Any-Subgroup Equivariant Network (ASEN)

• ASEN is a design principle enabling neural networks to realize equivariance to arbitrarily chosen subgroups through mechanisms like subgroup decomposition, symmetry breaking, and graph automorphisms.
• It leverages algebraic group theory, neural architecture search, and representation theory to construct layers that maintain subgroup invariance, enhancing performance on structured data.
• The framework provides theoretical guarantees and demonstrates practical trade-offs in efficiency and parameter sharing when balancing full symmetry versus subgroup restrictions.

Searching arXiv for the cited ASEN papers and closely related work to ground the article. An Any-Subgroup Equivariant Network (ASEN) is a neural-network construction intended to realize equivariance with respect to a chosen subgroup rather than a single, fixed symmetry class selected a priori. Across the cited literature, the term is used for several related constructions: subgroup selection by group decomposition and neural architecture search in finite groups (Basu et al., 2021), finite-subgroup equivariance via graph automorphism groups and bilabelled-graph bases (Pearce-Crump et al., 2023), subgroup restriction by symmetry breaking inside a fully permutation-equivariant model (Goel et al., 19 Mar 2026), encoder–decoder latent decompositions separating invariant and equivariant components (Winter et al., 2022), and subgroup restriction of general equivariant architectures on reductive Lie groups (Batatia et al., 2023). Taken together, these works define ASEN less as a single canonical architecture than as a family of mechanisms for making subgroup equivariance selectable, constructible, or learnable.

1. Terminological scope and core definition

The central object shared by these works is equivariance. Let $G$ be a group acting on feature spaces $\mathcal X$ and $\mathcal Y$ through linear representations $\Gamma_x:G\to GL(\mathcal X)$ and $\Gamma_y:G\to GL(\mathcal Y)$. A map $\phi:\mathcal X\to\mathcal Y$ is $G$-equivariant if

$$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$

This formulation appears explicitly in the group-decomposition treatment of autoequivariant networks (Basu et al., 2021).

Within that common formalism, ASEN refers to architectures that can realize equivariance to arbitrary subgroups drawn from a larger ambient symmetry class. In the finite-group setting, this may mean composing subgroup-equivariant layers associated with factors $H_1,\dots,H_m$ whose semidirect or direct products reconstruct a larger group (Basu et al., 2021). In the permutation setting, it may mean taking a fully $S_n$-equivariant base model and restricting equivariance to $\mathcal X$0 by appending an auxiliary structure $\mathcal X$1 whose automorphism group is $\mathcal X$2 or $\mathcal X$3 (Goel et al., 19 Mar 2026). In graph-based constructions, it may mean realizing any finite subgroup $\mathcal X$4 through a graph $\mathcal X$5 with $\mathcal X$6, then building all linear $\mathcal X$7-equivariant maps between tensor-power features (Pearce-Crump et al., 2023). In representation-theoretic settings for reductive Lie groups, it means restricting a $\mathcal X$8-equivariant layer construction to any closed subgroup $\mathcal X$9 by representation restriction and intertwiner-based layer design (Batatia et al., 2023).

A plausible implication is that ASEN is best understood as a design principle: start from a symmetry-rich model class and introduce mechanisms that permit exact or approximate restriction to a specified subgroup.

2. Group decomposition and architecture search

One formulation of ASEN originates in the result that equivariance to a large group can be reduced to equivariance to smaller groups from which it is constructed (Basu et al., 2021). For a semidirect product $\mathcal Y$0, with restricted actions $\mathcal Y$1 and $\mathcal Y$2, the paper states that a linear map $\mathcal Y$3 is $\mathcal Y$4-equivariant under $\mathcal Y$5 if and only if it is $\mathcal Y$6-equivariant under $\mathcal Y$7 for $\mathcal Y$8 (Basu et al., 2021). By induction, if $\mathcal Y$9 decomposes as a chain of semidirect or direct products of subgroups $\Gamma_x:G\to GL(\mathcal X)$0, then $\Gamma_x:G\to GL(\mathcal X)$1-equivariance is equivalent to equivariance to each factor.

