases of the embedding problem.

Provided insights relevant to invariant machine learning applications.

Abstract

Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry is motivated by applications to invariant machine learning. We introduce several new theoretical tools before using them to tackle various fundamental instances of this problem.