Inherent Trade-offs when Enforcing Correct Data Symmetries in Multi-head Self-attention
cs.LG, math.AG, math.RT
Submitted: 2026-08-31
Updated: 2026-10-15
License: http://creativecommons.org/licenses/by/4.0/
The gist: We give a complete characterization of equivariant multi-head self-attention (MHSA): if an MHSA layer is equivariant to a symmetry group G, then G can only act by permuting head-clusters, with QK and
Terminology
Abstract
We give a complete characterization of equivariant multi-head self-attention (MHSA): if an MHSA layer is equivariant to a symmetry group G, then G can only act by permuting head-clusters, with QK and OV matrices satisfying an equivariance constraint tied to the group action. As a consequence, we prove that any fixed MHSA architecture that achieves exact equivariance by polynomially parameterizing unconstrained MHSA parameters inevitably leads to expressivity loss within the class of equivariant maps: the equivariance locus of unconstrained MHSA forms a union of extremely many Zariski-irreducible components in a reduced parameter space, and any single architecture covers at most one. For G=D 4 acting on C copies of the regular representation as the token feature space, we show that there are Ω(C 64) components for eight attention heads.
Sources
- Roto-Translation Covariant Convolutional Networks for Medical Image Analysis
- Flopping for FLOPs: Leveraging equivariance for computational efficiency
- Rank, Head-Channel Non-Identifiability, and Symmetry Breaking: A Precise Analysis of Representational Collapse in Transformers
- A General Theory of Equivariant CNNs on Homogeneous Spaces
- Group Equivariant Convolutional Networks
- An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale
- Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data
- Vanilla Group Equivariant Vision Transformer: Simple and Effective
- SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks
- Geometry of Lightning Self-Attention: Identifiability and Dimension
- LieTransformer: Equivariant self-attention for Lie Groups
- On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
- Swin Transformer: Hierarchical Vision Transformer using Shifted Windows
- Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networks
- Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry
- Quick ViTs: Speeding up Vision Transformers through Equivariance
- On Universality of Deep Equivariant Networks
- Group Equivariant Stand-Alone Self-Attention For Vision
- Identifiable Equivariant Networks are Layerwise Equivariant
- DINOv3
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