提出统一的等变神经网络框架，拓展对称性学习边界。
Categorical Equivariant Deep Learning: Category-Equivariant Neural Networks and Universal Approximation Theorems
- 用范畴论统一建模群、偏序集、图等对称结构
- 证明有限深度网络可逼近任意连续等变变换
- 适合研究对称性与结构化数据的学者
我们构建了范畴等变神经网络（CENN）的理论体系，统一了群/群胚等变网络、偏序集/格等变网络、图与层叠神经网络。等变性在具有Radon测度的拓扑范畴中以自然性形式定义。在范畴框架下构建线性与非线性层，证明了广义情形下的等变通用逼近定理：有限深度的CENN类在连续等变变换空间中是稠密的。我们系统地将该框架应用于群/群胚、偏序集/格、图和细胞层叠，推导出相应的通用逼近定理。范畴等变深度学习因此突破了传统群作用的局限，不仅涵盖几何对称性，也包含上下文与组合对称性。
原文摘要 · Abstract (English)
We develop a theory of category-equivariant neural networks (CENNs) that unifies group/groupoid-equivariant networks, poset/lattice-equivariant networks, graph and sheaf neural networks. Equivariance is formulated as naturality in a topological category with Radon measures. Formulating linear and nonlinear layers in the categorical setup, we prove the equivariant universal approximation theorem in the general setting: the class of finite-depth CENNs is dense in the space of continuous equivariant transformations. We instantiate the framework for groups/groupoids, posets/lattices, graphs and cellular sheaves, deriving universal approximation theorems for them in a systematic manner. Categorical equivariant deep learning thus allows us to expand the horizons of equivariant deep learning beyond group actions, encompassing not only geometric symmetries but also contextual and compositional symmetries.
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