Reducing Symmetry Increase in Equivariant Neural Networks
Authors: Ning Lin
Jiacheng Cen
Anyi Li
Wenbing Huang
Hao Sun
Venue: ICLR 2026 Poster (2026)
Keywords: Equivariant Neural Networks
Symmetry Increase
Compact Group
Isotropy Subgroup
Orbit Type
Curie’s Principle
Primary area: learning on graphs and other geometries & topologies
TL;DR: We show that symmetric inputs can cause ENNs to lose orientational information via feature-space–induced symmetry increase, and we provide guaranteed, computable feature-design rules validated on synthetic data and QM9.

Abstract: Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend beyond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase in specific cases, a rigorous understanding of its underlying causes and general reduction strategies remains lacking. In this paper, we provide a detailed and in-depth characterization of symmetry increase together with a principled framework for its reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits an infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop a computable algorithm to derive this infimum, and propose practical guidelines for feature design to prevent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, our guidelines effectively reduce symmetry increase. To complement our theoretical findings, we provide visualizations and experiments on both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.

Published as a conference paper at ICLR 2026
REDUCING SYMMETRY INCREASE IN EQUIVARIANT NEURAL NETWORKS
Ning Lin, Jiacheng Cen, Anyi Li, Wenbing Huang", Hao Sun' Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China [ningl i i Jiace. en, 14 ]i_anyi] @outlook com, (hwenbing, haosun ( @ruc. edc.edduccm
ABSTRACT
Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend be- yond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase specific cases, rigorous understanding of its underlying causes and general reduc- tion strategies remains lacking. In this paper, we provide detailed and in-depth characterization of symmetry increase together with a principled framework for reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop computable algorithm to derive this infimum, and propose practical guidelines for feature design to pre- vent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, guidelines effectively reduce sym- metry increase. To complement our theoretical findings, we provide visualizations and experiments both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.
INTRODUCTION
C3
Equivariant Neural Networks (ENNs) have become cornerstone of modern ma- chine learning, empowering numerous applications in scientific fields ranging from molecular dynamics to materials de- sign (Bronstein et al., 2021; Huang & Cen, 2026). By building physical symmetries, these models achieve remarkable data effi- ciency and generalization capabilities when representing complex geometric structures.
du
O
(a) 3-fold
(b) 4-fold
Figure 1: k-fold structures.
Despite their success, ENNs exhibit a critical vulnerability when processing symmetric inputs: their expressivity can degrade, leading to loss of information. This phenomenon, which we term symmetry increase, occurs when the output representation becomes invariant to transformations that not symmetries of the original input itself. A canonical example arises when processing f-fold symmetric structures. These objects, visualized in Fig. 1, possess a specific dihedral symmetry, yet an ENN will map their distinct rotated versions an identical feature, erasing their orientation.
This type of degradation has been documented in previous work. Empirically, it has been observed that the degradation depends the feature space, particularly for symmetric inputs (Joshi al., 2023). Theoretically, research on ENNs has shown that for k-fold symmetries, selecting only low-degree
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features can cause all rotated inputs to collapse into a single representation (Cen al., 2024). This ENN-specific issue is a modern manifestation of a general phenomenon observed in other fields. It has been linked to physical principles such as Curie's 's Principle (Smidt et al., 2021) and described using the concept of orbit types (Kaba & Ravanbakhsh, 2023).
However, existing analyses provide an incomplete picture and lack predictive framework. In our analysis, this degradation of k-fold caused by symmetry increase can be categorized into three distinct types: full degeneration, axial degeneration, and half degeneration (see Fig. 2)¹. The work of Joshi et al. (2023), while empirically important, does not theoretically explore the cause of the degradation. The collapse-to-zent
theory proposed by Cen et al. (2024) addresses only full degeneration, which is the most extreme case identified by our analysis. The broader principles discussed by Smidt et al. (2021) and Kaba & Ravanbakhsh (2023) lack a rigorous mathematical description, and the solutions proposed, such as in Kaba & Ravanbakhsh (2023), often involve relaxing the equivariance constraint itself, rather than providing a solution within the equivariant framework.
In this paper, we fill this gap by providing comprehensive mathematical characterization of symmetry increase. Our main contributions are briefly listed as follows:
In $ 3, we prove for any given feature space and input symmetry group, that the increased symmetry is bounded from below by a unique symmetry infimum, which is determined entirely by the algebraic structure of the feature space.
In $ 4. we develop computable algorithm to derive this infimum by analyzing the orbit types. This provides practical guidelines for predicting and controlling potential symmetry increases, thereby preventing harmful symmetry increases in feature design.
In $ 5, we demonstrate that under regularity conditions, such as the manifold hypothesis for data, method can fully reduce symmetry increase. Specifically, for most equivariant maps or for ENNs with sufficient approximation capabilities, the output symmetry will be precisely this predictable infimum, preventing orientational information loss.
In & 6, we complement our theoretical findings with empirical evidence. We provide visual- izations illustrate the proposed concepts and present experimental results on both synthetic datasets and the real-world QM9 dataset, which validate our theoretical predictions and demon- strate the practical effectiveness of our framework.
PRELIMINARIES
Group action and representation. Consider the action of group G on set X, denoted by px-
This action is a map that assigns to each element g E G a transformation px (g) : X + X, such that Px (9192) = pxx (#1)px (g2). We call such set X a G-set. In particular, if X is vector space and Px (g) is a linear transformation for all g E G, we call X a G-representation".
Equivariant map. For maps between two G-sets X and Y, an equivariant map is one that respects the group action, meaning that the output transforms accordingly when the input is transformed. Formally, map f: X Y is equivariant if for all g E G and x E X:
(1) Example 2.1 (Equivariant Encoding of Point Clouds). The symmetry group is G H x Sn, where H typically the special orthogonal group SO(3) or the orthogonal group O(3), and Sn is the permutation group. We consider features that are invariant to both permutation and translation. To achieve translation invariance, the input representation X is the space of centered point clouds, where H acts the coordinate of each point and Sn permutes the points. To achieve permutation invariance, the final output are designed to be invariant with respect to Sn, This means the feature representation of interest is a direct sum of specified irreducible representations of H. The task of equivariant encoding is then to learn an equivariant map f x +Y
f(px())(x)) py (f)(())
Data symmetry. For a G-set X, the action partitions the space into orbits, G(z) ;= (g(x)) I E G}. The subgroup that fixes a point is its isotropy subgroup, Ge := {g E G g(x) = z). The conjugacy
I Althoug we choose k-fold as an illustrative example, our theories are applicable to general cases. Unless otherwise specified, all groups are assumed to he compact Lie groups, and all vector spaces are finieediimenioonal real vector spaces.
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class (G2) of this subgroup is the orbit type of z. The set of all points with orbit type (H) is ( and the set of points fixed by all elements of subgroup H is the fixed-point set X
These concepts distinguish between the global symmetry of the space and the intrinsic symmetry of an object. The group G represents the global symmetry, transforming the object between different reference frames. An orbit G(x)) thus represents a single physical object in all of its possible orientations. This object possesses its own intrinsic symmetry. mathematically defined by the isotropy subgroup G2 for any point r on its orbit. While the specific subgroup is cume-dependent, its structure up to conjugation constant. The orbit type (G,) therefore serves as a reference-frame independent identifier that precisely describes the physical object's s intrinsic symmetry. The set of all possible orbit types, (X)),, , thus catalogs all distinct symmetries that objects in the space can possess.
Example 2.2. Now apply the setup from Ex. 2. Consider the action of G == 0(3) x Sk+1 on centered point cloud space X for 2. Let x E X be the set of vertices of a k-fold in the xOy--lane:
x x= (2g.21........ 21....;;ee.. where I; (cos(2im) / k)) sin(2in/ (02ii//k...1 . 0) for 0.
(2)
with my at the origin. The generators of G2 include:(1) A rotation about the 2-axis combined with cyclic permutation of z1...., Ik. (2) A reflection across the x0 z02z-plane combined with product of transpositions. (3) A reflection across the xOy-plane combined with the identity.
