Groupoid neural networks improve cnn accuracy on bounded domains
Theory for groupoid equivariant neural networks: an approach for steerable CNNs on bounded domains
Machine Learning
Summary
Convolutional neural networks often assume symmetries apply everywhere on their input, but this doesn't work well when the input has edges or boundaries. The authors develop a new theory using groupoids, which handle local symmetries that can vary across different parts of the input. This approach better accounts for boundaries and partial symmetries, especially in bounded planar areas. Their method significantly improves prediction accuracy compared to standard models that assume global symmetries.
What this means in practice
- •For computer vision engineers: Design CNNs that maintain symmetry properties on images with edges and irregular shapes for better accuracy.
- •For robotics software developers: Implement perception systems in robots that correctly handle partial symmetries in sensor data limited by physical boundaries.
Authors
Alberto Ibort, Maria Jimenez-Vazquez, Juan M. Perez-Pardo
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.