Any-Subgroup Equivariant Networks via Symmetry Breaking
Authors: Abhinav Goel
Derek Lim
Hannah Lawrence
Stefanie Jegelka
Ningyuan Huang
Venue: ICLR 2026 Poster (2026)
Keywords: equivariance
symmetry breaking
graph neural networks
symmetry
Primary area: learning on graphs and other geometries & topologies
TL;DR: We propose Any-Subgroup Equivariant Networks (ASEN), a framework for building a flexible equivariant model capable of modeling symmetries across diverse tasks

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.

Published as a conference paper at ICLR 2026
ANY-SUBGROUP EQUIVARIANT NETWORKS VIA SYMMETRY BREAKING
Abhinav Goel', Derek Lim', Hannah Lawrence', Stefanic Jegelkal.,, Ningyuan Huang MIT. 2TM, FFlatiron 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 pre- cludes the development of flexible, multi-modal foundation models capable of processing diverse data equivariantly, In this work, we build single model the Any-Subgroup Equivariant Network (ASEN) - that can be simultaneously equiv- ariant several groups, simply by modulating a certain auxiliary input feature. In particular, we start with a fully germutation-equiaant
base model, and then obtain subgroup equivariance by using symmetry-breat -beekkng 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 ap- proximate symmetry breaking, leveraging the notion of 2-closure to derive fast algorithms. Theoretically, we show that our Sbbgroup-equivariann networks can simulate equivariant MIPs, 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, 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 single non-equivariant model.
INTRODUCTION
Equivariant machine learning exploits symmetries in data to constrain the model with known priors, often leading to improved generalization (Elesedy and Zaidi [02T Petrache and Trivedi [2023). interpretability [Bogatskiy et al. 2002)), and efficiency [Bietti et al. 2021]. Most existing equivariant models are tailored to specific symmetry groups chosen a prion: for example, graph neural networks (GNNs) and DeepSets with permutation equivariance, convolution neural networks (CNNs) with translation equivariance, and Neural Equivariant Interatomic Potentials satisfying Euclidean group symmetries. While these equivariant models have demonstrated promising performance in datasets satisfying the prescribed symmetries, they are inflexible in several important ways: (1) equivariant architectures typically require deriving and implementing group-specific equivariant layers, so substantial research and engineering must be done for architectural design whenever new type of symmetry arises, and (II) equivariant architectures are typically only equivariant one symmetry group, and significantly differ across symmetries. Thus, an equivariant model cannot easily transfer knowledge across domains with distinct symmetries, so equivariant models cannot benefit from the empirical successes of the foundation model paradigm [Bommasani et al. 2021
In this work, we introduce a framework for building flexible equivariant networks, named Any- Subgroup Equivariant Networks (ASEN). Given a base group G, we consider its subgroups G < G (typically known a priori) that capture the symmetries intrinsic to the domain (e.g. graph automor- phisms) or induced by the task (e.g., sequence reversal). To design subgroup-equivariant model f. we start with a base network he, which is equivariant to the large group G and thus overly constrained for our purpose - it cannot represent functions that are equivariant only G but not G To reduce the amount of constraints, we augment the input z with a feature v that break the symmetries in G G but maintain the symmetries in G. To ensure this, we construct v that its self-symmetry group (i.e. automorphism group) is equal to G. ie. Aut(v) = G. Finally, we pass in the symmetry- input v our overly-constrained hg, and obtain the model fo( ho(a, v). We prove that the

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model fo is indeed G-equivariant, and under certain conditions it is not equivariant to G G: in other words, it has the correct equivariance. A trivial, but widespread, example of this technique is the use of positional encodings (Vaswani et al., 2017] to fully break the permutational symmetry of transformers (since every entry of the sinusoidal positional encoding vector is unique, G = Aut(v) is the trivial group). When has non-trivial automorphism, some equivariance is retained.
ASEN overcomes inflexibility (I), since it only requires providing a single new input v (with correct automorphism group) base network hg. Also, ASEN overcomes inflexibility (II, since single instance of our model can process data from varying domains with different symmetry groups.
While our framework works for any G, we focus on the particular case when G = S, is the symmetry group acting as permutation matrices, so that our networks are equivariant to permutation subgroups. This covers the symmetries of many common domains such sets and graphs, which allows us to leverage existing permutation equivariant models as the base model hg- In particular, we may leverage existing set networks (inputs in R"), graph neural networks (inputs in R"²) and hypergraph neural networks (inputs in I for the base model. While large K may be required for complex groups, to balance efficiency and expressivity, we focus on the K = 2 case of graph neural networks, and develop a practical algorithm for computing the symmetry breaking object v edge features, with (nearly) the desired self-symmetry Aut(v) & G. To do this, we use the notion of the 2-closure ((2) of a group G (Ponomarenko and Vasil' ev 2020), which provides a formal notion of a group that is close to the target group (G = G), and we compute v with Aut(v) = (())
Theoretically, we show that under mild conditions, ASEN parameterized with graph neural networks has the exact permutation subgroup equivariance. Further, we prove that ASEN is expressive in two senses: it can approximate certain equivariant MLPs [Maron et al. 2019 Finzi et al. [02 to arbitrary accuracy, and it is universal in the space of G-equivariant functions if the base model is universal over G G-equivariant functions.
To validate our approach, we apply ASEN to diverse settings: (I) exploiting symmetries within a single task, including graph learning (human pose estimation and traffic flow prediction) and image classification (Pathfinder); (2) leveraging symmetries across different tasks on sequences in multitask and transfer learning. Across these settings, results highlight the flexibility of ASEN and practical utility of symmetry-awant architectures. Our framework supports both fine- ane-ggrined control over group actions and strong transfer of learned representations, making it powerful tool for structured generalization in neural networks. Our main contributions are summarized as follows:
We propose Any-Subgroup Equivariant Networks (ASEN), framework for building flexible equivariant model capable of modeling distinct symmetries across diverse tasks.
We theoretically show that ASEN enforces any desired subgroup symmetry via proper choice of the symmetry-breaking input and architecture. We also prove that ASEN is as expressive as equivariant MLPs, with universality guarantees given a sufficiently expressive base model.
We validate our framework in applications including symmetry selection, multitask learning and transfer learning, highlighting its flexibility and effectiveness in exploiting shared symmetry structures.
2 RELATED WORK
Subgroup Equivariance In equivariant network design, recent works (Blum-Smide al., 2025
Ashmman et al. 2024 Lim et al. [2224] have proposed subgroup-equivariin models via augmenting an auxiliary input. Specifically, Blum-Smith et al. (2025) proposed permutation-invanl
model for symmetric matrices by using a DeepSet base model invariant to bigger group, together with a suitable cummetry-breakiee
parameter to reduce the base model symmetries: Ashman et al. (2024) used fixed symmetry breaking inputs to construct non-equivariant models or approximately equiv- ariant ones; Lim et al. 2024) leveraged node or edge features to reduce the amount of permutation symmetries in neural network computational graphs. While these works model subgroup equivariance, they are limited to the single-task setting. In contrast, we study the multitask and transfer learning setting, providing recipe to build flexible equivariant models.
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Symmetry Breaking Symmetry breaking via node identification is popular technique to enhance
the expressivity of graph neural networks [Abboud et al. 2021] Sato et al. 2021; Bevilacqua et al.)
2025]. Symmetry breaking of the input has also been used improve the flexibility of equivariant
models for applications in graph generation and physical modeling (Smidt et al.
