Equivariant Representation Learning in the Presence of Stabilizers

TL;DR

This paper introduces Equivariant Isomorphic Networks (EquIN), a novel method for learning data representations that are equivariant under complex group actions, including stabilizers, improving geometric understanding in data.

Contribution

EquIN extends equivariant learning to non-free group actions, grounded in the orbit-stabilizer theorem, enabling better geometric data representations.

Findings

EquIN effectively models data with rotational symmetries.

Accounting for stabilizers improves representation quality.

Empirical results show enhanced geometric structure extraction.

Abstract

We introduce Equivariant Isomorphic Networks (EquIN) -- a method for learning representations that are equivariant with respect to general group actions over data. Differently from existing equivariant representation learners, EquIN is suitable for group actions that are not free, i.e., that stabilize data via nontrivial symmetries. EquIN is theoretically grounded in the orbit-stabilizer theorem from group theory. This guarantees that an ideal learner infers isomorphic representations while trained on equivariance alone and thus fully extracts the geometric structure of data. We provide an empirical investigation on image datasets with rotational symmetries and show that taking stabilizers into account improves the quality of the representations.