Any-Subgroup Equivariant Networks via Symmetry Breaking

Abhinav Goel; Derek Lim; Hannah Lawrence; Stefanie Jegelka; Ningyuan Huang

cs.LGMar 19, 2026

Any-Subgroup Equivariant Networks via Symmetry Breaking

Authors: Abhinav Goel, Derek Lim, Hannah Lawrence, Stefanie Jegelka, Ningyuan Huang

Organizations: MIT · TUM · Flatiron Institute

Abstract

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.

Figures & tables

Figure 1: Example symmetry breaking objects as positional features $A^{(1)}$ and edge features $A^{(2)}$ for encoding subgroup symmetries in $4$ -node paths. These symmetries are explored further in Sec. 5 .

Figure 2: ASEN Architecture to model any permutation subgroup-equivariant functions.

Table 4

Figure 3: Pairwise distances of learned positional encodings of Transformer on Pathfinder-64, with test accuracy shown at the top: imposing local permutation symmetry improves performance.

Task	Input	Equivariant Output	Invariant Output	Group $G\leq S_{n}$
Intersect	$[a,b,c,\mid b,c,d]$	$[0,1,1,\mid 1,1,0]$	$2$	$S_{n/2}\times S_{n/2}\times S_{2}$
Symmetric Difference	$[a,b,c,\mid b,c,d]$	$[1,0,0,\mid 0,0,1]$	$1$	$S_{n/2}\times S_{n/2}\times S_{2}$
Palindrome ( $k=3$ )	$[a,b,c,b,d]$	$[0,1,1,1,0]$	True	Sequence Reversal
Monotone Run ( $k=3$ )	$[3,1,2,4,1]$	$[0,1,1,1,0]$	True	Sequence Reversal
Cyclic Sum ( $c=3$ )	$[7,1,2,9,8]$	$[1,0,0,1,1]$	$24$	$C_{n}$ (cyclic shifts)
Cyclic Product ( $c=3$ )	$[1,2,4,0,5]$	$[1,1,0,0,1]$	$10$	$C_{n}$ (cyclic shifts)

Table 3: Examples of synthetic tasks: input, equivariant and invariant output, and the target group.

Figure 6: (Top) Multitask versus single-task performance on Task Intersect with varying training set sizes (

r

units), showing the benefit of multitask training particularly on low-data regimes, even when matched to the same total training time as the single-task setting (dashed horizontal lines); (Bottom) Multitask performance with varying number of tasks (

n_{\text{task}})

, showing the diminishing returns of adding more tasks as the training size increases.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Group $G$	Params.	Test Acc. (mean $\pm$ std)	Train Acc. (mean $\pm$ std)
1D-PE	$I$	370.5k	0.656 $\pm$ 0.135	0.816 $\pm$ 0.093
2D-PE	$I$	116.6k	0.818 $\pm$ 0.005	0.831 $\pm$ 0.006
	$(S_{4})^{1024}$	112.5k	0.824 $\pm$ 0.000	0.848 $\pm$ 0.006
	$(S_{9})^{455}$	111.2k	0.827 $\pm$ 0.018	0.852 $\pm$ 0.013
	$(S_{16})^{256}$	110.4k	0.814 $\pm$ 0.020	0.840 $\pm$ 0.009

Appendix

Table 4: Performance of Transformers in Pathfinder64 tasks, using different positional encoding (PE): 1D-PE (learnable), 2D-PE (learnable, separate for rows and columns) with different group

G

. The accuracy is averaged over

3

runs.

Synthetic Tasks
Task	Equivariant	Invariant	Symmetry
Intersect	Given two sequences of length $\frac{n}{2}$ , determine which elements of each sequence are present in the other	Determine the size of the intersection of the two sequences	$S_{n/2}\times S_{n/2}\times S_{2}$ : sequences can be reordered, and the two sequences can be swapped
Symmetric Difference	Given two sequences of length $\frac{n}{2}$ , determine which elements of each sequence are not present in the other	Determine the size of the difference of the two sequences
Palindrome	Given a sequence of length $n$ , determine where the sequence has a contiguous subsequence that is a palindrome of length $k$ , if one exists	Determine if the sequence has a contiguous subsequence that is a palindrome of length $k$	Sequence reversal
Monotone Run	Given a sequence of length $n$ , determine where the sequence has a contiguous subsequence that is strictly increasing or decreasing of length $k$ , if one exists	Determine if the sequence has a contiguous subsequence that is strictly increasing or decreasing of length $k$
Cyclic Sum	Given a sequence of length $n$ , determine which cyclic contiguous subsequence of length $k$ has the largest sum	Find the largest sum of a cyclic contiguous subsequence of length $k$	$C_{n}$ (cyclic shifts)

Appendix

Table 5: Description of synthetic tasks, equivariant and invariant learning set-up, and their corresponding symmetry group.

