Piecewise Linear Equivariant Maps for Compact Groups

Abstract:Motivated by equivariant neural networks, we study piecewise linear equivariant maps between finite-dimensional real representations of compact groups. We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially, while equivariance forces linearity on the corresponding orthogonal complements. As a consequence, we obtain a compact-group analogue of the finite-group existence criterion of Gibson--Tubbenhauer--Williamson for non-zero equivariant piecewise linear maps between irreducible representations, with the identity component giving rise to a rigidity phenomenon absent from the finite-group case.

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

arXiv:2608.21645 [math.RT]

(or arXiv:2608.21645v1 [math.RT] for this version)

https://doi.org/10.48550/arXiv.2608.21645 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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