Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions

TL;DR

This paper introduces Equivariance by Contrast (EbC), a method for learning equivariant embeddings from unlabeled data with finite group actions, demonstrating high-fidelity equivariance and theoretical identifiability across various groups.

Contribution

EbC is the first encoder-only approach to learn equivariant embeddings from group action pairs without group-specific biases, handling non-abelian and product groups.

Findings

Embeddings faithfully reproduce group operations in latent space.

Validated on synthetic data with structured transformations.

Theoretically proven to be identifiable.

Abstract

We propose Equivariance by Contrast (EbC) to learn equivariant embeddings from observation pairs $(\mathbf{y}, g \cdot \mathbf{y})$, where $g$ is drawn from a finite group acting on the data. Our method jointly learns a latent space and a group representation in which group actions correspond to invertible linear maps -- without relying on group-specific inductive biases. We validate our approach on the infinite dSprites dataset with structured transformations defined by the finite group $G:= (R_m \times \mathbb{Z}_n \times \mathbb{Z}_n)$, combining discrete rotations and periodic translations. The resulting embeddings exhibit high-fidelity equivariance, with group operations faithfully reproduced in latent space. On synthetic data, we further validate the approach on the non-abelian orthogonal group $O(n)$ and the general linear group $GL(n)$. We also provide a theoretical proof for identifiability. While broad evaluation across diverse group types on real-world data remains future work, our results constitute the first successful demonstration of general-purpose encoder-only equivariant learning from group action observations alone, including non-trivial non-abelian groups and a product group motivated by modeling affine equivariances in computer vision.