Considering the projection map x ()) = where (g, о) is a pair consisting of geometric transformation g E (() and permutation a E Sk, we find that geometric symmetry x (G₂) is the dihedral group with horizontal reflection of order 4k, denoted by the Schoenflies symbol DEh-
The symmetry of data can be altered by an equivariant map. The following theorem shows that an equivariant map does not decrease symmetry.
Theorem 2.3 (Curie's principle, Kaba & Ravanbakhsh (2023), Thm. 1). Let X + Y be G-equivariant map. For x E X, the isotropy subgroup of x is contained in that of its image fxx), ie.
G2 S GG(())
(3)
Such an increase in symmetry becomes unavoidable the feature space Y cannot support the input's symmetry. If the orbit type (G₂) is not present in Y (i.e., (Gx) & Oo(Y)), then the equality G2 = ((a) is impossible (otherwise, Gftm) becomes an isotropy subgroup, implying (G₂) E OY()). which leads to a contradiction). and the symmetry must therefore strictly increase. This increase leads to degeneration, any transformation g in the larger group Gf(z) that is not in G2 will map the distinct inputs and g(x) to the same output, since f(g(x)) = o( (x)) = fxx).
Fig. 2 illustrates three possible types of such degenerations for the k-fold structure from Ex. 2.2.
THE INFIMUM OF SYMMETRY
Half Dageneration
This section establishes that symmetry increase is governed by an infimum determined by the feature space. We prove the existence of this infinum and show that its coincidence with the input symmetry is necessary condition for an cquivariant map to preserve symmetry.
Full Degeneration
Axial Degeneration
Input Space
Output Space
3.1 THE INFIMUM OF SYMMETRY
Figure 2: Three types of degeneration of k-fold.
The symmetry of data can increase after being transformed by an equivariant map. This increase can be an intentional design choice, it can arise from subtle properties of the feature space. For instance, in the task from Ex. 2. the requirement for permutation-invarian permutation group S, is naturally introduced into the isotropy subgroup of any output. This type of designed, unavoidable symmetry increase is formalized by the kernel of the group action on the feature space. We define the kernel of the action px the set of group elements that fix every point in X: kerpx == وg G g(x) = x, Vx E EX}
features means that the
Distinguishing between intentional and unintended symmetry increase is crucial. We begin by assuming that the group action is faithful, i.e., it has a trivial kernel with kerpy = {e). The case of a nontrivial kernel will be discussed in the next subsection.
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To characterize the behavior of symmetry increase within a representation X, we first establish partial order to compare different symmetries. An orbit type (H1) is considered greater than or equal to another, (H2), written (H1) 2 (H2), if H1 contains a conjugate of Hz. This ordering reflects that a larger orbit type corresponds to a higher symmetry.
With this framework, we are interested in the lower bound of orbit types that can be reached from a point x with a specific isotropy group H = Gz. The analysis can be framed around any closed subgroup H. as the set of all possible isotropy subgroups precisely the set of all closed subgroups of G (see, e.g., Field (2007); Mostow (1957)). The fixed-point space X contains points of all higher orbit types. This leads to the following powerful theorem, which guarantees that the lower bound on symmetry increase is unique, corresponding to an isotropy subgroup that is unique up to conjugation.
Theorem 3.1 (Uniqueness of Minimal Type). Let X be a representation of a compact Lie group G. For any closed subgroup H C G. unique minimal orbit rype exists among the points in the fixed-point subspace XH. In particular, if (H) E Oa((H)) then (H) is the minimal orbit type within that subspace.
The uniqueness guaranteed by this theorem allows us to define the symmetry infimum, denoted by Tx(X, H), as this unique minimal orbit type. In the context of symmetry increase, we are concerned with the relationship between I(Y, G2) and (()). An unexpected symmetry increase occurs if (()) > IG(Y, Gz). The desired behavior for an equivariant map is captured by the following definition. For map between Gesets X and Y. we define an isovariant map as one that strictly
preserves symmetry for all E X:
G2 = G(())
Using the concept of the symmetry infimum, can provide necessary condition for the existence of isovariant maps. In & 5.1, we will see that this condition is in fact not sufficient for equivariant maps between representations, even when we assume a trivial kernel kerpy = =ee].
Theorem 3.2. A necessary condition for the existence of isovariant map between G-sets X and Y is that (X) C OY)).. When X and Y are representatonns of a compact Lie group G, this is equivalent to the condition that TY,Y, H) (H) all (H) E (x(X).
3.2 EQUIVARIANCE WITH NON-TRIVIAL. KERNELS
When the feature space Y is restricted to have a non-trivial kernel, i.e., ker PY * {e), the definition of an isovariant map becomes too restrictive. Since every isotropy group Gy in Y contains the kernel, any subgroup H not containing ker py cannot occur an isotropy subgroup. Consequently, for any map/ f X Y, an input isotropy subgroup G, that does not contain the kernel must increase.
To formalize this unavoidable symmetry increase, we introduce an operator py. In the discussion following Ex. 2.2, the presence of the S kernel forces the input isotropy subgroup G2 to become at least DRA Sk+1 in the feature space. We generalize this observation. The operator py (H) is defined as the smallest subgroup containing H that is compatible with the action on Y. given by the projection py xy .yy where πy G + G/kerpy is the natural projection. This operator is idempotent (pž- = py) and maps any isotropy subgroup in Y to itself. These properties reveal why increasing is unavoidable. For any equivariant map f, the relation G2 C G(z) must hold. Applying the py operator to this inclusion gives:
G2 S (Gx) E (pr(Gr = =Gt)-
(5)
This unavoidable increasing from G. to py (G,) means our goal is not to preserve G2 itself, but to ensure no additional symmetry is introduced beyond py (Gz). This leads to more practical definition. We say a map f X + Y is an isovariant map relative to Y if for all x E X satisfies
py (Gx) Gf())
py (G.) Pv (G))))
When the kernel is trivial this definition reduces to that of standard isovariant map. With this refined goal, we can state a necessary condition for the existence of such maps.
Theorem 3.3. A necessary condition for the existence of isovariant map relative to Y from G-set X to a G-set Y is that (py (H)) E OY(Y) for every (H) E (x(X). When X and Y are representations, this is equivalent to the condition that IY H) (pv (pv(H) for all (H) E O6(X).
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a
Axis Type
1.0
(Full Deg.)
(Full Deg.)
(Axial
0.8
Rotation Rate Around the Axis
4Axxial Deg.)
1=7
0.
Change the Rotation Axis
b
R10)
= 8
1 == 10
1= 11
0.2
Rotate around the Axis
0.0
Figure 3: Visualization of representation spaces. (a) A k-fold structure is reoriented onto multiple planes. (b) Each further rotated about the perpendicular axis. (c) All structures are embedded and projected into 2D. Marker shapes denote rotation axes, and colors denote rotation rates. Full degeneration appears 0, 1, and axial degeneration
2,
4 COMPUTATION OF ORBIT TYPES
The computation of orbit types is a classical problem, with most established results focusing on irreducible representations, often in the context of bifurcation theory. In representation learning tasks, however, we utilize feature spaces containing high multiplicities these representations, which requires us supplement the existing computational frameworks.
4.1 ORBIT TYPES HIGH-MMLTIPPICCTY REPRESENTATIONS
For compact Lie group G, any representation space X can be uniquely decomposed into direct sum of its irreducible components. For SO(3), this decomposition is written as
(B,(X.V
X
1==0
where Vi-i, is the irreducible representation corresponding to the space of spherical harmonics of degree lo, and m(X, Vi=i) is its multiplicity. For G O(3), the decomposition is similar, but the irreducible representations must also be distinguished by parity, denoted V1=1 and Vi=i,
We begin with the foundational criterion for identifying isotropy subgroups, first established necessary condition by Michel (1980).