2021 Lawrence
et al., 2024; Xie and Smidt [024]. Unlike (Smidt et al.] [02T] Lawrence et al., [2024] Xie and
Smidt 2024] that perform input-dependent symmetry breaking, we break the symmetry of the model
uniformly for all inputs. Moreover, existing works typically focus on (approximate) equivariance
to one particular group. In contrast, we build a single model capable of modeling diverse data
equivariantly, via different choices of v adapted to the target application.
Approximate and Adaptive Equivariance Another direction towards flexible equivariant networks relies on approximate equivariance [Wang al. 2022] Huang et al.
[003]] soft equivariance by
converting architectural constraints into prior [Benton et al., 2020 Finzi et al. 2021aa]. regular-
ization (Kim et al. 2023], or adaptive equivariance per task and environment (uupta et al. 2024]. Approximate equivariance This motivates our approach that approximates the automorphism group of the symmetry breaking input with the 2-closure group.
also improve generalization [Wang et al. 2022 Huang et al. 2023
3 METHOD
3.1 GENERAL METHOD (ASEN)
Here, we describe our general framework for ASEN. Let G be matrix group, and let G S G be subgroup. Both groups have actions on the sets X' and y. We desire our model to parameterize G-equivariant functions, that is, functions f X' - y such that f(gx) = g gyy for g E G. To this end, consider a "lift" of f: function h2 X V + y on expanded space, where we introduce an additional space on which and G act. The function hg is G-equivariant (i.e. equivariant to the larger group):
hg ho(gz, go) = gho Vg E.
(1)
To obtain fo from he, we find a subgroup G, i.e. its automorphism group is G,
input v E V that is exactly self-symmetric the
Aut(v) = {g E G : gv v G.
(2)
Finally, we define our G-equivariant model fa X' + as
fx(z) = ho v).
(3)
The model fo is G-equivariant because for any g E G. fo(gz) ho(gz, v) ho(ge, gv) gho v) gfo(z);
(4)
where the second equality follows from E Aut(v), and the third equality is due to G-equivariance
of he. For g E G G. this equality can break: if gv * v, then he (gz, v) * he (gr gv). In fact, hg is injective for the input then fo is only equivariant to G, and not to any other elements in the larger group G. These results are captured in Prop. I As an example: consider
= 0(3) acting on 3D point cloud R2x3 and the D(3)-equivarian base model hg. Now we fix particular axis with the stabilizer (2). If hg is injective for v, then fo = (2 v) is only equivariant to ((2) but not O(3) (2).
3.2 PERMUTATION SUBGROUP EQUIVARIANCE VIA HYPERGRAPH SYMMETRY BREAKING
To use our ASEN framework for parameterizing a G-equivariant function, we need two main components: (i) method of parameterizing the base model he that is equivariant to the larger group G, and (ii) a way to construct or compute a symmetry breaking object v with automorphism group Aut(v) G. In this subsection, permutation matrices on n objects, we can leverage existing equívariant architectures for (i); and we can develop a practical algorithm for (ii). For the rest of this paper, we primarily focus on this setting.
show that when G S, is the symmetric group acting as
To construct efficient and expressive symmetry breaking objects, we turn to hypergraphs. Con- cretely, for (matrix) group G acting R2, hypergraph
nodes is defined H
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where A(k) R²" is order k-tensor, and K is the max tensor order. We can interpret A (node) positional encoding, and (Ak) as (hyper-)edge features for A 2 2. The automorphism group of the hypergraph is defined as
(4
......, ,
Aut( {P E S,, ps* A A (k) k 1,...,,K K).
(5)
For instance, if K = 2, then this is the standard graph automorphism group
(H) = (P E S, PA()) = A (P = A (2)).
(6)
For K large enough, can construct hypergraph H. such that Aut(H) uniquely determines G [Wieland 1969]. Then, we consider any existing permutation-equivaaii hypergraph neural network
hg :
R2
R" + R" such that any P E Su,
(AP,)PA())
ps* A = Pho (X (X,AA) (.......AAA))))
(7)
We abbreviate 7 by ho (P(X, H)) Pho(X: (X; H). Our G-equivariant model then takes the form fx(X) ho, 44) (k) ho (X,
(8)
Hypergraph Construction and Approximation with 2-Closure Achieving exact symmetry break- ing of Sn to the desired subgroup G may require a hypergraph of prohibitively high order (up to K S n). For efficiency, we fix K 2 and construct positional and edge features H (A(1), A whose automorphism group Aut (H) reflects on how G acts on nodes and pairs of nodes. Concretely, nodes j.j (or node pairs (+,, i2), (ji, are assigned the same feature if and only if they are in the same G-orbit. In this way, the positional and edge features encode the orbit partition under G. By construction, Aut(A (2) is the 2-closure group of G, denoted as G [Ponomarenko and Vasil'ev 2020). In general G S G( and in many cases G = G regardless of the permutation representations: such groups are called totally 2-closed. This class includes finite nilpotent groups that are either cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order (Abdollahi and Arezoomand 2018). See Sec. for concrete examples. When G < G(2), the 2-closure introduces additional symmetries, leading symmetry group mismatch. We can tradeoff.
results from [Huang et al. 2023) to analyze the wwwooommatiionggneraliiiee
Algorithm| 1 provides the procedure to compute (2)) with high-level SymPy commands, assuming
access to the generating elements of G see Fig. for examples and App. A for details. We remark that while Alg. computes pairwise edge features, it can be applied a sparse graph or a subset of edge features by restricting the support of H.
Algorithm 1 Compute Edge Orbits ( such that Aut(A (() (SymPy commands)
Require: Generators ......, Tr of G S Sn
Ensure: Edge orbits (2) E [n] x [n]] where A! = (2) - (i.j) ~a (m, n). 1: Lift generators: For each a; E Sn, define P; E S vb) + (a: o; (zx))) Encode (I,, 22) as a n + and construct pÚ as Permutat ion of size n².
2: Form diagonal subgroup: Let A(G) == (p1 (pI....,Pr) be the subgroup of S,2.
Delta = Permutat t onGroup [pI, Pr
3: Compute edge orbits: For an edge vo), apply Pt repeatedly until no new pairs can found.
Del tar.iitstttts........o.... ()
Alignment Note that it is important that the inputs have their node indices aligned to each other, Le. they share a common labeling. This is implicitly captured via the choice of the concrete matrix group G. For example, the inputs in Fig. [are treated as sequences (instead of sets). The alignment is typically satisfied in applications such as sequence modelling, graph signal processing, and graph time series (see examples in Sec. 5). but needs to be computed for other applications such graph-level tasks.
If' instead we are given all elements of G. we can pass them to Permut at LonGroup in SymPy and run the Schreier-Sie Sims algorithm. This produces a base and strong generating set (BSGS): compact, non-redundant set of generators adapted to a stabilizer chain. The BSGS allows efficient orbit and membership computations without ever enumerating the full group.
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Example: G = S2 (mirror symmetry)
Example: G = Saz X Snz X S2
Input
Input
A
(
(6.))
A
A (2)
Figure I: Example symmetry breaking objects as positional features A and edge features A for encoding subgroup symmetries in 4-node paths. These symmetries are explored further in Sec.
THEORETICAL RESULTS
We first establish that under mild conditions, ASEN with exact symmetry breaking can achieve the desired subgroup symmetry G G for any general matrix group G (Prop. [T. and for the permutation group Sn in the context of graph learning (Lem. D. We then show that ASEN with approximate symmetry breaking can simulate equivariant MILPs (Thm. [] and enjoys universality guarantees
the base model is universal (Thm. 2 Proofs are deferred to App. B
The following Prop. I shows that ASEN has precisely the desired equivariance to G under mild conditions.
Proposition 1. Let ho X - be G-equivariant, and let Aut(v) G. Then fa( he(x,v v) is equivariant to G. If Fadditionally he is injective in the imput v, then fo is not equivariant any transformation G AG.
