Theory for groupoid equivariant neural networks: an approach for steerable CNNs on bounded domains
Abstract
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.
Related
- Schauder estimates for the linearized very fast diffusion equations in bounded domains
- Absence of bounded-orbit wandering domains
- On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative
- Global Well-Posedness of Landau Equation in Bounded Domains
- Componentwise rigidity of holomorphic isometric maps between bounded symmetric domains with negative conformal factors
BibTeX
@misc{ibort2026theory,
title = {{Theory for groupoid equivariant neural networks: an approach for steerable CNNs on bounded domains}},
author = {Alberto Ibort and Maria Jimenez-Vazquez and Juan M. Perez-Pardo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25987},
eprint = {2609.25987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25987},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Theory for groupoid equivariant neural networks: an approach for steerable CNNs on bounded domains
AU - Alberto Ibort
AU - Maria Jimenez-Vazquez
AU - Juan M. Perez-Pardo
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25987
UR - https://arxiv.org/abs/2609.25987
AB - 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.
ER -