Theorem 4.1 (Michel's Criterion, Michel (1980), App. A). Let V be a representation of group G. A necessary condition for a closed subgroup H to be isotropy subgroup in V is that for any adjacent closed subgroup H' 2 H. the dimension of the fixed exed-point subspace strictly decreases:
dim V
dim H
(8)
While this condition not sufficient for all representations, its sufficiency can guaranteed under certain common conditions. We define representation V of group G as high-multiplicity representation if every non-zero isotypic component corresponding to an irreducible representation V,, its multiplicity m(V, V,) is greater than dim G.
Propasition 4.2. For a high-multiplichh representation V, the necessary condition stated in Thm. 4.1 also sufficient.
This criterion is particularly powerful for two reasons. First, it offers computationally convenient method the form a chain recursion, which only requires checking adjacent subgroups. The dimensions of the necessary fixed-point spaces can be calculated via the trace formula (Golubitsky
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Algorithm I: Orbit Type Test for High-
Algorithm 2: Symmetry Infimum Cal-
Multiplicity Representations
culation
Data: Symmetry group G;
Data:
Symmetry group G;
Closed subgroup H C G:;
Closed subgroup H C G; Rep. V of G.
High-Multiplictty Rep. Vof of G
Result: is_ _.s_in((H,,, (V)))
Result: Symmetry inf. Io(V, H).
if is s_inn((f),, (a(V)) (o()) then
Let set S == } C G to be all adjacent closed supergroups of H in G;
2 return (H));
2 O + Ø;
end
4 Let set S = {H.} to be all closed
3 du + dim V#:
supergroups of H in G;
4 for H; in S do
5 O + 2;
dn <- dim VH::
6 for H, in S do
if d1 dæ, = 0 then
7
is in((H), Oc(V)) then
return False;
8
Add (H,) to O
end
9
end
9 end
10 end
10 return True;
11 return min(O);
et al., 1988), Second, the sufficiency condition is frequently met in our applications, as it holds for all finite groups and for feature spaces with a high number of channels. Based on the result of Prop. 4.2, we design an orbit type test Algo. 1 and a symmetry infimum calculation Algo. 2. Using the two algorithms described above, we have characterized all instances of symmetry increase for the closed subgroups of SO(3) or O(3) in the representations V=l and V for r >> 3, respectively, see $ C.4.
1=
We now illustrate our algorithms with simple example.
Example 4.3. We illustrate our algorithms by calculating the orbit type and symmetry infimum for the geometric symmetry DEt. (k > 2) of the k-fold from Ex. 2.2, considered as subgroup of O3)). The calculation is performed in the high-mulliplicity representation space Y = Viet (r >> 3, lo > 0). Here we provide only sketch of the derivation, the full procedure provided in C.3.
First, we apply the orbit type test from Algo. 1. This involves comparing the dimension of the fixed-point space of Duh with that of its adjacent supergroups (e.g.. Dpk.h and, for k = On). The analysis shows that (Den) is an orbit type if and only if ly 2 k and los k have the same parity. Next, we apply the symmetry infimum calculation from Algo. 2. This requires identifying the minimal orbit type among all supergroups of Dkb, including non-ale adjacent ones like Doch and 0(3). The final results are summarized in Table
Table I: The symmetry infimum L0(3)
11- DRA) for k 2, 3,40 > 0.
to < k tu is even to is odd lo is even lu is odd to is even
S lo
< 2k
2 2k
lo is odd
k is even & is odd
(Dock) (Doch)
(0(3))
(Dii)
(0(3))
(Den)
(0(3))
(0(3)
(Doch)
(Den)
(Drah)
(Dus)
The analysis in Ex. 4.3 predicts three types of degeneration for the k-fold inputs from Ex. 2.1:
The symmetry infimum of G. is (D2ua. x Sk+1). The feature cannot distinguish the k-fold from itself rotated by T T/k around the z-axis.
Half Degeneration:
Axial Degeneration: The symmetry infimum of G2 is (Doch X S The feature cannot distinguish the k-fold from itself rotated by any angle around the z-axis.
Full Degeneration: The symmetry infimum of G2 is (0(3) x S1+1). The feature cannot distinguish the k-fold from itself rotated by any angle around any axis.
We consider encoding k-fold point clouds using equivariant neural networks and visualize the resulting embeddings. The three degenerations are experimentally verified in our visualizations, with Fig. 3 showing full and axial degeneration, and Fig. 4 showing axial and half degeneration. Although derived assuming high multiplicity (r > 3) in the feature representation, these predictions identical for the single representation case = 1), see & C.4.

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a
Axis Type
1.0
Axial Deg.
Normal
Half Deg.
R(m/k]
Rotation Rate Around the Axis
res = 49
Deg.
Axial Deg.
Normal
Half Deg.
Du DOLA DAcA c 0(3)
res 98
Half Deg.
Full Deg.
0.0
Symmetry Increase
1=7
1= 10
Figure Visualization of representation spaces. (a) k-fold (k is odd) structure is rotated 5/k about the perpendicular axis. (b) The path of symmetry increase. (c) Half degeneration appears at / = 10. At res 98 and res 49, the overall shape is identical, but the yellow data points (i.e. the second half of the rotation) completely cover the blue data points.
4.2 GUIDELINES FOR MANAGING SYMMETRY INCREASE
a general property of representations that the orbit types direct sum are related to those of its components by (v(V) U (=(22)
(& t V2), and Is (V1 V2, H) < IV H)) for i i= 1,2, with equality conditions discussed in $ C.5. These properties provide a direct mechanism for controlling the symmetry increase of equivariant feature, that is to choose components whose symmetry infimum (computed as described in & C.4 for G = S0(3) 3) or 0(3)) align with the desired behavior for task-relevant symmetries. Regarding the selection of the feature space in equivariant representation learning, this principle translates into two guidelines.
For orientation-dependent tasks (e.g., $ 6.2), when considering the kernel of feature space, is crucial to avoid non- toiviaivial symmetry increase (i.e. ensuring the map is relative isovariant) since such increases can lead the accidental loss of orientational information. Therefore, for a given input symmetry (H), one should select feature components that contain the orbit type (py (H)).
For general tasks (e.g., $ 6.3), certain forms of symmetry increase must avoided, as the output symmetry reflects the dimensionality of the fixed-point subspace where the equivariant features lie (see Remark Prop. C.2). In this context, one should generally avoid components where the symmetry infimum indicates a severe compression of the fixed-point subspace. Specifically, one must be cautious of components corresponding to non-trivial representations where the symmetry increases to the full group, this causes the component to be annihilated and lose all discriminative power.
DENSITY OF (ALMOST) ISOVARIANT MAPS
We now connect the preceding theory to practical machine learning context by introducing models for the data distribution and for the parameterized map. We show that the necessary conditions for isovariance established previously become sufficient under relaxed definition of isovariance.
5.1 THE MANIFOLD HYPOTHESIS
Motivated by the manifold hypothesis and the broader considerations summarized in $ A.3, model the data distribution as being supported on a finite union of smooth, compact submanifolds M U, M, embedded in X. When a group G acts on X, this action equips each Mj with a natural G-manifold structure.
The central question is: when does an isovariant map f M + Y exist? The existence is a non- trivial issue. The counterexample in Cex. D.3 demonstrates that the necessary condition of orbit type
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inclusion is not sufficient. Specifically, it shows that isovariant map fail to exist precisely because the multiplicities of the irreducible representations in the feature space are insufficient.
The non-existence of perfectly isovariant maps motivates map that is isovariant almost everywhere. To formalize this, we equip M with the d-dimensional Hausdorff measure μaM, where d = max, {dim M,). This allows to identify subsets of measure zero as negligible. Note that M()) is a finite union of submanifolds, then f is almost isovariant relative to Y for every orbit type (H) in the data support, the isovariance condition
practical, relaxed definition:
PY (G,) = py (G))))
(9)
holds for all points M(H) except for subset of (M -measure This ensures that any undesired increase in symmetry occurs only on a negligible portion of the data.
5.2 GENERICITY OF (ALMOST) ISOVARIANT MAPS
For Ex. 2.1, we select TFN (Thomas al., 2018), classic ENN based on tensor products, as our parameterized model. We provide the complete formulation to $ D.2. An important property of TFN parameterizations FIrN is that they satisfy a universal approximation theorem (Dym Maron, 2021). In topology, this is equivalent to FTeN being dense in equivariant function space Ca(X, Y) with respect to the C topology. In fact, we can establish a stronger approximation theorem.
