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.

By inferring latent groups in the training data, recent works introduce invariant learning to the case where environment annotations are unavailable. Typically, learning group invariance under a majority/minority split is empirically shown to be effective in improving out-of-distribution generalization on many datasets. However, theoretical guarantee for these methods on learning invariant mechanisms is lacking. In this paper, we reveal the insufficiency of existing group invariant learning methods in preventing classifiers from depending on spurious correlations in the training set. Specifically, we propose two criteria on judging such sufficiency. Theoretically and empirically, we show that existing methods can violate both criteria and thus fail in generalizing to spurious correlation shifts. Motivated by this, we design a new group invariant learning method, which constructs groups with statistical independence tests, and reweights samples by group label proportion to meet the criteria. Experiments on both synthetic and real data demonstrate that the new method significantly outperforms existing group invariant learning methods in generalizing to spurious correlation shifts.

By inferring latent groups in the training data, recent works introduce invariant learning to the case where environment annotations are unavailable. Typically, learning group invariance under a majority/minority split is empirically shown to be effective in improving out-of-distribution generalization on many datasets. However, theoretical guarantee for these methods on learning invariant mechanisms is lacking. In this paper, we reveal the insufficiency of existing group invariant learning methods in preventing classifiers from depending on spurious correlations in the training set. Specifically, we propose two criteria on judging such sufficiency.