Structuring Representations Using Group Invariants

A finite set of invariants can identify many interesting transformation groups. For example, distances, inner products and angles are preserved by Euclidean, Orthogonal and Conformal transformations, respectively. In an equivariant representation, the group invariants should remain constant on the embedding as we transform the input. This gives a procedure for learning equivariant representations without knowing the possibly nonlinear action of the group in the input space. Rather than enforcing such hard invariance constraints on the latent space, we show how to use invariants for "symmetry regularization" of the latent, while guaranteeing equivariance through other means. We also show the feasibility of learning disentangled representations using this approach and provide favorable qualitative and quantitative results on downstream tasks, including world modeling and reinforcement learning.