No Equivariant Architecture Covers All Equivariant Attention

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 $\Omega(C^{64})$ components for eight attention heads.

Machine Learning (cs.LG); Algebraic Geometry (math.AG); Representation Theory (math.RT)

arXiv:2608.30417 [cs.LG]

(or arXiv:2608.30417v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.30417 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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