Tīkun Ông | Searcharxiv

arXiv2026

Inherent Trade-offs when Enforcing Correct Data Symmetries in Multi-head Self-attention

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. We illustrate this with experiments on artificial tasks and show that the predicted head-cluster structures can arise naturally when the attention layer is pressured to become equivariant.

cs.LG↗

arXiv2026

A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of $\mathrm{O}(2)$

Vision transformers have become a dominant architecture for visual recognition. However, standard models do not explicitly encode the planar symmetries that arise in many vision domains. We introduce a family of vision transformers equivariant to arbitrary discrete subgroups of $\mathrm{O}(2)$, providing a unified framework that generalizes prior flipping- and $D_4$-equivariant transformer architectures. Our construction yields equivariant analogues of the core transformer components, together with expressivity guarantees for the resulting layers. In particular, we show that whenever $H \le G$, the class of $G$-equivariant ViTs embeds naturally into the class of $H$-equivariant ViTs. We also prove that, in the single-head setting, the corresponding equivariant self-attention layer realizes every $G$-equivariant self-attention map representable by ordinary self-attention. We further construct a $D_6$-equivariant model based on hexagonal patches, making the architecture compatible with six-fold rotational symmetries. We evaluate the resulting models on the PatternNet aerial image dataset in artificially data-scarce regimes across subgroups of $D_4$ and $D_6$. Our experiments compare two equivariant attention mechanisms and analyze how the choice of homogeneous-space configurations used in the nonlinearities affects performance. Preliminary results under matched parameter budgets indicate that equivariance can improve recognition accuracy, motivating further study of how discrete symmetry groups shape transformer-based visual recognition models.

cs.CV↗