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

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.