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

Tomas Karella; Emily Shinkle; Alice Elisabeth Anastasia Allen; Pieter J. Swart; Nicholas Lubbers; Roxana Bujack

cs.LGOct 12, 2026

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

Authors: Tomas Karella, Emily Shinkle, Alice Elisabeth Anastasia Allen, Pieter J. Swart, Nicholas Lubbers, Roxana Bujack

Organizations: CAI-3: Information Sciences Los Alamos National Laboratory · Machine Learning in Molecular Dynamics Max Planck Institute for Polymer Research · T-5: Applied Mathematics and Plasma Physics Los Alamos National Laboratory

Abstract

Respecting a-priori known symmetries of underlying data is a key principle in designing efficient neural architectures. Convolutional neural networks are a canonical example, encoding translation equivariance. Extending this inductive bias to richer symmetry groups such as $\mathrm{SE}(2)$ in the most effective way possible is an ongoing architectural and computational challenge. Building on classical invariant theory, we derive invariant feature maps that are provably optimal: minimal and information-preserving with respect to the symmetry group. Our construction is designed to integrate seamlessly into standard architectures, requiring no modification to nonlinearities or normalization layers. As a concrete implementation, we introduce a plug-and-play module for convolutional neural networks that enforces $\mathrm{SE}(2)$ invariance, capturing both translations and rotations within standard architectures. The resulting models are both efficient and expressive. Experiments show competitive or improved accuracy compared to state-of-the-art equivariant methods at significantly lower computational cost. Finally, we provide a theoretical and empirical analysis of the relationship between equivariance and invariance, offering insight into their roles in controlling robustness and flexibility.

Figures & tables

Figure 1: Overview of the proposed invariant convolutional architecture. (a) The overall network architecture obtained by stacking the proposed blocks. (b) The invariant convolution block, combining a moment layer and an invariant layer within a residual structure. (c) Illustration of the invariant layer, showing how moment responses are combined to produce rotation-invariant features.

	Dataset			Ours			E2-CNNs
Dataset	Reso.	Cls	Size	Acc	Prm.	$\mathrm{RCI}$	Acc	Prm.	$\mathrm{RCI}$
MNIST RotTest	$28$	10	50k	$\mathbf{99.42}{\scriptstyle\pm.07}$	119k	$\mathbf{0{\scriptstyle.01}}$	$97.20{\scriptstyle\pm.14}$	362k	$0{\scriptstyle.89}$
rot MNIST	$28$	10	12/50k	$\mathbf{99.08}{\scriptstyle\pm.04}$	119k	$\mathbf{0{\scriptstyle.01}}$	$98.82{\scriptstyle\pm.01}$	362k	$0{\scriptstyle.64}$
RESISC45	$256$	45	31.5k	$\mathbf{85.77}{\scriptstyle\pm.47}$	230k	$\mathbf{1{\scriptstyle.66}}$	$84.02{\scriptstyle\pm.27}$	365k	$5{\scriptstyle.13}$
EuroSAT rgb	$64$	10	27k	$\mathbf{95.29}{\scriptstyle\pm.40}$	149k	$\mathbf{0{\scriptstyle.71}}$	$93.90{\scriptstyle\pm.34}$	363k	$1{\scriptstyle.60}$
DTD ∗	$\tiny{\leq}640$	47	5k	$67.14{\scriptstyle\pm.27}$	336k	$\mathbf{0{\scriptstyle.01}}$	$67.18{\scriptstyle\pm.81}$	469k	$1{\scriptstyle.97}$

Table 1: Quantitative comparison between our Mom/Inv CNN and

E(2)

-CNNs. Results are reported as mean

\pm

standard deviation over multiple runs. For DTD, we report top-five accuracy, denoted by ∗ .

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Flexible benchmark and its invariant representation. Left: data generation from rotated three-fold and four-fold structures, with labels determined by their minimal distance. Right: projection into Flusser complex moments, showing the structured embedding induced by invariant features. Inspired by graph neural network benchmarks [ 66 ] .

Dataset	$\mathbf{m_{1,1}}$	$\mathbf{m_{2,0}}$	$\mathbf{m_{2,1}}$	$\mathbf{m_{3}}$	$\mathbf{m_{4}}$
Histology	$8.7\mathrm{e}{-5}$	$2.7\mathrm{e}{-4}$	$2.1\mathrm{e}{-4}$	$7.1\mathrm{e}{-5}$	$5.1\mathrm{e}{-5}$
MNIST	$1.4\mathrm{e}{-6}$	$1.4\mathrm{e}{-6}$	$1.4\mathrm{e}{-6}$	$1.4\mathrm{e}{-6}$	$1.4\mathrm{e}{-6}$
Satellite	$1.8\mathrm{e}{-5}$	$2.2\mathrm{e}{-4}$	$4.9\mathrm{e}{-4}$	$2.3\mathrm{e}{-6}$	$1.4\mathrm{e}{-6}$

Appendix

Table 3: Conditional probability that another moment is nonzero given that the reference moment

m_{1,0}

is zero, measured across channels and spatial locations. Larger values indicate cases where flexible reference selection could avoid degeneracies caused by a vanishing reference moment. The consistently small probabilities indicate that situations where an alternative reference moment could be used are exceedingly rare in these CNN feature representations.

Figure 6: Relative pose relationships are essential for object composition. All panels contain the same local facial components, but only the outer panels preserve a coherent global face configuration. The inner panels disrupt the relative positions, orientations, and scales of the parts, illustrating that independent invariance of local elements can discard information needed to recognize the object as a face. Taken from Romero et al. [ 68 , Fig. 1] , based on Schwarzer [ 69 ] .