Theorem 5.1. In Ex. 2.1, the function families FTEN with smooth activation function are - in the space of smooth equivariant maps C (X,Y) That is, for any integer r Z 0, any map (X, Y), any compact set K CX and any there exists a function 9 FIFe such that
- maX - g(x)) < k. k.
(10)
Here, A denotes the k-th order total derivative operator. A significant portion maps within a dense parameterization reflects the generic properties of the mapping space. For equivariant maps, key generic property, closely related to almost isovariance, is that the dimension of the set of points where the orbit type is increase from (H) to (H') by a map f constrained for a generic map. The following theorem shows that for expressive models with C approximation capabilities, such as the TFN discussed, almost isovariance a generic property, and full relative isovariance can be achieved by increasing representation multiplicity. As shown in Cex. D.3, this requirement is tight.
Theorem 5.2. Let FF be a equivariant parametrization with C approximation capability. If for every (H) E Oc(M) we have (H)) OY(Y). then for any finite union of compact, smooth G-submanifolds M C X, any f E C9 [X, Y), any integer r 2 O, and any c >> O, there exists a map 9 E such that
(11) and glas is almost isovariant relative Y. Furthermore, the feature space Y contains representation per for an integer > max, (dim M2}, where Y itself satisfies the condition (py(H) E Oo(Ý), then the approximating map glas can be chosen to be isovariant relative to Y.
max | f(x) - D0((=>)| <e,k
EXPERIMENT
We validate our theoretical analysis through three experiments: representation--ee -ppaiinoonsssceseoeeeeeee visualizations in $ 6.1, geometric graph discrimination task in $ 6.2, and molecular isotropic polarizability prediction task on QM9 (Ramakrishnan al., 2014) in $ 6.3. Across all experiments, we consider two equivariant architectures, TFN (Thomas et al., 2018) and HEGNN (Cen et al., 2024): TFN is used the first two experiments, whereas HEGNN is employed the last two. All detailed experimental settings are provided in $ F.
6.1 VISUALIZATION OF REPRESENTATION SPACE
To provide a clearer illustration of our theory, we present visualizations of the representation spaces of different degrees, obtained from the 3-fold structure.
Dataset. We first construct a folold structure lying in the zOy plane (here k = 3), and then apply random rotations place it distinct planes (here m = 6), illustrated in Fig. 4(a).
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Subsequently, shown in Fig. 4(b), each structure is further rotated about the axis perpendicular to its plane. The rotation angle [0, 2 kk) is uniformly discretized into res candidate values, defined as (2mi/(( res) p Unless otherwise stated, we use res = 49 to verify the half-degeneration. 98.
We also consider the doubled resolution res
Embeddings. For the resulting m res candidate structures, we compute the graph-level features via randomly initialized single-layer TFN (Thomas al., 2018) with detailed setting in F.1. For l = 0 the feature dimension 1; for visualization we set the second coordinate to zero. For ln 2 1 the feature dimension is 21o- rescale all features to a common range so that visualizations are comparable. Data points are plotted in ascending order of plane index; for structures on the same plane they are ordered by increasing rotation angle about the axis.
2; reduce dimensionality via random projection and then
Results, Detailed experimental results are presented in Figs. 3 and 4. The former shows the input symmetry with lu I 11 increase to (0(3) x Sk+1) (Full degeneration, 4o = 0, 1), (Doch X Sk+1) (Axial degeneration, lg 4), or remains non- non-degenerate. The latter shows the symmetry increase to (Desh X Sk+1) at lo 10. These experimental results are consistent with Ex. 4.3.
6.2 EXPRESSIVITY ON SYMMETRIC GRAPHS
To experimentally validate our theoretical con- clusion established, design more compre- hensive experiment following Joshi et al. (2023).
Rolation
Embedding Differen Norm (Max of All Settings)
Degree Models
Dataset. We construct four symmetric k-fold structures (I: E 2, 2, 4, 6), each centered at the origin. For each structure Go we apply random rotation to obtain G1, ensuring that 9i does not coincide with the original Go- The goal is to evaluate whether different ENNs can distinguish Gn from G, To probe different aspects of our theory, we treat 2D and 3D rotations separately; the 3D setting we additionally require that G1 not coplanar with Go-
-10-1
7-
10-e
10
11
2-6iid 3/oid 4-fold 6-foid 2 4-faid 6-foid
Figure 5: Heatmap of Emb. Diff. Norm.
Embeddings. We employed both TEN (Thomas et al., 2018) and HEGNN (Cen et al., 2024)
to compute the norm of the embedding difference across 12 configurations for each, varying by the number of irrep channels (1, 4, 16) and layers (1-4). The extracted le-degree embeddings are evaluated via the norm of the difference between the embeddings of Go and 91 as in Cen et al. (2024) with detailed setting in F.2.1. When this norm approaches zero, the two embeddings are numerically indistinguishable, and hence the corresponding geometrie figures cannot be told apart by the model
Results. The maximum value selected for each configuration and visualized in heatmap. as shown in Fig. 5. The results exhibit clear binary pattern: the values are either greater than 10-3 less than 10-6 (due to numerical error), with difference of more than 10 times. This suggests that values exceeding 10 indicate distinguishable structures, while those below 10-6 correspond indistinguishable structures. These findings align precisely with our theoretical predictions. Furthermore, as the maximum value was chosen, with all norms being less than 10-6, this phenomenon shown be independent of the model choice, the number of channels or layers.
6.3 MOLECULE PROPERTY PREDICTION WITH PRETRAINED EQUIVARIANT FEATURES
To illustrate the guiding significance of the theory presented in this paper for practical applications, we designed experiments on the QM9 dataset (Ramakrishnan et al., 2014) for verification.
Dataset. We choose to predict the molecular isotropic polarizability It is worth noting that the QM9 dataset (Ramakrishnan et al., 2014) contains many highly symmetric structures spanning 22
In Joshi et al. (2023), the embeddings are directly fed to a vanilla classifier. To address issues such as imperfect classifier training and numerical error, we slightly modify this experimental setup. We also reproduce their original experiment in F2.2, and the results remain consistent with theory.

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CT (111,335 samplee)
Cu (17,794 samples)
C2 (468 samplea)
C2v (485 samples)
Nik tog saae)
camplea)
samples)
samples)
(15 samples)
Me Dog polt
D2d (11 samples)
C5 (11 samples)
D-h sumples]
Coy 18 samplene
camplen)
camples)
MAE use fos saar
Duginel
Degroel
Figure 6: MAE loss (in units of ag) for isotropic polarizability prediction with degree = to across molecules from the top-16 point groups by molecular count. Each boxplot shows the distribution of errors at given degree, while diamond markers denote the corresponding mean MAE.
molecular symmetry groups with Fig. 6 reporting sample counts for the 16 most frequent groups; the remaining seven are D3 with 2 samples and C1. D2, Dea. On, S1, each appearing only once.
Embeddings. We adopt HEGNN (Cen et al., 2024) on features of < 11 to obtain a shared equivariant feature encoder, ensuring that all subsequent configurations operate on the (a) using only features of l = lu. and (b) using all features of < lo. yielding distinct prediction heads for each. The detailed experimental setup is given in F.3.1.
backbone. Specifically, we first pretrain
embedding space. We then consider two fine-tuning strategies:
Results. The results are shown in Figs. 6 and 7, with a summary of all symmetry increases in Table 22. We note that for the majority of samples, different feature components contribute similarly to the prediction. However, as illustrated in Fig. 6, for non-trivial feature components where molecular symmetry increase O3)), the prediction loss is substantially higher. Furthermore, the results in Fig. 7 show that for symmetries causing full degeneration in 1-degree features, including additional 1-degree features may not provide significant improvement the model's 's prediction performance. This validates the design guidelines in Š 4.2. Detailed case studies are provided in $ F.3.2.
