Constructing Bilipschitz Invariant Embeddings of Quotient Metric Spaces

Core Concepts

The core message of this article is to identify necessary and sufficient conditions for embedding the quotient of a Hilbert space by a subgroup of its automorphisms into a Hilbert space via a bilipschitz map, and to construct such embeddings that minimally distort the quotient distance.

Abstract

The article focuses on embedding the quotient of a Hilbert space V by a subgroup G of its automorphisms (isometric linear bijections) into a Hilbert space via a bilipschitz map. This is motivated by the analysis of data that resides in such an orbit space, where the data is naturally identified with the other members of its orbit under the group G.
The key highlights and insights are:

The article defines the metric quotient V/
/G, whose points are the topological closures of G-orbits in V. This matches the intuition that salient features are continuous functions of the data.

The article shows how a bilipschitz invariant on the sphere can be extended to a bilipschitz invariant on the entire space, and proves that every bilipschitz invariant map is not differentiable.

The article constructs bilipschitz invariants for finite G ≤ O(d) from bilipschitz polynomial invariants on the sphere, which exist precisely when G acts freely on the sphere (hence rarely).

The article uses a semidefinite program to show that some of the constructed extensions of polynomial invariants deliver the minimum possible distortion of V/
/G into a Hilbert space.

The article estimates the Euclidean distortion of infinite-dimensional Hilbert spaces modulo permutation and translation groups.

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Key Insights Distilled From

Towards a bilipschitz invariant theory

by Jameson Cahi... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2305.17241.pdf

Deeper Inquiries

How can the results be extended to more general group actions beyond finite subgroups of the orthogonal group

The results can be extended to more general group actions beyond finite subgroups of the orthogonal group by considering continuous group actions on the Hilbert space. For continuous group actions, the orbit space can still be embedded into a Hilbert space by a bilipschitz map, and constraints on such embeddings can be identified. By studying the metric quotient and homogeneous extensions in the context of continuous group actions, one can generalize the results to a broader class of group actions. This extension allows for the construction of bilipschitz invariant embeddings with minimal distortion for a wider range of group actions on Hilbert spaces.

What are the implications of the non-differentiability of bilipschitz invariants for practical applications in data analysis and machine learning

The non-differentiability of bilipschitz invariants has significant implications for practical applications in data analysis and machine learning. In many machine learning algorithms, differentiability is a crucial property for optimization and model training. The non-differentiability of bilipschitz invariants can pose challenges in incorporating these invariants into existing machine learning frameworks. It may limit the applicability of bilipschitz invariants in certain optimization algorithms and gradient-based methods commonly used in machine learning. Researchers and practitioners need to carefully consider the non-differentiability aspect when utilizing bilipschitz invariants in practical applications to ensure compatibility with existing algorithms and frameworks.

Are there other approaches beyond homogeneous extension that can be used to construct bilipschitz invariant embeddings with minimal distortion

Beyond homogeneous extension, other approaches can be used to construct bilipschitz invariant embeddings with minimal distortion. One such approach is to consider lifting the output space into an extra dimension, similar to the technique used in Lemma 14. By lifting the output space, one can modify the map to achieve bilipschitz bounds that preserve the distortion while ensuring the map remains well-defined and continuous. This technique allows for the construction of bilipschitz invariant embeddings that maintain optimal distortion properties while addressing non-differentiability issues. Additionally, exploring different types of transformations and mappings that preserve the bilipschitz property can provide alternative methods for constructing invariant embeddings with minimal distortion.