Bases of Steerable Kernels for Equivariant CNNs: From 2D Rotations to the Lorentz Group

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.

We introduce Conditional Clifford-Steerable CNNs (C-CSCNNs), a unified framework that incorporates equivariance to arbitrary pseudo-Euclidean groups and significantly improves the expressivity of standard CSCNNs. We show that the kernel basis of the standard formulation is incomplete, limiting model capacity. To address this, we augment the kernels with equivariant representations of the input feature field. We derive the equivariance constraint for these input-dependent kernels and show how it can be solved efficiently via implicit parameterization. We empirically validate on multiple PDE forecasting tasks, including fluid dynamics and relativistic electrodynamics, where our method consistently outperforms standard CSCNNs and performs on par with state-of-the-art baselines.

Steerable convolutional neural networks (Steerable-CNNs) guarantee SE(3)-equivariance by parameterizing kernels as linear combinations of steerable basis functions, but their deterministic nature precludes uncertainty quantification - limiting their use in settings where confidence estimates are essential. We propose a Bayesian Steerable-CNN that places posterior distributions over the basis coefficients, yielding stochastic kernels while preserving equivariance exactly. The loss function of the model is obtained via variational inference and minimized by Bayes-by-Backpropagation. The framework admits a decomposition of predictive uncertainty into epistemic and aleatoric components. Empirically, the model attains competitive classification accuracy alongside an expected calibration error of 0.0263 and outperforms its deterministic counterpart by up to 6.17% under distributional shift induced by additive Gaussian noise. Furthermore, we leverage the model's uncertainty estimates to enhance its performance significantly, achieving a notable gain - approximately 4% higher accuracy across 84% of the test dataset. A statistically significant negative correlation between epistemic uncertainty and prediction error confirms that the learned posterior variance is semantically meaningful. The framework unifies Bayesian uncertainty quantification with the inductive bias of equivariant CNNs.