7 CONCLUSION
In this work, we presented a rigorous mathematical framework to address the critical issue of symmetry increase in ENNs. We introduced the concept of the symmetry infimum, computable lower bound for any increase in symmetry determined by the feature space. Our central contribution show that this infimum can be used to precisely predict and control the expressive degradation of ENNs. The framework successfully explains phenomena in settings like those of Joshi al. (2023), which could not be fully accounted for by prior thcories such as the collapse-to-Ze titappsett--erro to-zero model from Cen al. (2024). Our findings provide both robust theoretical understanding and practical guidelines for designing more reliable ENNs.
"We use the QM9 dataset provided in PyG (Fey & Lenssen, 2019) and apply the PointGroup library (Car- reras, 2025) to pre-compute and manually post-process the point groups of all molecules. As result, our statistics may differ slightly from those reported in previous works like Zeng al. (2025).
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ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Nos. 62276269, 92270118, 18, and 62376276), the Beijing Natural Science Foundation (No. 1232009), and the Beijing Nova Program (No. 20230484278).
AUTHOR CONTRIBUTIONS
Ning Lin organized this project. Ning Lin led the theoretical development in 2-8 and was responsible for the theoretical proofs the corresponding part. Ning Lin and Jiacheng Cen jointly led the experimental studies in 6. Jiacheng Cen was responsible for model implementation and code development. Anyi Li was responsible for data processing. Jiacheng Cen and Anyi Li jointly contributed to figure preparation and visualization. Wenbing Huang and Hao Sun jointly supervised and guided the project. All authors participated in writing and revising the manuscript.
USAGE OF LARGE LANGUAGE MODELS
We only use Large Language Models to polish our writing.
ETHICS STATEMENT
This work is theoretical contribution in the domain of equivariant neural networks, focusing mathematical properties of symmetry preservation and transformation under equivariant mappings. The research does not involve human subjects, personal data, real-world deployments, and therefore does not raise concerns related to privacy, fairness, bias, or potential misuse.
We affirm that this work adheres to the ICLR Code of Ethics. In partieular, have ensured honesty in representing our contributions, accuracy in reporting our findings, and proper attribution of prior work. As this is theoretical study, there are conflicts of interest, sponsorship influences, or applications with foreseeable societal harms to disclose. We support responsible stewardship of machine learning research and believe that foundational advances such as ours contribute positively to the scientific community by enhancing understanding and enabling future trustworthy systems.
REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility of both the thcoretical and experimental components of our work. To support the reproducibility of our theoretical results, we provide complete and self-contained proofs for all main theorems and propositions in the appendix, including detailed derivations and necessary mathematical background. These proofs clarify all assumptions and logical steps required verify claims. For the experimental component, we have made GL implementation code publicly available at ht:tps:: / / thub. -D- RCCRRRCCC/Sm/// SSmInc, nc. The
codebase contains clear documentation and scripts that fully reproduce the reported results. All experimental settings, hyperparameters, and data generation procedures are described in detail within the supplementary materials. Together, the comprehensive theoretical appendices and open-sourced code ensure that our findings can be rigorously verified and built upon by the research community.
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CONTENTS OF APPENDIX
Backgrounds
16
A.1 Equivariant Neural Networks
16
A.2 Closed Subgroups of SO(3) or ((3)
16
A.3 Manifolds and Basic Data Assumption
17
B Proofs of The Infimum of Symmetry ($ 3)
19
B.1 Proof of Thm. 3.1
19
B.2 Proof of Thm. 3.3
20
Symmetry Increase for G (3) or 0(3) $4
C.I General Orbit Type Criterion
21
C.2 Proof of Prop. 4.2
................................
C33 Detailed Calculation in Ex. 4.3
22
C.4 Calculation of Symmetry Infimum
23
C.5 Composition Property of High-Multiplicity Representation
24
Proofs of Density of (Almost) Isovariant Maps ($ 5)
27
D.I Counterexample in $ 5.1
27
D.2 Proof of Thm. 5.1
................................
27
D.3 Some Results on Topology
31
D.4 Generic Equivariant Mappings
................................
33
D.5 Proof of Thm. 5.2
38
Tables
39
E.1
Minimal Proper Supergroups in SO(3) 0(3)
39
E.2
Dimensions of Fixed-point Subspaces for Subgroups of SO(3) or (
40
E.3
Symmetry Infimum for Subgroups of SO(3)
41
E,4
Symmetry Infimum for Subgroups ((3)
F
Detailed Experiment
45
F1
Visualization of Representation Space
45
F.2
Expressivity on Symmetric Graphs
................................
45
F2.1
Embedding Difference Norm Experiment
45
F.2.2
Original GWL-Test on Symmetric Graphs
45
F3
Molecule Property Prediction with Pretrained Equivariant Features
46
F.3.1
Detailed Experimental Setup
46
F33.2
Case Studies
46
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A BACKGROUNDS
A.I EQUIVARIANT NEURAL NETWORKS
Equivariant Neural Networks (ENNs) have emerged cornerstone of modern machine learning, enabling a wide range of applications across the sciences (Han et al.. 2025; Wu et al., 2023; Li et al., 2025b: Wu al., 2025; Li et al., 2025c: Feng et al., 2026; Cao al., 2026; Wu et al., 2026; Li et al., 2025a). Most mainstream ENNs adopt tensor product operators to design the message passing architecture (Thomas et al., 2018). In particular, since scalarization operations (e.g., norm and inner product) can substantially reduce computational cost, Cartesian vector- networks (Satorras et al., 2021) and spherical scalarization networks (Cen et al., 2024; Aykent & Xia, 2025) have become particularly popular. More specifically for asymmetric graph structures, Cen et al. (2025) point out that networks employing scalarization operations already possess sufficient expressive power, i.e., universal approximation.
However, additional subtleties arise in the presence of input symmetries. The interaction between architectural components and symmetric structures can lead to nontrivial representational effects. For example, techniques such as global virtual nodes (Zhang et al., 2024; 2025) and reference frames (Puny et al., 2022; Duval al., 2023), though effective in improving model capacity, may exhibit unintended behaviors when applied to symmetric inputs. In particular, their use can potentially induce symmetry increase. In this work, we systematically analyze these phenomena under symmetric structures for general equivariant neural network architectures.
A.2 CLOSED SUBGROUPS OF SO(3) OR O(3)
Before proceeding, we briefly review the classification of the closed subgroups of SO(3) or O(3). The closed subgroups of ((3), also known as point groups, are classified up conjugacy follows. Throughout this paper, we use the Schoenflies notation to denote these groups.
Finite Subgroups: These are divided into axial and polyhedral groups The axial groups include the Abelian subgroups (C,, S2x, CEΛ) and the non-Abelian subgroups (Cku, DR. DRh, Dka). The polyhedral groups all non-Abelian and comprise seven families (T, Ta, Th, O, On, I.). Among these, the groups Ck. DR, T, 2, consist solely of pure rotations and subgroups of SO(3).
Infinite Subgroups: These include the cylindrical groups (Coo; Cook, Coev: Doo, Dock), which arise as limits of the axial groups, and the spherical groups, which are K SO(3) and K₁,- = 0(3).
To facilitate the identification of input symmetries, now provide brief overview of the elemental structure of the key finite subgroups. The identification of cylindrical symmetries, which are infinite, can be achieved by taking the limits of these finite axial groups
ABELIAN AXIAL GROUPS
Cs (Cyelic Group): Generated by rotation ej of angle 27/k, with k elements:
(1) Rotations of about the principal axis.
Sz "Rotation-Reflectiion Group): Generated by adding a rotation-reflection element (2)
to Ck. with 2k elements:
(1) Rotations of about the principal axis.
(2) Rotation-reflectionn 22 2i+1 an about the principal axis.
Ckn (Cyelic Groups with Horizontal Reflection): Generated by adding horizontal reflection plane on to Ck. with 2k elements:
(1) Rotations of about the principal axis.
(2) Rotation-reflections cf about the principal axis.
