We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.
Equivariant convolutional neural networks are usually built from a group acting globally on the space of signals. This hypothesis is inappropriate for many bounded or stratified domains: an ambient rigid motion may be admissible only on part of the domain, and the boundary introduces geometric types that are invisible to a transitive group action. We develop a theory of groupoid-equivariant neural networks in which the symmetry datum consists of a groupoid, a selected pseudogroup of local bisections, a measure, and input and output representation bundles. For integral channels on the object space, we prove a bisection-equivariant kernel theorem: equivariance is equivalent to a transport constraint on the two-point kernel, and its solutions are classified by one joint-stabilizer intertwiner on each orbit of pairs. As a case study we apply the theory to bounded planar domains. The resulting architecture is implemented through offline nullspace bases and sparse gather--transform--scatter operations. A Poisson--Dirichlet kernel study is used separately to assess boundary-aware inductive bias; the exact inverse is shown to preserve the global symmetries of the rectangle but not general proper local bisections. The numerical results show that the proposed architectures provide significant advantages when symmetries cannot be globally implemented by group actions and provide an accuracy improvement of at least one order of magnitude with respect to the models tested.
Alberto Ibort, Maria Jimenez-Vazquez, Juan M. Perez-Pardo
Symmetry is everywhere in nature and society. Geometric deep learning exploits symmetries in data to improve the performance and efficiency of deep learning systems. In this paper, we extend geometric deep learning to utilize richer symmetry structures. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks (for which no UAT was known before). We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In addition, we show that OENN can be extended further to CENN, Category-Equivariant Neural Network, which gives the general form of equivariant neural networks as well as of equivariant universal approximation theorems, allowing us to leverage categorical symmetry in data (e.g., non-invertible symmetries on multiple objects with compositional relations on those symmetries).
Nonlinear activations can create equivariant interactions between irreducible representations that linear maps cannot. We use the Gaussian degree decomposition to extend ordinary polynomial degree to such nonlinear maps, and prove that for a fixed coordinatewise equivariant layer each degree factors into a polynomial determined by the linear maps and a scalar determined by the activation. This separates three distinct obstructions, coming from symmetry, coordinates, and activation.