On Making $\mathrm{SE}(2)$-Invariant Networks Optimal

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.

The inclusion of symmetries as an inductive bias, known as equivariance, often improves generalization on geometric data (e.g. grids, sets, and graphs). However, equivariant architectures are usually highly constrained, designed for symmetries chosen a priori, and not applicable to datasets with other symmetries. This precludes the development of flexible, multi-modal foundation models capable of processing diverse data equivariantly. In this work, we build a single model -- the Any-Subgroup Equivariant Network (ASEN) -- that can be simultaneously equivariant to several groups, simply by modulating a certain auxiliary input feature. In particular, we start with a fully permutation-equivariant base model, and then obtain subgroup equivariance by using a symmetry-breaking input whose automorphism group is that subgroup. However, finding an input with the desired automorphism group is computationally hard. We overcome this by relaxing from exact to approximate symmetry breaking, leveraging the notion of 2-closure to derive fast algorithms. Theoretically, we show that our subgroup-equivariant networks can simulate equivariant MLPs, and their universality can be guaranteed if the base model is universal. Empirically, we validate our method on symmetry selection for graph and image tasks, as well as multitask and transfer learning for sequence tasks, showing that a single network equivariant to multiple permutation subgroups outperforms both separate equivariant models and a single non-equivariant model.

Convolutional Neural Networks (CNNs) are widely assumed to be translation-invariant, yet standard architectures exhibit a startling fragility: even a single-pixel shift can drastically degrade performance due to their reliance on spatially dependent fully connected layers. In this work, we resolve this vulnerability by proposing a lightweight 'Online Architecture' strategy. By strategically inserting Global Average Pooling (GAP) layers at various network depths, we effectively decouple feature recognition from spatial location. Using VGG-16 as a primary case study, we demonstrate that this architectural modification achieves a massive 98% reduction in trainable parameters (from 5.2M to just 82K) and a 90% reduction in total network size (138M to 14M). Despite this drastic pruning, our variants maintain competitive Top-1 accuracy on ImageNet (66.4%) while doubling translational robustness, reducing average relative loss from 0.09 to 0.05. Furthermore, our analysis identifies a fundamental limit to invariance: while GAP resolves macroscopic sensitivity, discrete pooling operations introduce a residual periodic aliasing that prevents perfect pixel-level stability. Finally, we extend these findings to Perceptual Image Quality Assessment (IQA) by integrating our invariant backbones into the LPIPS framework. The resulting metric significantly outperforms the retrained baseline in generalization across the KADID-10k dataset (Spearman 0.89 vs. 0.75) and achieves a near-perfect alignment with human psychophysical response curves on the RAID dataset (Spearman 0.95). These results confirm that enforcing architectural invariance is a far more efficient and biologically plausible path to robustness than traditional data augmentation. Data and code are publicly available. The data and code are publicly available to facilitate validation and further research.