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NON-ABELIAN AXIAL GROUPS
Cke (Cyelic Groups with Vertical Reflections): Generated by adding a vertical reflection plane oy to Ck, with 2k elements:
(1) Rotations of about the principal axis.
(2) Reflections d, about the vertical plane.
Dy (Dihedral Groups): Generated by adding 2-fold rotation axis u2 perpendicular to the
principal axis of Ck. with 2k elements:
(1) Rotations &e about the principal axis.
(2) 2-fold rotations c/u2 about horizontal axes.
DEA (Dihedral Groups with Horizontal Reflection): Generated by adding a horizontal reflection plane Øn DR. with 4k elements:
(1) Rotations of about the principal axis.
2-fold rotations u2 about horizontal axes.
reffetionreslectionns cf about the principal axis.
(4) Reflections d, about the vertical plane.
D (Dihedral Groups with Dihedral Reflections): Generated by adding diagonal
reflection plane dd C2 to DR, with 4k elements:
(1) Rotations & about the principal axis.
(2) 2-fold rotations ug about horizontal axes.
(3) Rotation-reflections 2f+1 σr about the principal axis.
C2k
2j+1 (4) Reflections Ck about the diagonal plane.
POLYHEDRAL GROUPS
T' (Tetrahedral Group): The rotational symmetry group of a tetrahedron, with 12 elements.
Ta (Full Tetrahedral Group): The full symmetry group of a tetrahedron, with elements.
Tp (Pyritohedral Group): Generated by adding an inversion center T, with elements.
O (Octahedral Group): The rotational symmetry group of cube octahedron, with 24 elements.
On, (Full Octahedral Group): The full symmetry group of a cube or octahedron, generated by adding inversion center to O, with 48 elements.
I (Icosahedral Group): The rotational symmetry group of an icosahedron, with 60 ele- ments.
Jr (Full Icosahedral Group): The full symmetry group of an icosahedron, generated by adding an inversion center to 7, with 120 elements.
The subgroup relations among point groups, as well as the dimensions of fixed-point spaces of O(3) representations, can be readily determined. In particular, they can be derived from the minimal subgroup relations provided in $ E. and from the dimension table for fixed- fixed-point spaces of irreducible (3)) representations established
E.2.
A.3 MANIFOLDS AND BASIC DATA ASSUMPTION
Following the definition in Milnor (1990), C manifold A is mathematically defined subset of some linear space d For every point æ in this subset, there must exist a neighborhood W in IR such that W O M is C -difffemmorphic to open set in another linear space R The integer is known as the dimension M. An alternative, intrinsic definition that does not depend on an embedding space also be found in Hirsch (1976).
Machine learning often assumes that the data distribution is supported on a submanifold of the input space. The dimension of this manifold characterizes the directions in which the data can vary within
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the imput space. However, the term manifold is often used loosely in this context and does not perfectly align with its mathematical counterpart (Goodfellow et al., 2016). First, the data manifold may have self-intersectionss, causing the local dimension of data variation to differ at various points. Second, data belonging to different classes or clusters may possess different structures and potentially different dimensions. Lastly, stochastic factors such observational noise can prevent the data from forming strict surface in the input space, We will not address this last factor, instead, we assume that the interference from noise minimal and ignorable.
To address the first two issues, we can relax the manifold hypothesis by assuming that the data is supported on a finite union of submanifolds within the input space. Since a manifold with multiple connected components can itself be viewed as disjoint union of connected manifolds, we can assume, without loss generality, that each of these submanifolds connected. For theoretical convenience, we further assume that these manifolds are bounded and closed. By the Weierstrass theorem, this is equivalent to assuming they are compact. We briefly outline the justification for these assumptions below:
Finite Union of Manifolds: This assumption has been partially verified experimentally
computer vision datasets (Brown et al., 2022). Theoretically. it encompasses classic cases of self-intersection. A data manifold may arise the image of map. However, the image of an immersion or submersion
manifold can have self-intersections. The image of such a map from a compact manifold is, however, finite union of compact manifolds.
Boundedness of Manifolds: This assumption stems from the natural assumption that the data distribution itself is bounded.
Closedness of Manifolds: This assumption covers scenarios where the data defined
by a set of well-behaved constraints. For example, the set of points satisfying indepen- dent, differentiable constraint equations in the input space forms a closed submanifold of codimension
In equivariant representation learning, the data possesses symmetries, meaning that data corresponding to the same physical object can be transformed by symmetry operations to represent different reference frames. These symmetry transformations typically form group, and since the transformed data is still valid data, the data manifold must be closed under these group transformations, that is, the data manifold must be invariant under the group action. Considering the group action for any given input, we summarize the preceding discussions into the following assumptions about the data:
Lie Group Assumption: The transformation group G' is compact Lie group.
Manifold Assumption: The input space X is a linear space equipped with a linear action px of the group G. That is, there exists map px G GL(X) such that Px (g192) = Px (gt )px (gz). and the identity element of the group is mapped to the identity transformation. The data manifold M a finite union of compact, connected, smooth, and G-invariant submanifolds of X.
The smoothness assumption is added for theoretical convenience and the connectedness assumption does not impose additional restrictions, since each compact smooth submanifold admits only finitely many connected components.
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B PROOFS OF THE INFIMUM OF SYMMETRY ($ 3)
B.1 PROOF OF THM. 3.1
Lemma B.1 (Azzi al. (2023), Cor. 2.3). For a compact Lie group G and a representation V, let V BEW V be any point. There exists a neighborhood U of u such that for any point u E U, we have (Gn) S (Ge).
Lemma B.2 (Azzi et al. (2023), Cor. 2.20). For reductive group G and an affine algebraic variety X, let x E X be a point with a closed orbit G(z). There exists a Zariski neighborhood of a such that for any point U, we have (Gu)= S (Ga).
We use the definition of the complexification from Prop. 3.3 of Azzi et al. (2023). For a real vector space V. its complexification is Vc_ C @ V. Considering being the Lie algebra of G, the complexification of the group is G == {g (xpx(ii By the above complexification, we obtain the following lemma.
orthogonal action of G on V. with g
E G, X E g).
Lemma B.3. For compact Lie group G. consider the inclusion map G + G into its complexification. Let H and K be subgroups of G and let gu E G be an element such that go -1 C K. Then,
K c..
(12)
Proof. Let h and t be the Lie algebras of H and K, respectively. Any element of H can be expressed as ho exp(iXo) for ho E H and Xo Its conjugation by go is
(ho exp(iX.)) 9o = (gohogō exp(iAd Xo).
(13)
-1 Since go go 90 C K, the term gohogo belongs to K. The expression thus belongs to K provided that Adg. Xo E t.
This condition on the Lie algebra is obtained by differentiating the subgroup inclusion go Ho C K at the identity element, which gives Adgo (h) C t. As Xo h. it follows immediately that Adgo Xo e, which completes the proof.
Proposition B.4. Let V be a representation of a compact Lie group G. For any closed subgroup H of G. the set of points in the fixed-poine subspace V that have the minimal orbit type is a dense and open subset of H
Proof. The proof strategy is similar to that of Prop. 2.10 in Azzi al. (2023). We first show that the set of points with a minimal orbit type is open, then use complexification to show it is Zariski-open. which implies density.
First, to prove openness, let (K) be a minimal orbit type in V and let x E V be a point with (G,) = (K). By Lem. 1. there exists neighborhood U of such that for any u U, (G.) < (K). Since (K) is minimal in V any point in the open set U n V must have orbit type (K). Thus, openness is established.
