Estimating the Euclidean distortion of an orbit space

2025-06-04

AI Summary

This paper addresses the optimal metric embedding of the orbit space $V/G$—induced by a compact isometric group $G$ acting on a finite-dimensional inner product space $V$—into a Hilbert space, with the objective of minimizing distortion between the quotient metric and the Euclidean distance. We develop an original analytical framework unifying Lie group representation theory, metric geometry, and invariant theory. This yields the first systematic characterization of Euclidean embeddability criteria for $V/G$, along with tight lower bounds on distortion. For canonical compact groups—including orthogonal and cyclic groups—we derive exact distortion bounds, surpassing prior heuristic approaches that rely solely on empirically designed invariant feature embeddings. Our results establish the first rigorous geometric foundation for invariance-aware machine learning, enabling provably robust construction of invariant feature embeddings.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Representation Learning for Vision
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Estimating Euclidean distortion of orbit spaces
• Embedding orbit spaces into Hilbert spaces
• Preserving quotient metrics for invariant learning

Innovation

• Embed orbit space into Hilbert space
• Preserve quotient metric accurately
• Use new theoretical tools