This theorem leads directly to a constructive ASEN recipe for MLP-like layers. A fully connected layer with input index set $\Gamma_x:G\to GL(\mathcal X)$2, output index set $\Gamma_x:G\to GL(\mathcal X)$3, and weight matrix $\Gamma_x:G\to GL(\mathcal X)$4 is made $\Gamma_x:G\to GL(\mathcal X)$5-equivariant by tying together all entries $\Gamma_x:G\to GL(\mathcal X)$6 lying in a common orbit of the joint action of $\Gamma_x:G\to GL(\mathcal X)$7 on $\Gamma_x:G\to GL(\mathcal X)$8 (Basu et al., 2021). Instead of computing orbits under the full group $\Gamma_x:G\to GL(\mathcal X)$9, the proposed fast construction computes a combined orbit index using subgroup actions only, with complexity

$\Gamma_y:G\to GL(\mathcal Y)$0

rather than

$\Gamma_y:G\to GL(\mathcal Y)$1

The resulting weight matrix is defined through shared parameters $\Gamma_y:G\to GL(\mathcal Y)$2, and layerwise nonlinearity preserves equivariance (Basu et al., 2021).

The same work treats subgroup choice as a discrete architecture-search problem. The search state is a binary mask over candidate subgroups, possibly extended with augmentation choices and channel-size options in a GCNN setting. Actions toggle subgroup inclusion or modify augmentation/channel flags. A deep Q-network $\Gamma_y:G\to GL(\mathcal Y)$3 is trained with Bellman updates,

$\Gamma_y:G\to GL(\mathcal Y)$4

and the reward can combine validation accuracy with a parameter-count penalty (Basu et al., 2021). In this usage, ASEN is not only a layer-construction method but also a search space over equivariance patterns.

3. Symmetry breaking from full permutation equivariance

A later formulation begins from a fully permutation-equivariant base model

$\Gamma_y:G\to GL(\mathcal Y)$5

satisfying

$\Gamma_y:G\to GL(\mathcal Y)$6

for every $\Gamma_y:G\to GL(\mathcal Y)$7 (Goel et al., 19 Mar 2026). In practice, that work takes $\Gamma_y:G\to GL(\mathcal Y)$8 and implements each layer as a standard Message-Passing Neural Network with injective MLPs $\Gamma_y:G\to GL(\mathcal Y)$9 and an injective multiset aggregator $\phi:\mathcal X\to\mathcal Y$0 (Goel et al., 19 Mar 2026).

Subgroup restriction is then induced by a fixed auxiliary symmetry-breaking input. The model appends a “hypergraph” $\phi:\mathcal X\to\mathcal Y$1 whose automorphism group is exactly or approximately the desired subgroup $\phi:\mathcal X\to\mathcal Y$2, and defines

$\phi:\mathcal X\to\mathcal Y$3

If $\phi:\mathcal X\to\mathcal Y$4, then for all $\phi:\mathcal X\to\mathcal Y$5,

$\phi:\mathcal X\to\mathcal Y$6

while in general $\phi:\mathcal X\to\mathcal Y$7 is not equivariant outside $\phi:\mathcal X\to\mathcal Y$8 (Goel et al., 19 Mar 2026). The mechanism is therefore explicit symmetry breaking inside an ambient $\phi:\mathcal X\to\mathcal Y$9-equivariant architecture.

Because exact symmetry-breaking inputs are computationally hard to find, the paper relaxes the construction through 2-closure. Given $G$0, its 2-closure $G$1 is the largest subgroup of $G$2 acting identically to $G$3 on unordered pairs $G$4. If $G$5 is partitioned into $G$6-orbits $G$7, then the edge-orbit labelling tensor $G$8 satisfies

$G$9

and equivalently

$$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$0

(Goel et al., 19 Mar 2026). The construction computes pair orbits of the diagonal subgroup $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$1 via Schreier–Sims; orbit decomposition is stated as $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$2 time for generators $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$3, with preprocessing polynomial in $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$4 and $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$5 (Goel et al., 19 Mar 2026).

This symmetry-breaking view differs conceptually from group decomposition. Rather than assembling subgroup-equivariant components, it starts with maximal permutation equivariance and uses an auxiliary input to lower the symmetry of the realized function.

4. Finite-subgroup realization through graph automorphisms

A distinct ASEN construction is based on the observation that finite groups can be realized as graph automorphism groups (Pearce-Crump et al., 2023). For a graph $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$6 with $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$7, the automorphism group is

$$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$8

where $$\phi(\Gamma_x(g)x)=\Gamma_y(g)\phi(x)\qquad \forall g\in G,\;x\in\mathcal X.$$9 is the adjacency matrix. This group acts on $H_1,\dots,H_m$0 and on tensor powers $H_1,\dots,H_m$1 through

$H_1,\dots,H_m$2

(Pearce-Crump et al., 2023).