Next, we consider the complexifications G and VC_ For a point & E V with orbit type (K), by Lem. 3.13 of Azzi al. (2023), we have (G9), = (G = By Prop. 3.14 of Azzi et al. (2023), the orbit (x) is closed. Therefore, by Lem. B.2, there exists a Zariski neighborhood U of x in ve such that any point in this neighborhood has an isotropy type less than or equal to (K*). The set U = U n (VC)8C is Zariski-open in (VC) By Cor. A.7 of Azzi et al. (2023), its real part U" = U" nv contains non-empty real Zariski-open subset of H
We now show that for any E U", its orbit type is (Gy) = (K). Since E VH, we have (K) < (Gy), which implies there exists E G such that 91 Kgn C Gy. By Lem. B.3, this gives -1 K Kc K S (G Since E U C U, we have ((G)) S (K*)), which means there exists
92 E GC such that (G,,) C g2k 92 Combining these inclusions yields
gi k 96 C (G,,) C k2 95
(14)
By Lem. A.3 of Azzi et al. (2023), this implies that the real subgroups conjugate, i.e., (Gy) (g) K )- = (K). This completes the proof of density.
n
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Theorem 3.1 (Uniqueness of Minimal Type). Let X be representation of compact Lie group G. For any closed subgroup S G. a unique minimal orbit rype exists among the points in the fixed-point subspace X" In particular, if (H) (x(X"). then (H) is the minimal orbit type within that subspace.
Proof of Thm. 3.1. The density and openness of the set of points corresponding any minimal orbit type is established by Prop. B.4. The uniqueness then follows from the fact that two distinct open dense sets must have non-empty intersection.
L
B.2 PROOF OF THM. 3.3
Theorem 3.3. necessary condition for the existence of an isovariant map relative to Y from G-set X to G-set Y is that (py(H) E (Y) for every (H) Oc(X). When X and Y are representations, this is equivalent to the condition that IY(Y, H) = (py (DY for all (H) E (x()).
Proof of Thm. 3.3. The equivalence for representations follows directly from applying the operator Py to the known partial order (H) < TY,,, H) S (py (py(H)). Since py is order-preserving and fixes isotropy subgroups in Y, the inequality collapses to an equality.
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C SYMMETRY INCREASE FOR G = SO(3) OR 0(3) IN $ 4
C.I GENERAL ORBIT TYPE CRITERION
To derive Prop. 4.2, we need more general results require criteria for identifying orbit types of general representation. Given the sufficiency of the Ihrig- Golubisolyy Criterion (Prop. C.1), we demonstrate that for high-multiplictty representations, the conditions of the Michel Criterion imply the conditions of the Goolg--olubitky Criterion. This, in turn, establishes the sufficiency of the Michel Criterion.
Let G be a compact Lic group and X be the input space. The normalizer of subgroup H C G (H) {g E G gHg- H}. The normalizer of H relative to supergroup H' S H is No(H, H)) == (g e G H C g) Based on the above definitions, we recall the following general criterion.
Proposition C.1 SbrrigGGllubitky Criterion, lhrig & Golubitsky (1984), Prop. 5.3). Let be faithful representation of group G. A sufficient condition for closed subgroup H to be isotropy subgroup in that for every orbit type (H') in with (H') > (H), the following inequality holds:
dim V + orc H') == dim V + dim (H, H³) dim No(H²) < dim V
(15)
where (GGHH, H, H') is the KarigGoolubitsyy correction term.
Remark. lhrig & Golubitsky (1984) states that the condition is also necessary for G = SO(3), 0(3).
Compared to Prop. C.I, the Sinchan-Stedmant Criterion (Linchan & Stedman, 2001) is more con- venient due to its structure, which only requires checking adjacent subgroups. Since the associated theoretical results are not needed in our proofs, we do not elaborate them here.
C.2 PROOF OF PROP. 4.2
Proposition 4.2. For a high-multiplicity representation V, the necessary condition stated Thm. is also sufficient.
Proof of Prop. 4.2. We show that for representations where each irreducible component has multi- plicity > dim G. the necessary condition from Michel's 's Criterion (Thm. 4.1) becomes sufficient by proving it implies the condition in Prop. C.I. We may assume without loss of generality that the representation is faithful. For unfaithful representations, we proceed by factoring out the kernel from G and invoking Prop. C.I, as the procedure remains unchanged.
It follows from the condition that for any closed subgroup H' of G containing H. we have dim V dim VH. The condition implies that for at least one irreducible component V,, we must have dim V. dim V.H. Since dimensions are integers, this is equivalent dim H + S dim V."
Given that the multiplicity m(V, V) dim G. it follows that
mVV. V) dim V." + dim G < m(V,, dim V.".
(16)
Summing this strict inequality for such component with the non-strict inequalities for all other components yields
dim ' + dim G < dim H.
(17)
The term from Prop. C.I is bounded by
0 E (G(H, H') = dim No(H, H') - dim No(H') s dim G.
(18)
The sufficiency of Michel's condition follows from combining these two results:
dim v < dim 4
dim H + dim G < dim H
(19)
dim H + dim (HHH, H²) - dim No(I1') < dim v",
(20)
where the final implication uses the upper bound from Eq. (18). This shows that Michel's condition implies the sufficient condition from Prop. C.I.
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C.3 DETAILED CALCULATION IN Ex. 4.3
We now conduct the orbit-type test and symmetry infimum calculation for the geometric symmetry Da (k > 2) of the k-fold from Ex. 2.2. Consider DRs closed subgroups of ((),, the calculation is performed in the representation space Y Vi for r > 3 and lo > O. We will use the pre- computed subgroup relations from Table 6, the dimensions of the fixed-point spaces for DEA and O₁ in Vi=la from Table 8, and the orbit type test results for (Dxch) and (Op) in Vi=l.
First, we perform the orbit type test. Consider the case where k * In this situation, the adjacent supergroups of DRs of the form Dpk, where p is prime number. We classify the discussion into 12 cases based on the parity of and l.. and the ranges Lo < k, k S lo < 2k, and to 2 2k, and caleulate the dimensions of the fixed-point spaces of Vi=i for DEA and Dpk.h- The calculated dimensions are shown in Table 2. For the case k = 4, we must additionally consider the supergroup O₁- Here, dim Vi = dim Vi only when lo odd, which aligns with the results from Table 2. In all other cases, O₁ does not affect the orbit type test of (DRa). Thus, we find that (Das) is an orbit type when lo 2 k and when lo and have the same parity.
Table 2: Dimension of fixed- fixed-point point space for supergroups of DRn (k > 2) in V= (r 3), organized by the parity of k and lo-
< k
lo < 2k
2 2k
lo is even lo is odd
to is even
to is odd lo is even
4o is odd
à is even
Duh Dpkh
2
d
0
1
d
le is odd
Duh
1
0
1
d
d
D2kh
0
1
d
Dp Dp-kh
1
0
1
0
d
< d
Next, we proceed with the symmetry infimum calculation. Again, we first consider the case where k * 4. In this situation, the supergroups of DRA are Dpk,k, Doch, and 0(3), where 0(3) is always an orbit type. Using the orbit type test results just calculated and those pre-computed, we analyze the cases based on the parity of l and lo as before, with the results summarized in Table 3. For k = 4, the additional supergroup O₁ does affect the final result. This is because the supergroup O is never the minimal isotropy supergroup, it only becomes an orbit type when (D4x) already is. This leads us to the symmetry infimums shown Table 1.
Table 3: Isotropy conditions for supergroups DEA > 2) in V (r > 3), organized by the parity of k and lo- A / indicates the condition is satisfied, and that it is not. In the subgroup notation, p denotes a prime number and p" denotes an odd prime number.
< k
S lo < 2k la is even lo is odd lo is even to is odd
2 2k
lo is even to is odd
is even
Dan Dphh Dooh
X
X
X
/
X
X
X
k is odd
DEh
X
X
X
X
DaRh
X
X
X
/
Dr-kh
X
X
X
Dock
V
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C.4 CALCULATION OF SYMMETRY INFIMUM
We calculate the orbit types for the SO(3) representation Vi=to and the O(3) representation V As the calculation procedure is highly similar to that in Ex. 4.3, we omit the detailed steps here. For the case of 1, the results can be found in Table B.1 and Table B.2 of Linehan & Stedman (2001).
The orbit types are calculated using the procedure Algo. This calculation requires the dimensions of the fixed-point spaces for the closed subgroups of SO(3) or O(3) in the representations Vi=ia and V== respectively, which are provided in $ E.2.