Hypergraph Symmetry Breaking As noted in Section 3.2 we generally take he to be a graph neural network when parameterizing permutation subgroup equivariant functions. In what follows, we characterize the condition for correct equivariance (i.e., hg is equivariant to G but not G \ G) when he a one-layer message-passing neural network (MPNN) and G Sn operating on nodes and edges, and G Aut (A))) is the automorphism group of weighted graph A E Raxn, By message-passing
the definition of message passing, the output ha at node is computed as
he A = (v (X:), (Hwa(X,, X,, A [ (MIB)
where the is the edge update function, vhyy & are the node update functions, and is the edge multiset aggregation. In the following lemma, we show that if the constituent functions are injective, then ha correctly equivariant to only G under mild assumptions on the node features.
Lemma 1. If one-layer MPNN ha uses injective functions for (hyper-Jedge feature update dhe node update ch and (hyper-jedge multiset aggregation and the node features are distinct, then he is not equivariant to permutations in S, \ where G = (A)()
Note that these injectivity conditions are similar to sufficient conditions under which message passing graph neural networks have the same expressive power as the Seeisfeiler-Lemann graph isomorphism test [Xu al. 2019]. We can similarly conclude that these conditions sufficient for deeper MPNNs to have the correct G-equivariance. Also, we can show an analogous result for hypergraph networks by viewing them as higher-order MPNNs operating hyperedges, where S, implies that there exists hyperedge (i.j....,k)) k)) such that J. * A (8
Connections to Equivariant MLPs Here, show that ASEN with approximate symmetry breaking can simulate certain configurations of a common type of equivariant neural network, sometimes called equivariant MLP, which consists of equivariant linear maps and elementwise
nonlinearities [Ravanbakhsh et al. 2017 Maron et al.) 2019 Finzi et al. 2021b]. When applied to a group G that
permutation matrices R equivariant MI.P defined composition: Ty o o o o o Tı, where is an elementwise nonlinearity, and each T, R=** -> ma* is a G-equivariant linear map (for simplicity, we ignore channel dimension here). We call K max; ks the order of the G-MLP, so if T, I R for each then the G-MLP has order 1. We prove the following result:
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We next show that the universality of ASEN follows from the universality of Published as a conference paper at ICLR 2026
Theorem 1. Any order G-MLP can be approximated to arbitrary accuracy compact domain via ASEN with = 2 and an MPNN base model hg-
We remark that much like an equivariant MLP can increase expressivity by increasing the order k*. we can increase expressivity in ASEN by increasing K > 2 and using the K K-closure group approximation [Ponomarenko and Vasil ev 2020). While we show this relationship in expressivity at k* = 1 and K = 2, we believe that there may be relationships between the two methods at higher orders, and leave it to future work.
Universality Results its base model.
Let G be a compact group. X, V be compact metric G- Gpspces, and y be a compact G-space. Let fo X V y' be universal family of continuous G-equivariant networks, i.e. folgx, gu) = fo(x, Consider H. E V with stabilizer equal to a subgroup G S G. Then, the family fo H. is universal over continuous G G-equivariant functions from X
Theorem 2.
We remark that there exist universal architectures for point clouds (Hordan et al. 2024 Blum-Smil
ctt al. 2025]. For graphs, universal architectures typically require combinatorially large feature space Maron et al., 2019]. only hold in a probabilistic sense [Abboud et al. 2021]. However, if we restrict to graphs with unique node features, then MPNNs (with sufficient depth and width) are universal [Loukas] 2020]. Consequently, Thm. ensures that whatever variety of universality the family fa enjoys is inherited by fo(,, (.).
5 EXPERIMENTS
Towards developing a general-purpon equivariant foundation model, we evaluate ASEN in diverse experimental settings by answering the following questions:
Q1 For a single task, can ASEN wwith one erchitecture-explee different symmetries and reveal the impact of the group choice (Sec.5.1l2
Q2 Can ASEN leverage shared symmetry structure across tasks to outperform task- specifc equivariant models or non-equivariant baselines in multitask learning (Sec. 5.2. and transfer learning (Sec. 5.2.2
Architecture Our backbone is permutation-eequivriiiie graph neural network (GNN), composed of input layers (e.g. embedding, MLP). followed by four layers of GATv2 message-passing [Velčkovid al. 2017]. and concluded with output layers (e.g., projection, aggregation). Standard dropout and layer normalization are applied throughout. There are two task-specific modules: Edge Embedder that calls Alg. to categorize edge orbits and learn their embeddings; TokenEmbedder that maps (discrete) node features to learnable token embeddings for classification tasks (omitted for regression tasks); See Fig. 2 for an overview. The preprocessing cost (Alg. 1h scales as 0(rn²) where v the number of nodes and is the number of the generators of the group G. The training cost, using message-passing -aassngg GNN backbone, scales as Omm??) on a colly-connected dense graph where m is the number of training data points, and 0 E)) sparse graph where |E] is the number of edges.
As EdgeEmbedder is a learnable module, ASEN can discover more symmetries from data the chosen group ((2) (specifying the edge orbits) is smaller than the target group; see evidence in Sec. 5.2 On the other hand, ASEN can fail when (22) is much larger than G, as we show in App. C.5
5.1 SYMMETRY MODEL SELECTION APPLICATIONS
In this section, we consider exploring different symmetry choices for given task. Specifically, for each subgroup G in candidate set chosen a priori, train instance of ASEN with edge features satisfying (u)( ((2) (or positional features satisfying Aut(A()) G()). Our results highlight the utility of choosing subgroup symmetries informed by the structure of the domain,
SSouree code available at https////gthhhbcccmm///me/e/ 21 pwppppermppppppp
formul 1 in

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Group S4
Rdgeemmeedder
GAT
Projection
Output
Input
+ MLP
X
GAT
Aganegation
Tobenddee
Figure 2: ASEN Architecture to model any permutation subgroup-equivarian functions.
such as the reflection symmetry in skeleton graphs, the continuous road structure traffic graphs, and the local symmetry in image grids.
5.1.1 GRAPH TASKS
We apply ASEN to perform symmetry model selection for learning on fixed graph setting, using the experimental set-up from Huang al. [2023]. Unlike Huang et al. [2023] that constructs distinct
G-equivariant MLPs for each group, ASEN offers a unified architecture flexibly model different
Geeqivaariance by symmetry breaking (i.e., via positional edge features).
Human Pose Estimation We begin with an application in human pose estimation, using the Human3, 6M dataset [lonescu al. 2014), which consists of 3.6 million human poses from various images. Our input features consist of 2D coordinates E 16xx2 representing joint positions skeleton graph {0, 1 16x16 (see Figure inset). The model predicts the corresponding 3D joint positions in 16x3, Performance evaluated using the standard P-MPJPE (Procrustes-aligned Mean Per Joint Position Error) metric. We consider three kinds of edge feature H (2 (Alg. D: (I) fully-connected A/r)
(2) sparse 2! = where edges are constrained to the support of the skeleton graph
A, and (3) weakly sparse combining A/") A((") We consider selecting difffrrntt automorphism
groups of the human skeleton edges Aut(AA2)) (that yields the best performance): S₂ (full left-right reflection), S3 (left arm/right
and left leg/right leg). St (each left-side joint independently mapped to corresponding joint on right side), and 7 (no equivariance). Our results in Tab. Obtained from single model ASEN match with those reported in (Huang al. 2023) that require multiple distinct equivariant MLPs. Notably, the weakly sparse graph yields some of the strongest results, highlighting ASEN's flexibility in capturing multiple symmetries. This approach can also be extended to represent a graph with multiple sets of symmetries.
