研究紧群表示间的分段线性等变映射，发现非线性仅出现在平凡作用子空间。
Piecewise Linear Equivariant Maps for Compact Groups
- 通过分段线性构造等变映射，揭示非线性行为的局限范围
- 证明紧群恒同分支上非零等变映射存在性条件
- 揭示紧群特有的刚性现象，适用于表示论与对称学习
受等变神经网络启发，我们研究紧群有限维实表示之间的分段线性等变映射。结果表明，所有真正非线性的分段线性行为均局限于群恒同分支平凡作用的子空间，而等变性强制在正交补空间上保持线性。由此得到紧群情形下类比于有限群的Gibson-Tubbenhauer-Williamson存在性准则，其中恒同分支引发有限群中不存在的刚性现象。
原文摘要 · Abstract (English)
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.
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