ASEN (Loss $\downarrow$ )	Invariant			Non-invariant
Task	1	1.5	2	1	1.5	2
Intersect	0.076 $\pm$ 0.002	0.061 $\pm$ 0.009	0.060 $\pm$ 0.005	0.081 $\pm$ 0.002	0.078 $\pm$ 0.0004	0.077 $\pm$ 0.001
Palindrome	0.250 $\pm$ 0.04	0.170 $\pm$ 0.01	0.125 $\pm$ 0.05	0.270 $\pm$ 0.04	0.250 $\pm$ 0.04	0.132 $\pm$ 0.04
Detect Capital	0.053 $\pm$ 0.005	0.056 $\pm$ 0.007	0.068 $\pm$ 0.016	0.075 $\pm$ 0.011	0.066 $\pm$ 0.004	0.063 $\pm$ 0.007
Longest Palindrome	0.138 $\pm$ 0.014	0.110 $\pm$ 0.009	0.110 $\pm$ 0.009	0.152 $\pm$ 0.007	0.130 $\pm$ 0.012	0.130 $\pm$ 0.010

Appendix

Table 6: Performance of two ASEN variants across tasks for varying data amounts.Within each model variant, columns indicate data used (in units): 1, 1.5, and 2.

Figure 9: Invariant experiment ablation: test loss of ASEN using invariance (orange) versus its non-invariant counterpart (blue) across varying training data sizes (1 unit refers to 8,000 datapoints) and different tasks.

Method	Cyclicsum	Palindrome
Single-task	0.3869	0.510
Multi-task	0.3839	0.537

Appendix

Table 7: Multi-task Test Losses (0.08 units)

Figure 10: Compute cost in multitask training (measured as training time per epoch on an NVIDIA A100 GPU device) with respect to the number of training set sizes

r

and the number of tasks

n_{\text{task}}

Figure 11: Invariant pretraining ablation: test error of ASEN for finetune-only (blue bars) versus pretrain-finetune (orange bars) across varying input sequence lengths. Each subplot shows the pretraining effect for a different finetune task.

Explore similar work

Feb 23, 2026

Quantitative Approximation Rates for Group Equivariant Learning

The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of

α

-Hölder functions

f: [0,1]^N \to \mathbb{R}

. The goal of this paper is to provide similar quantitative approximation results in the context of group equivariant learning, where the learned

α

-Hölder function is known to obey certain group symmetries. While there has been much interest in the literature in understanding the universal approximation properties of equivariant models, very few quantitative approximation results are known for equivariant models. In this paper, we bridge this gap by deriving quantitative approximation rates for several prominent group-equivariant and invariant architectures. The architectures that we consider include: the permutation-invariant Deep Sets architecture; the permutation-equivariant Sumformer and Transformer architectures; joint invariance to permutations and rigid motions using invariant networks based on frame averaging; and general bi-Lipschitz invariant models. Overall, we show that equally-sized ReLU MLPs and equivariant architectures are equally expressive over equivariant functions. Thus, hard-coding equivariance does not result in a loss of expressivity or approximation power in these models.

Jonathan W. Siegel, Snir Hordan, Hannah Lawrence +2

Department of Mathematics, Texas A&M University, College Station, TX, 77840, USA · Faculty of Mathematics, Technion- Israel Institute of Technology, Haifa, 3200003, Israel · Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, 02139 USA

Equivariant Neural Networks Neural Network Approximation Theory Permutation-Invariant Learning

Ask PDF

Jun 26, 2026

A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of $\mathrm{O}(2)$

Vision transformers have become a dominant architecture for visual recognition. However, standard models do not explicitly encode the planar symmetries that arise in many vision domains. We introduce a family of vision transformers equivariant to arbitrary discrete subgroups of

\mathrm{O}(2)

, providing a unified framework that generalizes prior flipping- and

D_4

-equivariant transformer architectures. Our construction yields equivariant analogues of the core transformer components, together with expressivity guarantees for the resulting layers. In particular, we show that whenever

H \le G

, the class of

G

-equivariant ViTs embeds naturally into the class of

H

-equivariant ViTs. We also prove that, in the single-head setting, the corresponding equivariant self-attention layer realizes every

G

-equivariant self-attention map representable by ordinary self-attention. We further construct a

D_6

-equivariant model based on hexagonal patches, making the architecture compatible with six-fold rotational symmetries. We evaluate the resulting models on the PatternNet aerial image dataset in artificially data-scarce regimes across subgroups of

D_4

and

D_6

. Our experiments compare two equivariant attention mechanisms and analyze how the choice of homogeneous-space configurations used in the nonlinearities affects performance. Preliminary results under matched parameter budgets indicate that equivariance can improve recognition accuracy, motivating further study of how discrete symmetry groups shape transformer-based visual recognition models.

Tīkun Ông, Georg Bökman

Independent Researcher · University of Amsterdam

Vision Transformer Equivariant Neural Networks Equivariant Representation Learning Image Classification

Ask PDF

Oct 12, 2026

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

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.

Tomas Karella, Emily Shinkle, Alice Elisabeth Anastasia Allen +3

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

Invariant Representation Learning Equivariant Neural Networks Convolutional Neural Networks

Ask PDF

Abstract

Explore similar work

Quantitative Approximation Rates for Group Equivariant Learning

A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of O(2)\mathrm{O}(2)O(2)

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

A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of $\mathrm{O}(2)$

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