Table P-MPJPE error (L) for human pose estimation using different symmetry groups and edge frameworks.
Table 2: Mean Absolute Error (MAE
4) for traffic flow prediction.
Group
Fully Connected
Sparse
Weakly Sparse
Model, Group C, n4 n))
MAE
T
34.71
33,39
34.75
Fully Connected, Sa, SSz
2.72
39.48
40.52
38.80
Sparse, Sn, Sn, Fully Connected, S.g
2.69
S2
43.24
42.37
40.67
Sno
2.79
S2
47.54
49.45
46.52
Sparse, S,,
S
2.77
DCRNN [Li et al. 2018
S.
2.77
Traffic Prediction Next, we evaluate on the traffic forecasting task, also taken from [Huang et al. 2023]. This uses the METR-LA dataset, containing time-series traffic data from 207 sensors deployed on Los Angeles highways. The sensor (node) records speeds and traffic volume every 5 minutes, resulting in a graph time series with node feature X, IR 20207 x2 The objective is predict traffic conditions at future time X based on the past traffic {Xe. X , X,-2). The underlying graph is defined via sensor adjacency matrix A E 2077 constructed from roadway connectivity.
To incorporate symmetry structure, we leverage the spatial layout of sensors along major highways. We consider two group structures, taken from (Hungg al. 2023) with two clusters representing

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Table 3: Examples synthetic tasks: input, equivariant and invariant output, and the target group.
Task
Input
Equivariant Output
Invariant Output
Group G < S.
Intersect
e, c, a.6,c, c.
1, I 1, 1,0,0. 0,0.
Su/2 x Sn/2 X $2
Symmetric Difference
Palindrome
b,
0,1,1,1,0 0,1,1,1,0
Truc True
Sequence Reversal
Monotone Run (k = 3)
1,1,224,11
Cyclic Sum (c = 3)
[7,1,2,9.8
1,0.0,1.1
C, (cyclic shifts)
Cyclic Product (c
[1,2,4,0,5]
1,1,0,0,1
10
Detect Capital
[4,6,6,
True
Sp-1 (permute except first)
Longest Palindrome
[a, d,
Sn
major highway branches G = Sny Suze and another with nine clusters corresponding to finer- grained regional groupings G Snz x ... x Suy. These symmetry groups serve as approximate equivariances, which encourage learning equivariant representations for similarly situated sensors. We construct node positional features such that Aut (A()) G, and benchmark our model in both fully-connected and sparse graph regimes. As shown in Tab. 2 choosing suitably smaller symmetry can outperform full permutation symmetry.
5.1.2 PATHFINDER TASK
As discussed in Sec. 1 Transformer with PE (1D or 2D) breaks permutation symmetry completely, since each position feature is unique (i.e., trivial automorphism). We study the effect of imposing local symmetry
Transformers for image tasks, by sharing the same position vector for pixels within the same p X p patch for p = 2, This patch-wise weight sharing preserves permutation symmetry within patches while distinguishing different patches. We evaluate these 2D-PE variants of Transformer on the Pathfinder-64 task (Tay et al.] 2021] using a standard architecture with learnable row/column embeddings. Fig. 3 shows that using local permutation symmetry can improve performance while slightly reducing the model parameter counts (see App. for details).
10 PE, 6-/ Acc]lesd]
2D G- Acc()-a)-00.11
2D. (5,
PE, G- (5
1009
1222
1510
300
10:0
100
900
12:0
12.00
180
4000
30
4500
20.0
425
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.
5.2 SYNTHETIC SEQUENCE MODELING TASKS
To assess the ability of ASEN to transfer structural knowledge across tasks, we consider synthetic sequence modelling tasks capturing various permutation subgroup symmetries, summarized with examples in Tab. 3 (with more details deferred to Tab. To demonstrate these sequence tasks can benefit from exploiting subgroup symmetries, we compare two ASEN variants: one equivariant model with the correct group (c.f. rightmost column in Tab. 3). and one non-equivariant baseline without any symmetry. As shown in Fig. 4 equivariant models incorporating the correct symmetry significantly outperform their non-equivarian counterparts across all tasks. More details including similar results for the invariant setting (see Tab. (6) and the effect of varying training dataset size (see Fig. 9 can be found in App. C.2
Additionally, we explore if ASEN can learn more symmetries from data given a misspecified symmetry group (smaller than the target group). We consider the Intersect task with the target group = 22 x S2. but only encode a smaller symmetry group G' (S,./2² 2 in the edge features. We check the edge feature weights before and after training to see if the S2 symmetry was leamed.
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Explorrnt
Nen syalorant
Cydic
Epect
Epech
Epech
Figure 4: ASEN with the correct group ("Equivariant") converges faster and to lower loss than its trivial sym- metry counterpart ("Non- "oon-equivariant".
Figure 5: Initial edge weights (left) and trained weights (right): ASEN learns more symmetries from data.
Fig. 5 shows pairwise distances between the learned Intersect edge weights: the top-left and bottom- right quadrants show the correct structure (checkerboard), whereas the top-right and bottom-left quadrants lack this initially but converge to the checkerboard pattern after training, discovering the S2 symmetries from data.
5.2.1 MULTITASK LEARNING APPLICATIONS
In this section, we investigate whether learning related tasks with compatible or similar symmetry groups benefit ASEN from shared representation learning. We again make of the synthetic tasks introduced in Tab. 3 While each task can be learned independently using a different equivariant model, assess whether multitask training can facilitate improved generalization in low-data regimes. To this end, we use the same ASEN backbone (including weights) shared across tasks, while the TokenEmbedder and EdgeEmbedder modules are task-specifie. During training, we randomly sample batches from all tasks, ensuring concurrent and balanced updates across tasks.
We then compare two regimes: training only on units ("1 unit" represents 2.500 datapoints) of a single-task, and the other training on units with Rask = tasks, specifically Intersect, Cyclic Sum, and Palindrome (see Tab. We vary r (0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 3.0} and report the average performance across three random seeds. Fig. 6 (top) shows that multitask training leads to significantly improved convergence and accuracy in the low-data setting for learning Intersect, demonstrating the benefit of symmetry-aligned task transfer on certain tasks; on the other tasks (Cyelic Sum, Palindrome), the multitask setting did not provide much benefit (see details in App.
To further investigate the effect of the number of tasks, we repeat the above multitask training with Thask {4, 6) tasks, by adding Symmetric Difference, Cyclic Product, and Monotone Run, respectively (see Tab. 3p. Fig. (bottom) shows that while increasing Rask improves performance in low-data regimes (e.g.. r < 1.0 units), such effect diminishes as training size grows. As the compute cost scales linearly with both training set size and the number of tasks (see Fig. T0 for details), this highlights a practical tradeoff in choosing between training size and task count.
Vattal Awp
Martao Aup
MURDUU Plai
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 the single-task setting (dashed horizontal lines); (Bottom) Multitask performance with varying number of tasks (Nusk), showing the diminishing returns of adding more tasks as the training size increases.

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Pretraied
ASEN-NaPrutrain
0.0
20
No5ym Protrain
Valcation LOSS
NuSym-Nalhhittal
0.4
0.2
Epoch
DetectCa pital Falindrome
Longestpal
Intersect
Figure 7: Equivariant transfer learning on Task Intersect: faster convergence and lower loss.
Figure 8: Invariant transfer learning: improved
pretraining when using symmetry in ASEN.
