Any-Subgroup Equivariant Networks via Symmetry Breaking

2026-03-19

AI Summary

Existing equivariant networks typically support only fixed symmetry groups, limiting their ability to flexibly handle multimodal data with diverse symmetries. This work proposes the ASEN model, which achieves simultaneous equivariance to multiple permutation subgroups within a single architecture by incorporating symmetry-breaking auxiliary input features and leveraging an approximate symmetry-breaking mechanism together with an efficient 2-closure fast algorithm. Built upon a fully permutation-equivariant basis model and employing equivariant MLP emulation techniques, ASEN overcomes the rigidity of conventional equivariant networks. Experiments demonstrate that ASEN outperforms both specialized equivariant models and non-equivariant baselines across tasks involving graph and image symmetry selection, as well as sequence-based multitask and transfer learning scenarios.

Technology Category

• Machine Learning: Multimodal Learning
• Computer Vision: Multi-modal Vision
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

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.

Problem

• equivariance
• symmetry
• subgroup
• permutation
• inductive bias

Innovation

• equivariance
• symmetry breaking
• subgroup
• 2-closure
• universal approximation