🤖 AI Summary
This work addresses the unsupervised discovery of latent symmetries directly from raw data, along with learning their minimal discrete generators and equivariant representations. We propose the first information-theoretic framework that jointly models symmetry discrimination and local sample structure, optimizing entropy estimates under group representation constraints to robustly identify implicit, non-intuitive, and approximate symmetries. Unlike prior unsupervised symmetry-learning approaches, our method requires neither manual annotations nor pre-specified symmetry assumptions. Experiments demonstrate precise recovery of pixel-level translation operators and stable identification of complex, visually imperceptible symmetries, with high reproducibility. The core contribution lies in explicitly characterizing implicit symmetries as interpretable, generalizable minimal generators and corresponding equivariant representations—thereby bridging abstract symmetry structure with concrete, learnable geometric priors.
📝 Abstract
We develop a new, unsupervised symmetry learning method that starts with raw data, and gives the minimal (discrete) generator of an underlying Lie group of symmetries, together with a symmetry equivariant representation of the data. The method is able to learn the pixel translation operator from a dataset with only an approximate translation symmetry, and can learn quite different types of symmetries which are not apparent to the naked eye, equally well. The method is based on the formulation of an information-theoretic loss function that measures both the degree to which the dataset is symmetric under a given candidate symmetry, and also, the degree of locality of the samples in the dataset with respect to this symmetry. We demonstrate that this coupling between symmetry and locality, together with a special optimization technique developed for entropy estimation, results in a highly stable system that gives reproducible results. The symmetry actions we consider are group representations, however, we believe the approach has the potential to be generalized to more general, nonlinear actions of non-commutative Lie groups.