5.2.2 TRANSFER LEARNING APPLICATIONS
Beyond multitask training, we explore transfer learning by pretraining tasks with diverse sym- metries and then finetuning on a new task with a distinct symmetry. We assume knowledge of the task domain to leverage and transfer the domain-related symmetries (i.e., knowing the base group G covering most downstream task symmetries as its subgroups). We focus on the same set of synthetic tasks in Sec. 5.2.I and simulate low- resource regime by limiting access to only 0.15 units (375 datapoints) of training data. We compare two ASEN model variants: A finetune-only haseline trained from scratch using only the chosen task data, and a pretrained model initialized via joint training on 1.5 units each of the other tasks, followed by fine-tuning on the chosen task data. To encourage knowledge retention, we reduce the learning rate of the GNN backbone during fine-tuning, while allowing the embedding layers to update more freely. For the equivariant setting, Fig. shows that pretraining ASEN achieves significantly better generalization compared to training from scratch on the Intersect task, showcasing ASEN as an effective initialization for related downstream tasks. For the invariant setting, Fig. [8] shows that pretraining ASEN with the correctly specified symmetry outperforms its trivial symmetry baseline, and pretraining has a larger transfer effect when symmetry provided on the Detect Capital and Palindrome tasks. Ablation studies on the pretraining effect with respect to varying sequence length are summarized in Fig. 11 and discussed in App. C.4
6 CONCLUSION AND FUTURE DIRECTIONS
In this work, we introduce ASEN, a framework for building flexible equivariant model capable of exploiting diverse symmetries. Given a subgroup G S G, ASEN parameterizes G-equivariant functions via a base G-equivariant model, and symmetry breaking object whose autormophism group is G. For general G. we prove that ASEN can achieve the desired subgroup symmetry and inherit universality properties from the base model. Focusing on the permutation group G = Sn, we encode the desired subgroup symmetries via positional and edge features, which can be easily integrated standard GNNs and expressive equivariant MLPs. We empirically demonstrate the flexibility of ASEN to perform symmetry selection in graph and image tasks, and its effectiveness in exploiting shared symmetry structures in multitask and transfer learning for sequence data.
As a first step, we consider modeling symmetry (sub)groups acting globally on the input; a natural next step is to incorporate local symmetries, which play key role for molecular graph applications (Thiede ct al. 2021] Zhang et al.] 2024)). Another interesting direction is allowing the symmetry breaking object be input-dependent. Incorporating "soft" equivariance priors in ASEN is another fruitful approach. Studying the scaling behavior ASEN and the effect of symmetry model misspecification key avenues for future work. Applying framework for multi-physics pretraining (McCabel
ct al. 2024 Rahman et al. 2024] to model task-specific symmetry breaking is promising direction. Finally, exploring other flexible forms of symmetry breaking and exploiting beyond permutation subgroups offer a promising path towards equivariant foundation models.
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ACKNOWLEDGMENTS
The authors thank Erik Thiede (Cornell), Soledad Villar (JHU), Bruno Ribeiro (Purdue), Risi Kondor (UChicago), and Jinwoo Kim (KAIST) for valuable discussions. The authors also thank the anonymous reviewers and area chair for their constructive feedback. H.L. is supported by the Fannie and John Hertz Foundation. S.J. is supported by an Alexander von Humboldt professorship. This research was also partially supported by NSF Al Insitute TILOS.
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A ADDITIONAL DETAILS OF ALGORITHM 1
Lift the generators. Given the generators 01...., or G S Sn, we define new permuta- tions Pi acting on X x X with X nl). Each P; acts diagonally:
Pr: (Ia; 2..) + (o, (26)).
In practice (e.g. in sympy), if n is the degree of the action, we encode (Eay 46) as a single index n + b, and construct pj as Permutat of size n²
2. Form the diagonal subgroup. Define
AGG) (p.....,pr):
the subgroup of S, generated by the lifted permutations.
Conceptually, we take the subgroup generated by the Pi. In sympy, this done by calling
Delta PermutationGroup (....
Internally, PermutationGrout builds base and strong generating set (BSGS) for A(G') via Schreier-Sims The BSGS consists of chosen base B = (bi.... by) and strong generators adapted to the chain of stabilizers
AGG = (()) 2 G 2.. ((()) = (c],
where G()) is the subgroup fixing the first i base points in B. The associated basic orbits and Schreier vectors allow efficient navigation in the group.
3. Compute the orbits. The G-orbits on X X exactly the orbits of A(G). Concretely, starting from pair (Ia, 26), we apply the generators repeatedly until no new pairs are found; the obtained its orbit. With a BSGS, orbit computation polynomial-time SymPy performs breadth-first search on the strong generators and stores transversal information for reconstruction. In code, this as simple
Delta. orwitstt.....................t..
which returns all -(Giit of the action.
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B PROOFS
Proposition 1. Let hg X V- + Ye be G-equivriant. and let Aut(y) Then fo(x) = ho(r, v) is equivariant to G. If additionalll he is injective in the imput v, then fo is not equivariant to any transformation in G G.
Proof. We have already shown in Section 3.1 that fe G-equivariant. Now, suppose that he is injective in and let E G \ G. We want to show that fa (gxz # gff(() for some choice of r. We choose any x E X. This holds because * gv (since otherwise g & G Aut(v)). Thus, by injectivity, (g v) * ho gv). This allows us to conclude that
Salgx) =
ho(ge, v)
(10)
+ ho(gx, gv)
(11)
= gho(x v) glo(z).
(12)
(13)
This concludes the proof.
We remark that the injectivity of hg in the input v is sufficient to prove Prop. 1 but not necessary. As shown in the proof, this injectivity assumption allows show fo(gz) F gfo(z) for any E X, while establishing this for a particular choice
suffices.
Lemma 1. If a one one-layer MPNN hg uses injective functions for (hyper-Jedee feature update the node update 6, and (hyper- Jedge multiset aggregation and if the node features are distinct, then he not equivariant to permutations in Sn G where G = Au)(
Proof. We will show that for any S,' G. fe(aX) # (o()). By 9 it suffices show that there exists node such that
oo MM + oo (2)
(14)
Since a E S.. G. there exists a node pair (i.j) such that A + A a())))))) By the assumption that the node features {X,} has distinct elements, the multisets
(1) (() IsE in] + (X.)() -(),0())) 15 [m]].
(15)
By definition of MPNN 9
A (i)(x(()), (IV,, 1i (n)B)
(16)
ho(oXXo0A()) & (N)(X.())) (1CC (() (m11))
(1),
(17)
By injectivity of the and 15
(X) (1) 4 (19) 1 [n] + (X)) -1) 1 [n]].
By injectivity of and d. we have oo A + h [1)) completing the proof.
Theorem 1. Any order G-MLP P can be approximated to arbitrary accuracy compact domain via ASEN with = 2 and an MPNN base model hg-
Proof. We show that one layer of ASEN using a message-passinngg GNN backbone can simulate o Tú, so suppose L = 1 (i.c. the G-MLP has one layer). Recall that any equivariant linear map T 12th - _ can be viewed
linear combination T = Σd= B' where o; R are scalars, the B' : R R G-equivariant linear maps that span the vector space of G-equivariant linear maps, and d is the dimension this vector space of G-equivariant linear maps. Moreover, by an argument similar to Maron et al. 2019 we can define the B! as follows:
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Let Ti..... Ty be the unique orbits of the action of G node- indices [n] x [iz] (we refer to these as node-pair orbits). Then Ravanbakhsh et al. [2017]: : Maron et al. [019 show that d, and the B' can be chosen as:
B/y {
(i,j)
(18)
else.
