圏同変深層学習：圏同変ニューラルネットワークと普遍近似定理
Categorical Equivariant Deep Learning: Category-Equivariant Neural Networks and Universal Approximation Theorems
圏論を用いて群・グラフ・層など様々な対称性を統一的に扱う同変ニューラルネットワークの理論を構築し、普遍近似定理を証明した。
分類: cs.LG, cs.AI, cs.CV, cs.RO
原文アブストラクト
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.