A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning

Symmetry is present throughout nature and continues to play an increasingly central role in machine learning. In this paper, we provide a unifying theoretical and methodological framework for incorporating Lie group symmetry into machine learning models in three ways: 1. enforcing known symmetry when training a model; 2. discovering unknown symmetries of a given model or data set; and 3. promoting symmetry during training by learning a model that breaks symmetries within a user-specified candidate group only when the data provide sufficient evidence. We show that these tasks can be cast within a common mathematical framework whose central object is the Lie derivative. We extend and unify several existing results by showing that enforcing and discovering symmetry are linear-algebraic tasks that are dual under the bilinear pairing induced by the Lie derivative. We also propose a novel way to promote symmetry by introducing a class of convex regularizers, built from the Lie derivative with a nuclear-norm relaxation, that penalizes symmetry breaking during training. We explain how these ideas can be applied to a wide range of machine learning models including basis-function regression, dynamical-systems discovery, neural networks, and neural operators acting on fields.