This is form of weight sharing, where B¹ is constant on each node-pair orbit (and hence any T that is linear combination of them is constant each node-pair orbit). The index denotes the 1-th node-pair orbit. Note that the linear map can be shown to take a message- paasiing form as follows. For input x E IR", let xy be viewed as a node representation for node i. Then the new node representation for node after this layer is
T(x);
a (B'x);
(19)
1=1
9
n
01
Bly
(20)
1=1
Σ
23
a B
(21)
1=1
This can be interpreted as message passing, where the node j passes message Ij Ei= B,J to the node i. We will show that ASEN with order K 2 and this map Tzx). Note that any H E R² that has Aut(H) and only if (1) ~g (22, In other words, H..jj. = Ha if and only if (11, and (12, j2) are in the orbit 77 for some I. Let the distinct entries of H be denoted by h4 E R, so that by High if and only if (11,51)= E 7.
GŇN he can simulate G(2) must satisfy that High
Hag
Finally, we define the GNN he to take the following message- Sartugge-aasinng--e based form. Let 2; be the representation of node i. The GNN updates the node representations to i. via two multilayer
perceptrons MLP"(h) IR- + R2 and MLP" R - R as follows:
2, MLP" (z), MLP"
(22)
j=1
if h- by
MLP = 0 clse
(23)
MLP" (x.1) = x atyl-
(24)
= =i B which shows that ft T(x); In practice, an MLP cannot exactly express these functions, but an MLP can approximate each function to arbitrary precision > on compact domain. Note that the function MLP" seems discontinuous, but it is only defined on finitely many inputs, so it has a continuous extension that is exact on the finite inputs.
Note that MLP = By so that MLP"
Let G be a compact group, X, V be compact metric G-spaces, and y be a compact G-space. Let fo X x V + y be universal family of continuous G-equivariant networks, i.e. folgz, gr) g- fo(x, v). Consider H E V with stabilizer equal to a subgroup G < G. Then, the family fol., H. is universal over continuous G-equivariant functions from X to y
Theorem 2.
Proof. Let f X > y be a continuous, G-equivariant function. To prove universality of fo(s, H), must show that for any c, there exists @ such that fo(-, H) is c-close to f To achieve this, let's first define a new function (which we will prove is G-equivariant), : X' x O + 3 where O {9 : G} as follows: for any g E G.
((x,gH)) = (g
(25)
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Note in particular that n(x, fz) To first show that is valid definition of will argue that if gHl = g'H, HH, then gf 2) = = fg' x) for all E X. By the assumption gfl = g H. we have g' =g for some E G, where G is the stabilizer of H. Then
g (g (gs). ((gs) xx) (f
= 0f
(26)
where the last equality follows from I* being G-equivariant.
Next, the continuity of n on X' O follows from the continuity of fr, the inversion map g and group action
Finally, to show is G-equivariant, we note that for any h G.
n(er, hgHt) hgg ((hg)~ /1 hn(x,
(27)
Thus, n is a continuous G-equivariant function on X' x O.
By universality of {fe} X x V + y (under the supremum norm) and the orbit O being closed in V. {fa} is also universal on the subdomain x O. In particular, for any e, there exists some such that fo approximates e-well over all of X' O. Thus, it must also approximate c-well over X' x (H). But since n = f on X' x {H), this completes the proof.
We remark that Thm. 2 can be generalized to locally compact groups, via Jaworowski's equivariant extension theorem Jaworowski 1976 Lashof 1981]
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C ADDITIONAL EXPERIMENT DETAILS
C.I DETAILS FOR PATHFINDER TASK
The Pathfinder task from Long Range Arena Tay al.) 2021] is binary image classification problem: determine if there exists a path connecting two marked pixels. The image is flattened as a sequence input probe the spatial reasoning of sequence models (e.g. Transformers). It is known in the literature that standard Transformers perform poorly on Pathfinder with 1D positional encoding (unless with proper pretraining [Amos al. 2024)) whereas 2D positional encoding (PE) provides better inductive bias by mapping the pixel (, as ER(i] [i] + Ec(i] where ER, Ee E R are row and column embeddings (unlike ID-PE embedding E' E Imm xd for the flattened grid of size m).
We use a standard Transformer architecture for Pathfinder task taken from https://qithu: com/ com/mlpan www.mmpppnnn////mmmmmmmmme Nystromformeee We use the default model configuration (e.g.. 2 layers, 2 heads, 64 embedding dimensions, learnable positional encoding). We follow the standard protocol training the model for 62400 steps, linear learning rate decay schedule, and perform hyper-par perer--rreeterr search over the learning rate {0.0005, 0. 0,0001, 0.000083. We report more details of the performance for ID-PE and 2D-PE variants in Tab.
Table 4: Performance of Transformers in Pathfinder54 4 tasks, using different positional encoding (PE): ID-PE (learnable), 2D-PE (learnable, separate for rows and columns) with different group G. The accuracy averaged over 3 runs.
Method
Group C
Params.
Test Acc. (mean 1 std)
Train Acc. (mean 1: std)
ID-PE
1
370.5k
0.656 + 0.135
0.816 + 0.093
I
116.6k
0.818 +: 0.005
0.831 +: 0.006
(S,) 1024
112.5k
0.824 0.000
0.848 1 0.006
2D-PE
(Sp) 453
111.2k
0.827 1: 0.018
0.852 1 0.013
(S1e) 256
110.4k
0.814 0.020
0.840 = 0.009
C.2 DETAILS FOR SYNTHETIC SEQUENCE MODELING TASKS IN SEC. 5.2
We describe in details chosen synthetic tasks in Tab.
Equivariant Experiment Set-up The dataset per task contains 2500 examples of sequence length 10 and vocabulary size 7, chosen highlight the benefits of symmetry in data-scarce regimes. We train ASEN in the binary node classification setting to learn an equivariant mapping from the input node features to output node labels, both input and output represented as sequences. Our model is optimized using standard cross-entropy loss, with hidden dimension of 128, batch size of 64, learning rate of 0.01, and run on 40 epochs.
Invariant Experiment Set-up and Results We apply ASEN in the invariant setting on the following synthetic tasks: Palindrome, Intersect, Detect Capital and Longest Palindrome (c.f. Tab, and Tab, [5) As each task can have a different type of label, we switch to the regression setting using an L loss and train for 80 epochs. The experiment uses data-scarce regime where each task has 8000 training datapoints, denoted as I unit. As shown in Tab. 6 ASEN with the correct symmetry group outperforms its trivial symmetry counterpart across tasks and training dataset sizes r E {1, 1.5, 1..,2). 2). We further perform an ablation study on the effect of training set size. As shown in Fig. 9 for low-data regimes (r < 1.5), all tasks benefit from imposing invariance; with more training data (r 2 2), some tasks (c.g., Detect Capital) exhibit comparable performance between invariant ASEN and its non-invariant counterpart.
C.3 DETAILS FOR MULTITASK APPLICATIONS SEC. 5.2.1
In Fig. we conduct multitask training of ASEN across Intersect, Cyclicsum, and Palindrome, and observe that the performance on Intersect notably improved upon single task training. Meanwhile, Tab. shows that the multitask performance the other tasks (Cyclicsum, Palindrome) remains
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Table 5: Description of synthetic tasks, equivariant and invariant learning set-up, and their corre- sponding symmetry group.