Architecture	Accuracy	# Params
E2CNN	$98.82\pm.01$	362 K
Mom/Inv CNN	$\mathbf{99.00\pm.03}$	362 K
Equivariant Branch	$98.82\pm.06$	363 K

Appendix

Table 4: Comparison between invariant and equivariant variants on RotMNIST. Adding an explicit equivariant branch does not improve performance over the invariant architecture.

Variant	Test Accuracy	Val Accuracy
Monomials	$98.82$	$99.25$
Legendre polynomials	$98.80$	98.97
B-splines	$\mathbf{99.07}$	99.74

Appendix

Table 5: Comparison of different radial basis parameterizations. Performance remains consistent across variants and simplified radial constructions.

Ring Count	Test Accuracy	Train Accuracy	Val Accuracy
4	$\mathbf{99.15}$	99.93	99.87
7	$99.15$	99.98	$\mathbf{99.89}$
3	$99.13$	99.89	99.80
6	$99.13$	99.96	99.89
5	$99.09$	99.95	99.83
8	$99.02$	99.77	99.22

Appendix

Table 6: Effect of the number of radial rings. Performance is stable across a range of values, with a slight optimum at 4 rings.

Figure 7: Effect of filter size on validation performance without data augmentation. The left y-axis shows validation accuracy on upright digits, while the right y-axis shows validation accuracy on digits rotated by $45^{\circ}$ . Larger filters preserve very high upright accuracy while substantially improving performance on rotated inputs, illustrating the trade-off between local precision and rotational robustness. Error bars indicate one standard deviation across runs.

Figure 8: Distribution of SEM images across classes in the NFFA-Europe dataset. The dataset exhibits significant class imbalance, with some categories containing substantially fewer samples than others. This makes the benchmark particularly relevant for evaluating symmetry-aware architectures in low-data regimes.

Dataset	Ours	E2-CNNs	ResNet	H-Nets	ViT
MNIST RotTest	$\mathbf{99.42}{\scriptstyle\pm.07}$ (119k)	$97.20{\scriptstyle\pm.14}$ (362k)	$46.28{\scriptstyle\pm.26}$ (373k)	$99.05{\scriptstyle\pm.03}$ (33k)	$42.08{\scriptstyle\pm.64}$ (853k)
rot MNIST	$\mathbf{99.08}{\scriptstyle\pm.04}$ (119k)	$98.82{\scriptstyle\pm.01}$ (362k)	$98.55{\scriptstyle\pm.01}$ (373k)	$98.74{\scriptstyle\pm.03}$ (33k)	$88.72{\scriptstyle\pm.99}$ (853k)
RESISC45	$\mathbf{85.77}{\scriptstyle\pm.47}$ (230k)	$84.02{\scriptstyle\pm.27}$ (365k)	$84.32{\scriptstyle\pm.11}$ (377k)	$53.06{\scriptstyle\pm.02}$ (37k)	$63.18{\scriptstyle\pm.55}$ (928k)
EuroSAT rgb	$\mathbf{95.29}{\scriptstyle\pm.40}$ (149k)	$93.90{\scriptstyle\pm.34}$ (363k)	$79.50{\scriptstyle\pm.31}$ (377k)	$86.28{\scriptstyle\pm 1.13}$ (45k)	$76.74{\scriptstyle\pm.46}$ (924k)
DTD ∗	$67.14{\scriptstyle\pm.27}$ (336k)	$67.18{\scriptstyle\pm.81}$ (469k)	$56.66{\scriptstyle\pm.23}$ (377k)	$38.24{\scriptstyle\pm.87}$ (36k)	$37.95{\scriptstyle\pm.45}$ (928k)
NFFA	$87.02{\scriptstyle\pm.87}$ (227k)	$87.32{\scriptstyle\pm.30}$ (363k)	$85.08{\scriptstyle\pm.79}$ (11M)	–	$74.20{\scriptstyle\pm.63}$ (875k)

Appendix

Table 7: Quantitative comparison of our CNN to others. Entries are given in the format

\mathrm{Acc.}\pm\mathrm{\scriptstyle{Std.Dev.}}\scriptstyle{\mathrm{(Param.Count)}}

Parameter	Optimal	ESCNN
Stem kernel size	15	7
Stage kernels	$[11,11,11]$	$[5,5,5]$
Stage depths / blocks	$[1,2,2]$	$[1,2,2]$
Classifier hidden size	64	64
Dropout	0.2	–
Maximum type	3 (4 for DTD)	3 (4 for DTD)

Appendix

Table 8: Shared architecture hyperparameters.

Dataset	Resolution	Batch	Optimal LR	ESCNN LR	Preprocess
RotMNIST	$64^{2}$	16 / 32	$1.525\mathrm{e}{-3}$	$10^{-3}$	Upscaled to 64
MNIST-rot-test	$64^{2}$	64	$1.5\mathrm{e}{-3}$	$10^{-3}$	–
DTD	$96^{2}$	16	$1.5\mathrm{e}{-3}$	$1.5\mathrm{e}{-3}$	Circular Mask
EuroSAT	$64^{2}$	32	$1.5\mathrm{e}{-3}$	$10^{-3}$	–
Galaxy Zoo	$96^{2}$	64	$1.5\mathrm{e}{-3}$	$1.5\mathrm{e}{-3}$	Circular Mask
RESISC45	$96^{2}$	8	$1.5\mathrm{e}{-3}$	$1.5\mathrm{e}{-3}$	–

Appendix

Table 9: Dataset-specific training hyperparameters.

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Equivariant Neural Networks

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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.

Abstract

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