Learning Symmetries in Datasets

We investigate how symmetries present in datasets affect the structure of the latent space learned by Variational Autoencoders (VAEs). Understanding symmetries in data is essential because symmetries determine the true degrees of freedom, constrain generalization, and provide physically interpretable coordinates. We therefore study whether a standard, non-equivariant VAE can reveal symmetry-induced dimensional reduction directly from data, without imposing the symmetry in the architecture. By training VAEs on data originating from simple mechanical systems and particle collisions, we analyze the organization of the latent space through a relevance measure that identifies the most meaningful latent directions. We show that when symmetries or approximate symmetries are present, the VAE self-organizes its latent space, effectively compressing the data along a reduced number of latent variables. This behavior captures the intrinsic dimensionality determined by the symmetry constraints and reveals hidden relations among the features. Furthermore, we provide a theoretical analysis of a simple toy model, demonstrating how, under idealized conditions, the latent space aligns with the symmetry directions of the data manifold. We illustrate these findings with examples ranging from two-dimensional datasets with O(2) symmetry to realistic datasets from electron–positron and proton–proton collisions. Our results highlight the potential of unsupervised generative models to expose underlying structures in data and offer a novel approach to symmetry discovery without explicit supervision.