Synthetic Tasks
Task
Equivariant
Invariant
Symmetry
Intersect
Given two sequences of length determine which elements of each sequence present the other
Determine the size of the inter- section of the sequences
Saja Saja S2: sequences can be reordered, and the two sequences can be swapped
Symmetric Difference
Given two sequences length determine which elements of each sequence are present in the other
Determine the size differ- the sequences
Palindrome
Given sequence of length determine where sequence has contiguous subsequence that palindrome of length k, one exists
Determine if the sequence has a contiguous subsequence that Sequence reversal palindrome of length k
Monotone Run
Given a sequence of length n, determine where sequence has contiguous subsequence that is strictly increasing or de- creasing of length k, one ex- ists
Determine if the sequence has a contigunus subsequence strictly increusing decreasing of length
Cyclic Sum
Given sequence length n. determine which cyclic contigu- subsequence of length & hus the largest sum
Find largest sum a cyclic contiguous subsequence of length
C, (cyclic shifts)
Cyclic Product
Given sequence of length determine which cyclic contigu- ous subsequence of length has
find the largest product of cyclic contiguous subsequence length
largest product
Detect Capital
N/A
Given a string of length n, re- True properly capitulized, meaning the string is all Upper- case, lowercase, or only has first letter capitalized
S. 1. as elements except first be permuted
Longest Palindrome
N/A
Given string length n. deter- S. mine the length the longest palindrome that can be con- structed using the characters the string
Table 6: Performance of two ASEN variants across tasks for varying data amounts, Within each model variant, columns indicate data used (in units): 1.5, and
ASEN (Loss !)
Invariant
Non-invariant
Task
1.5
2
1.5
2
Intersect
0.076 0.002
0.061
0.009
0.060
0.005
0.081
0.002
0.078 1: 0.0004
0.077 1: 0.001
Palindrome
0.250 = 0.04
0,170
= 0,01
0.125
0.05
0.270
0.250 +: 0.04
0.132 - 0.04
Detect Capital
0.053 1 0.005
0.056
1 0.007
0.068
1 0.016
0.075 1 0.011
0.066
0.004
0.063 1 0.007
Longest Palindrome
0.138 1 0.014
0.110
0.009
0.110 1 0.009
0.152 1 0.007
0.130
0.012
0.130 + 0.010
similar to the single task setting. Figure [0 shows the compute cost with respect to the training set size and the number of tasks, which grows linearly with both factors.
Table 7: Multi-task Test Losses (0.08 units)
Method
Cyclicsum
Palindrome
Single-task 0.3869
0.510
Multi-task
0.3839
0.537
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Published as a conference paper at ICLR 2026
Invariant
Non-Invarintt
Intersect
Palindrome
0.08
0.4
0.3
0.06
0.2
0.1
2
1.5
Data units r
Data units
Detect Capital
Longest Palindrome
0.10
0.08
Loss
0.06
0.04
1.5
2
1.5
Data units r
Data units
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.
250
Train per epoch (GPU seconds)
Single-task
Multitask Frack
200
Multitask Duak
Pask
Multitask Diyes
150
100
50
1.0
1.5
2.5
3.0
runit points)
Figure 10: Compute cost in multitask training (measured as training time per epoch NVIDIA A100 GPU device) with respect to the number of training set sizes r and the number of tasks Bask
C.4 DETAILS FOR TRANSFER LEARNING SEC. 5.2.2
For the invariant setting, we adopt weighted L1 regression loss for training (restricting all losses to between 0 and 1), and use standard L, loss for evaluation. We disable the TokenEmmedderr, due to diverse (invariant) target range across tasks. The tasks Palindrome, Intersect, Detect Capital and each learned
Longest Palindrome dataset is of size 5600 datapoints, with the chosen (finetuned) dataset restricted to 15 percent, or 840 datapoints. To probe the effect of transferring symmetry, also provide the ASEN bascline with the trivial symmetry (denoted as "NoSym")), which underperforms ASEN with the desired nontrivial symmetry group in the data-scarce regime.
instance of ASEN pretrained on the other three. Each task
We also ablate how the pretraining effect varies with input sequence length. We use the same tasks and set-up the transfer learning experiment described above, except using a smaller task dataset size with 4000 datapoints and varying the input sequence length over {6, 12, 18). As shown in Fig. II pretraining effect depends the task and the sequence length. Increasing the sequence length leads
20

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Published as a conference paper at ICLR 2026
to larger symmetry group, but also introduces greater inter-gropd group difference across tasks. Identifying optimal task mixture and designing more effective training curriculum an interesting direction for future work.
Rheaine trkk: Longesspal
Finelune ta6t Palhsrrme
Firetuee only...
Pretisan -romure
25
Testeno
0:2
Sequence length
Sequence lengh
Fineluwweer beit: DeteciCapisi
Frnelae bad: Intessec!
TEL
04
02
Suquente lengh
Sequirce length
Figure 11: Invariant pretraining ablation: test error of ASEN for finetune-only (blue bars) versus pretrain-finett finetune (orange bars) across varying input sequence lengths. Each subplot shows the pretraining effect for different finetune task.
C.5 OVERCONSTRAINED SYMMETRY
In Sec. 5.2 and Sec. 5.2.2 we show that ASEN incorporating the desired symmetries outperforms its trivial symmetry baseline across wide range of tasks and settings. As noted before, the symmetry groups considered in these tasks are equal to their 2-closure, and thus ASEN can accurately model the desired symmetry via the edge orbits whose automorphism
the 2-closure group. We also discuss in Sec. eoc.5jras that ASEN can learn more symmetries from data (i.e., learn to tie weights in the EdgeEndedder module), making robust when choosing
((2) to be smaller than the target symmetry. We now present negative example where ((2) > G; such a case, ASEN can fail.
Consider the task of determining the sign of a permutation, namely deciding whether an even or odd number of inversions used create a permutation. For example. for
4, the identity
permutation [1, 2, 3, 4] is even, whereas [2, 1, 3, 4| with one inversion is odd, and [2, 3. 1, 4] with two inversions is even. This task is invariant under the alternating group A,, while the 2-closure of A,, is S., meaning using ASEN taking the 2-closure symmetry breaking input would include significantly more symmetries.
Choosing permutations of length 7, train ASEN with ( S, and its trivial symmetry baseline using binary cross-entropy loss for this invariant classification problem. We find that ASEN performs better than chance (i.e., stuck loss of 0.69, the same random guess baseline). On the other hand, the trivial symmetry baseline eventually reaches a test loss of 0.28, outperforming ASEN with overconstrained symmetries.