The paper gives a full characterization of linear $H_1,\dots,H_m$3-equivariant maps

$H_1,\dots,H_m$4

If $H_1,\dots,H_m$5 is the matrix of $H_1,\dots,H_m$6, equivariance is equivalent to

$H_1,\dots,H_m$7

with $H_1,\dots,H_m$8 flattened to $H_1,\dots,H_m$9 form (Pearce-Crump et al., 2023).

A classical result summarized there states that

$S_n$0

For a $S_n$1-bilabelled graph $S_n$2, the associated basis matrix $S_n$3 is indexed by $S_n$4 and has entries

$S_n$5

(Pearce-Crump et al., 2023). Consequently, each linear layer can be written as

$S_n$6

In practice one truncates to generating bilabelled graphs using the “no free components” result or Frobenius duality, then stacks these linear maps with pointwise nonlinearities to obtain an $S_n$7-equivariant network (Pearce-Crump et al., 2023).

The extension to arbitrary finite $S_n$8 uses Frucht’s theorem: every finite group $S_n$9 is isomorphic to the automorphism group of some finite graph $\mathcal X$00. The procedure is therefore to choose or construct such a graph and then apply the same homomorphism-matrix construction, yielding an $\mathcal X$01-equivariant network under the chosen embedding $\mathcal X$02 (Pearce-Crump et al., 2023). This places ASEN in direct contact with algebraic graph theory and invariant-theoretic characterizations of equivariant linear maps.

5. Representation-theoretic and encoder–decoder generalizations

Two further strands broaden ASEN beyond finite permutation groups.

For reductive Lie groups, an ASEN can be defined for any closed subgroup $\mathcal X$03 of a reductive Lie group $\mathcal X$04 by restricting finite-dimensional $\mathcal X$05-representations to $\mathcal X$06 (Batatia et al., 2023). If $\mathcal X$07 and $\mathcal X$08 are $\mathcal X$09-modules, their tensor product carries the representation $\mathcal X$10, and reductivity yields a Clebsch–Gordan decomposition

$\mathcal X$11

After restriction to $\mathcal X$12, every layer must intertwine the $\mathcal X$13-actions on input and output features (Batatia et al., 2023). The proposed ASEN layer on point-cloud or graph data consists of one-body basis functions $\mathcal X$14, aggregated one-body features $\mathcal X$15, $\mathcal X$16-body correlations, reduction to irreps through Clebsch–Gordan intertwiners, equivariant linear mixing, and norm-based gating. Group convolution on $\mathcal X$17,

$\mathcal X$18

is presented as the convolutional analogue of this construction (Batatia et al., 2023). This formulation treats ASEN as a subgroup-restriction principle internal to a general equivariant representation-theoretic framework.

A different generalization appears in unsupervised learning, where the ASEN framework separates a latent representation into an invariant code and a group-action component (Winter et al., 2022). Given a group action $\mathcal X$19, one encodes

$\mathcal X$20

with $\mathcal X$21 required to be $\mathcal X$22-invariant and $\mathcal X$23 predicting the group element used to reconstruct the input through

$\mathcal X$24

The basic training criterion is the reconstruction loss

$\mathcal X$25

optionally augmented by invariance and equivariance regularizers (Winter et al., 2022). The paper further states that $\mathcal X$26 must be strictly $\mathcal X$27-invariant, that $\mathcal X$28 is equivariant up to the stabilizer $\mathcal X$29, and that $\mathcal X$30 induces an isomorphism of the orbit $\mathcal X$31 with the coset space $\mathcal X$32 (Winter et al., 2022).

These two strands suggest a broader conceptual split. In one reading, ASEN is a layerwise subgroup-equivariant architecture built from intertwiners and symmetry-aware parameter tying. In another, it is a representation-learning framework that factors out invariant content and explicitly models the group action needed to reconstruct or realign the data.

6. Guarantees, empirical findings, and limitations

Theoretical guarantees differ across formulations. In the symmetry-breaking construction, an “ASEN Universality” theorem states that if $\mathcal X$33 is universal over continuous $\mathcal X$34-equivariant maps and $\mathcal X$35 has stabilizer $\mathcal X$36, then the restricted family $\mathcal X$37 is universal over continuous $\mathcal X$38-equivariant functions $\mathcal X$39 (Goel et al., 19 Mar 2026). The same paper states that any order-1 $\mathcal X$40-MLP composed of $\mathcal X$41-equivariant linear maps on $\mathcal X$42 plus entrywise nonlinearities can be approximated to arbitrary accuracy by ASEN with $\mathcal X$43 and an MPNN base (Goel et al., 19 Mar 2026). In the reductive Lie-group formulation, stacking the described equivariant blocks yields a universal approximator of any $\mathcal X$44-equivariant map, as cited there to Appendix E of the source paper (Batatia et al., 2023).

