Any-Subgroup Equivariant Networks via Symmetry Breaking
This addresses the limitation of rigid equivariant architectures for developing multi-modal foundation models, though it is incremental in extending equivariance to multiple subgroups.
The paper tackles the problem of designing flexible equivariant networks that can handle multiple symmetry groups simultaneously, by introducing Any-Subgroup Equivariant Networks (ASEN) which use symmetry-breaking inputs to achieve subgroup equivariance from a permutation-equivariant base model. The result shows that a single network equivariant to multiple permutation subgroups outperforms separate equivariant models and non-equivariant models in tasks like graph and image symmetry selection, as well as multitask and transfer learning for sequences.
The inclusion of symmetries as an inductive bias, known as equivariance, often improves generalization on geometric data (e.g. grids, sets, and graphs). However, equivariant architectures are usually highly constrained, designed for symmetries chosen a priori, and not applicable to datasets with other symmetries. This precludes the development of flexible, multi-modal foundation models capable of processing diverse data equivariantly. In this work, we build a single model -- the Any-Subgroup Equivariant Network (ASEN) -- that can be simultaneously equivariant to several groups, simply by modulating a certain auxiliary input feature. In particular, we start with a fully permutation-equivariant base model, and then obtain subgroup equivariance by using a symmetry-breaking input whose automorphism group is that subgroup. However, finding an input with the desired automorphism group is computationally hard. We overcome this by relaxing from exact to approximate symmetry breaking, leveraging the notion of 2-closure to derive fast algorithms. Theoretically, we show that our subgroup-equivariant networks can simulate equivariant MLPs, and their universality can be guaranteed if the base model is universal. Empirically, we validate our method on symmetry selection for graph and image tasks, as well as multitask and transfer learning for sequence tasks, showing that a single network equivariant to multiple permutation subgroups outperforms both separate equivariant models and a single non-equivariant model.