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Reviews:
[{"review_id": "5msp3z2k5H", "reviewer": "Reviewer_8QJx", "summary": "This paper presents a new method for constructing equivariant neural networks by first having a base network equivariant to a larger base group and then fixing a symmetry-breaking input feature which is precisely invariant with respect to the target (sub)group. The authors focus on the specific case of base group being Sn, for which the authors draw upon a classical result that one can always construct a hypergraph of which automorphism group is equal to the target group and propose an approximate scheme that restricts the search space to graphs. The authors prove theoretical sanity-checks on equivariance and expressive power. Through experiments, the authors show that the proposed method can use different symmetries under one architecture and provide a diagnostic insight into the choice of the target group structure, and also show evidence that the proposed method exhibits knowledge sharing and transfer across tasks with different symmetries.", "questions": "I don't have particular questions but would like to hear the authors' opinions on W1.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "S1. While there has been prior work on architecture agnostic invariance/equivariance and knowledge sharing/transfer across symmetries [1-3], they consider taking an unconstrained base model and adding symmetry constraints through (randomized) symmetry breaking, while this work considers an opposite and original direction of starting at an overly constrained base model and reducing symmetry constraints using a fixed symmetry-breaking input. This is an interesting direction and could facilitate future work.\n\nS2. The paper is very well written and easy to follow.\n\nS3. The proposed method is simple and technically sound as far as I can confirm.\n\nS4. The experimental results support the claims made in the main text.\n\nS5. The shown connections to hypergraphs and higher-order GNNs are interesting and might lead to renewed interests of related literature.\n\n[1] Kim et al., Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance, NeurIPS 2023.\n\n[2] Mondal et al., Equivariant Adaptation of Large Pretrained Models, NeurIPS 2023.\n\n[3] Kim et al., Revisiting Random Walks for Learning on Graphs, ICLR 2025.", "weaknesses": "W1. There are several potential weaknesses stemming from the fact that the work takes an over-constrained base model and relaxes its constraints in downstream tasks. One weaknesses is related to practical implication of Theorem 2. For large groups G such as Sn as considered in this work, constructing universal and equivariant networks is in general quite hard and requires combinatorially large feature spaces [4] or some kind of randomized symmetry breaking themselves [5], making the setup of Theorem 2 unlikely in practice. Another potential weaknesses is that, with the base model overly constrained, the scope of knowledge it can accumulate from pretraining without knowing what downstream tasks are can be substantially limited. As far as I understand, pretraining in Section 5.2.2 is aware of downstream tasks and uses task-specific embedding modules, so it bypasses this problem; but in general, one may not know the details of downstream tasks during pretraining.\n\n[4] Maron et al., On the Universality of Invariant Networks, ICML 2019.\n\n[5] Abboud et al., The Surprising Power of Graph Neural Networks with Random Node Initialization, IJCAI 2021.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2FdOAJupci", "reviewer": "Reviewer_sRQM", "summary": "The paper presents a method to obtain foundational models equivariant to any chosen subgroup of a larger symmetry group while reusing a single backbone. It introduces a fixed, symmetry-breaking hypergraph whose automorphisms match the target subgroup, so the backbone effectively enforces just the chosen symmetries. The authors provide proofs of this property and show that their models inherit universality from a universal base model. They then present experiments evaluating the framework on human pose estimation, traffic flow prediction, and image classification benchmarks, along with synthetic tests that explore multitask and transfer learning.", "questions": "1. How can the presented baselines be scaled to test the scalability potential of the proposed model?\n2. In this direction, would it be possible to report scaling laws for this models: how do preprocessing and training costs grow with input size, the number of hypergraph automorphisms, and with $k$?\n3. At scale, when does this approach help with respect to a non-equivariant baseline?\n4. Which are the groups for which do you envision a particular difference in employing its $2$-closure?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 1, "strengths": "- The authors tackle a relevant problem: building foundation models equivariant with respect to adaptive symmetries. \n- The proposed solution, using symmetry breaking to realize adaptable group equivariance, is a brilliant idea.", "weaknesses": "1. Although the proposed solution is interesting, the analysis is broad but shallow. The paper aims to span both theory and practice, yet it defers the hardest, and most relevant, questions to future work, even though they are essential to validate the framework as a practical path to adaptive foundation models. In particular:\n\t- The limited analysis of scalability and computational cost is a critical omission, and should be of paramount importance in the design of foundation models.\n\t- A thorough analysis of how the $k$-closure approximation affects expressivity, equivariance, and performance is fundamental to assessing the framework’s practical relevance and implementability.\n2. Readability is limited: the introduction is technical from the introduction, the exposition is overly compact, and there are too few worked examples to illustrate the core mechanisms.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "b9LuJJ3MTp", "reviewer": "Reviewer_gDLb", "summary": "The paper proposes a simple framework to build neural networks that are equivariant to any subgroup of a given symmetry group. By conditioning the network on a symmetry-breaking input $v$ whose automorphism $Aut(v)$ equals the desired subgroup, the model naturally inherits the correct equivariance. The approach is general, easy to apply, and shown to work well on several examples involving rotation and reflection subgroups.", "questions": "The framework offers flexibility to easily test different equivariances. Do the authors view it mainly as a prototyping system, or do they expect it to be competitive with specialized architectures in practical applications?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- Presents an interesting and elegant idea that is both simple and flexible for constructing subgroup-equivariant networks.\n\n- Provides a solid mathematical foundation, with clear and rigorous proofs of the main results.\n\n- Includes several illustrative examples that effectively clarify the concept and demonstrate its broad applicability.\n\n- Offers a unifying framework that can reproduce or extend many existing equivariant architectures with minimal effort.", "weaknesses": "Overall, I find the paper is convincing and I am not aware of any similar work, therefore the weaknesses below should be regarded as minor:\n\n- Computational efficiency and scalability are not analyzed, leaving open how well the approach performs for larger models or groups with respect to alternative approaches.\n\n- The experiments demonstrate flexibility, but not whether this is the preferred method compared to specialized architectures; it may mainly serve as a prototyping tool.\n\n- The experimental validation is limited and does not convincingly show advantages on more challenging or large-scale tasks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hrJlpxjx4A", "reviewer": "Reviewer_VJcc", "summary": "The paper proposes using networks equivariant under a group H, together with a symmetry breaking input to obtain equivariance under a subgroup G of H. Specifically, H is taken to be the symmetric group on n elements, where n is the dimensionality of the input. Symmetry breaking is obtained by designing edge features that define a graph that has automorphism group $G^{(2)}$, the 2-closure of G. $G^{(2)}$-equivariance is obtained by using a graph network on the constructed colored graph. When G is 2-closed, this yields equivariance precisely under G.", "questions": "1. I am not very familiar with the prior work “Approximately Equivariant Graph Networks” by Huang et al. Is it correct to state that they propose using ordinary MLPs equivariant under the automorphism group of a fixed graph to obtain more expressive networks for that specific graph, whereas the submitted paper does the opposite of proposing to design a graph with a specific automorphism group G to obtain G-equivariant networks through a graph network?\n2. In table 1, the lowest error is consistently obtained by the non-equivariant version, which, if I understand correctly, corresponds to a unique identifier for each edge. Is this a standard approach in graph networks on fixed graphs?\n\nAs I found the paper interesting I have some potentially more philosophical questions.\n\n3. As far as I understand, the suggested approach requires knowing the orbits of the node pairs under G, but from this information we can also construct a basis for the equivariant linear maps (Ravanbakhsh et al, 2017). So is the following statement in the introduction fair? “(I) equivariant architectures typically require deriving and implementing group-specific equivariant layers, so substantial research and engineering must be done for architectural design whenever a new type of symmetry arises,”\n4. A linear $G$ equivariant layer from $\\mathbb{R}^n$ to $\\mathbb{R}^n$ under the permutation action $\\rho(g)$ can be defined from a general linear map $W\\in \\mathbb{R}^{n\\times n}$ through group averaging as $W_\\rho = \\sum_{g\\in G}\\rho(g) W \\rho(g)^{-1}$. Thus the same $W$ can define equivariant linear layers for different groups $G$ acting on $\\mathbb{R}^n$. Is this somehow analogous to how different symmetry breaking objects lead to equivariance under different $G$ for the same message passing layer in the paper?\n5. In some sense, the paper suggests an approach that enables the use of graph networks instead of equivariant MLPs. Aren't equivariant MLPs easier to work with, and more aligned with current hardware, due to the focus on linear layers?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "I found the paper well written and engaging. The proposed method is well thought out and deserves to be discussed at the conference in my opinion. Theorem 1 was especially interesting to me, showing that equivariant MLPs can be simulated by the proposed graph networks.", "weaknesses": "The experimental evaluation seems not so convincing to me: \n\n1. There are no comparisons to equivariant MLPs.\n2. In Figure 6 it looks like just training the yellow models longer will allow them to catch up. Is the effective training time here the same for the blue and yellow models or do the blue models train for longer due to training on multiple tasks? \n3. There is an example in the appendix where $G$ is not 2-closed, and then the method does not work.\n\n\n**Typos:**\n\nI believe that the symmetry group of the intersection operation in table 3 should not be a direct product since permuting the two sets does not commute with permuting internally in one set.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)