Empirical results also vary by instantiation.

Setting	Reported finding	Source
G-MNIST, G-Fashion-MNIST, CIFAR10, SVHN, RotMNIST, ASL, EMNIST, KMNIST	AENs find the right balance between equivariance and network size; in high-performing GCNNs, group equivariance is the most dominating factor	(Basu et al., 2021)
Human-3.6M, METR-LA, Pathfinder-64, synthetic sequence tasks	A single network equivariant to multiple permutation subgroups outperforms both separate equivariant models and a single non-equivariant model	(Goel et al., 19 Mar 2026)
Pathfinder-64	1D-PE baseline ≈ 0.66; local 2D-PE with patch group $\mathcal X$45 ≈ 0.82	(Goel et al., 19 Mar 2026)
Synthetic multitask / transfer	multitask training yields up to 30 % faster convergence; transfer pretraining reduces fine-tune error by up to 40 %	(Goel et al., 19 Mar 2026)
TopTagger	AUC≈0.987 with ≈$\mathcal X$46 parameters for a Lorentz-group ASEN	(Batatia et al., 2023)
ModelNet10	~96% accuracy using only three ASEN layers in the $\mathcal X$47 example	(Batatia et al., 2023)

The limitations are likewise construction-specific. In the symmetry-breaking approach, when the target subgroup is not totally 2-closed, the relaxation can introduce additional symmetries; the paper states that ASEN learns extra symmetries from data if $\mathcal X$48 but fails if $\mathcal X$49, with parity detection as the explicit example where the target invariance is to $\mathcal X$50 but the 2-closure is $\mathcal X$51, leaving ASEN “stuck at chance” (Goel et al., 19 Mar 2026). The same source notes that too large a subgroup can over-constrain the model and degrade performance, while too small a subgroup slows convergence but may be recovered through learned ties in the edge embedder (Goel et al., 19 Mar 2026). In the reductive Lie-group formulation, applicability is limited to reductive Lie groups with finite-dimensional irreps, and basis truncation in body order and irrep range is a central expressivity–cost trade-off (Batatia et al., 2023).

A recurring misconception is that more symmetry is always preferable. The cited results do not support that simplification. One source explicitly reports that choosing a smaller subgroup $\mathcal X$52 can outperform full $\mathcal X$53-equivariance on traffic forecasting (Goel et al., 19 Mar 2026), and another frames the search itself as finding the right balance between equivariance and network size (Basu et al., 2021).

7. Conceptual synthesis and research directions

Across these works, ASEN occupies the intersection of equivariant deep learning, neural architecture search, algebraic graph theory, and representation theory. One line builds subgroup equivariance by decomposition of large groups into smaller factors and searches over subgroup sets (Basu et al., 2021). Another realizes finite subgroups through automorphism groups of graphs and characterizes all admissible linear layers via bilabelled-graph homomorphism counts (Pearce-Crump et al., 2023). A third treats subgroup equivariance as controlled symmetry breaking inside a universal $\mathcal X$54-equivariant base model, using auxiliary orbit-labelled structures and 2-closure relaxations (Goel et al., 19 Mar 2026). Further extensions connect the idea to invariant/equivariant latent decompositions in unsupervised learning (Winter et al., 2022) and to subgroup restriction for reductive Lie-group architectures built from Clebsch–Gordan reductions (Batatia et al., 2023).

This suggests that the most stable meaning of ASEN is operational rather than architectural: a network family in which the symmetry class can be specialized to an arbitrary subgroup by parameter tying, auxiliary symmetry-breaking inputs, graph-based realization of automorphism groups, or restriction of a larger equivariant representation. A plausible implication is that ASEN provides a route toward models that are not locked to a single hand-chosen symmetry, especially in settings involving grids, sets, graphs, sequences, or geometric data with heterogeneous or task-dependent invariances.

Another plausible implication is that the main unresolved issue is not only universality but subgroup specification: exact realizability, computational cost of subgroup-dependent preprocessing, and the approximation gap induced by relaxations such as 2-closure. The cited literature therefore places ASEN at a technically rich point between strict equivariant design and adaptive symmetry selection, with different formulations emphasizing exact algebraic characterization, efficient construction, or flexible deployment under changing symmetry requirements.