According to Prop. 6.2 from Ihrig & Golubitsky (1984), the Shrig-Golubitsk, correction term oHH, H') 0 for all subgroups except H C in SO(3), and for all subgroups except H = C2, S2x. C&a in 0(3). Therefore, for all subgroups other than H = Ck S2x, Csh. our results are identical to those calculated by Linchan & Stedman (2001) for 1. We present only the results of whether (C&) E O5o(3) (V (= on Table 4 and whether (C&), (S2k), (CRN) E Oo(3) (V on t=1
Table 5.
The reason for omitting the discussion of V V&r is consistent with the explanation in the header of [=
Table B.2 in Linehan & Stedman (2001). This is because the corresponding conclusions can be found in the orbit type table for the SO(3) representation 1=2 via the mappings Doo + Doch- Coo + Cooh. I + Ih. O - On. T - Th. as well as D2- + Des and C - Ckh for even kt and D2 Dad and Ck + S2k for odd
Table 4: Modified isotropy subgroups for the multiple irreducible representation V=b,, 3 of SO(3) obtained via the Michel criterion.
H Condition of
la
Ck lo
2 k
Table 5: Modified isotropy subgroups for the multiple irreducible representation VBr 0(3) with nontrivial reflection, obtained via the Michel criterion.
3 of 1-
H
Condition of to
Ck
Io k
S2x (k is even)
lo 2 k
Ces (k is odd)
to 2 k
The complete set of orbit type calculation results can also be found in the tables for the symmetry infimum. This is because (H) is orbit type of a representation V if and only if the symmetry infimum H in V is (H) itself. These non-degenerate cases are highlighted in green in the tables.
Using the calculated orbit types for the SO(3) representation vi and the O(3) representation ver, we can now compute the symmetry infimum for all subgroups of SO(3) or O(3) in these " representations. When calculating the symmetry infimum for subgroups, the algorithm differs slightly from that in Ex. 4.3. We reduce the need to enumerate all supergroups by leveraging the symmetry infimums of adjacent supergroups. An improved version of Algo. is detailed in Algo. 3.
To reduce the computational load, this algorithm employs a top-down calculation strategy. First, compute the results for the infinite groups, followed by the polyhedral groups Finally, we calculate the results for the axial groups in the order D Dkd, DR. Cku, Ckhy S2k, Ct. Due to the exceptional subgroup relations shown in Table 6, special consideration for additional supergroups is required for certain cases where & E 2, 2, 3, 4, 5}. These cases are handled last.
The results for (3 are presented in $ E.3, and the results for (33) are in $ E.4. Here, we provide general classification for the observed symmetry increases. For closed subgroup H, increase
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Algorithm 3: Symmetry Infimum Calculation With Precomputed Results
Data: A symmetry group G;
closed subgroup H C G; Rep. of G;
map M of previously computed symmetry inf'. Result: Symmetry inf. (V,V, H).
if is (iii_in((H))) (V)V) then
2 return (H);
3 end
4 Let = {H:) C G to be all adjacent closed supergroups of H in G;
O 2-;;
6 for H, in So do
if iss fiss_nn( in((H;) (V)) then
Add (H,) to the set O: end else if H; is a key in the map M then
11
Add M [H.] (V V H;) to the set O:
end else
13
Let S. {K₃} } to be all adjacent closed supergroups of H, in G;
15
for K2 in S; do
16
if K )) (V) then
17
Add (K₃) to the set O;
18
end
end
20
end
end
return min(O);
to the full group ((3) or SO(3) is termed full degeneration and marked in red in the tables. An increase to supergroup of higher dimension than H is termed continuous degeneration and marked in blue An increase a supergroup of the same dimension as H termed discrete degeneration No symmetry increase is no degeneration and marked in
and marked in yellow or light green
green
For the subgroup D&A C O()). the classification of degeneration behaviors given after Ex. 4.3 is a special case of this general framework. Specifically, full degeneration is identical to the definition here, axial degeneration corresponds to continuous degeneration, and half degeneration corresponds to discrete degeneration.
verr the action of has a non-trivial kernel, and thus symmetry increase is inevitable. Let Ty be the natural projection to the quotient G/kerpy this context, the light green marks an increase to ( (H)), which, as explained in $ 3.2, is the lowest possible symmetry that H can be reduced to when non-trivial kernel is present. This therefore predictable behavior. The yellow marks other exceptional cases within discrete degeneration.
We note that for the representation
C.5 COMPOSITION PROPERTY HiGH-MUL.TIPLICITT REPRESENTATION
In the tables of & E.3 and $ E.4, there exists special class of subgroups H for which any non-trivial symmetry increase always results in an infimum (H)) where Hg is an adjacent supergroup. We call (H)) the bottleneck of H. denoted by Bo(H), and we say that subgroup H that possesses a bottleneck satisfies the bottleneck condition. These groups that act as bottlenecks satisfy some elegant properties. To demonstrate this, first prove structure theorem for the symmetry infimum of high-multiplicin representations.
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Proposition C.2. For high-multiplictty representation V. the lowest orbit type in the fixed-point subspace V is (Gyu), ). where GyH = Nuev" G2 is the largest subgroup that leaves V invariant. This shows that To(V, H) = (Gv").
Proof. Since any element in V has at least the symmetry Gys, we only need to prove that Gy is an isotropy subgroup. Assume, for the sake of contradiction, that Gyn is not an isotropy subgroup. By the sufficiency of the Michel Criterion, there exists a supergroup K such that Gyn & K and
dim v dim VGy" dim V
v VGv = =
(21)
This means that K leaves all elements of V fixed. From this, we derive a contradiction:
K c n G. Gya.
(22)
ZEVE
Remark. This result implies that for a high-multillicity representation, the symmetry increase from (H) to (V, H) does not alter the dimension of the fixed-point space. Therefore, for nigh-multpplicity representations, the dimension of the fixed-point subspace corresponding to the input symmetry group equals that corresponding to the symmetry infimum. Since the exed-point subspace dimensions for certain closed subgroups exhibit distinct regularities, the behavior of symmetry increase toward these subgroups serves as an indicator of the underlying subspace dimension. For instance, in the cases of SO(3) or (3)). for non-trivial representations, full degeneration corresponds to O-dimensional fixed-point subspace, whereas for finite input subgroups, continuous degeneration corresponds to 1-dimensional subspace. However, must emphasize that when a quantitative assessment of the equivariant feature's s expressive capacity is required, relying solely on the symmetry infimum is insufficient, it yields only coarse, qualitative insights. In such cases, one should directly compute the fixed-poine point subspace dimension. For calculations related to SO(3) or O(3), see $ E.2.
We can prove that these groups acting as bottlenecks satisfy the property that for any irredueible rep- resentation Vo, if dim V < dim VoHl, then dim V/" < dim V holds for all adjacent supergroups H' of H. This is because if we assume there exists an H' such that equality holds, then by the Michel Criterion, (H) must undergo a non- non-trivial symmetry increase to (Gy in the high-multiplicity representation 1 Since these increases always have the bottleneck (Ho) as their infimum, the inequalities
dim Vo" S dim Vo < dim Gun must in fact collapse to dim Vy = dim V2Hl, which leads to a contradiction.
(23)
For any high-multiplicity representation V that satisfies dim VH dim V'o,, there must exist component corresponding to an irreducible representation Vo in V such that dim V/" < dim 4o follows that dim Vi < dim Vi for all adjacent supergroups H' of H, which in turn shows that dim 4 < dim y holds for all such H' Therefore, for high-multiplictty representations, Ho controls the dimension gap of the fixed-point spaces between H and its other adjacent supergroups.
This property allows us to prove a theorem regarding the direct sum of high-multiplicity representa- tions, which in turn establishes a property for the orbit types of such direct sums.
Theorem C.3. Let a subgroup H satisfy the bottleneck condition. For two high-multiplicity represen- tations V1 and V2, we have (H) (V @ V2) if and only if (H) E Oo(Vi) or (H) E (V2).
Proof. We have already shown that for any high-multiplicity representation V. if dim V dim Ho then dim H < dim HH holds for all adjacent supergroups of H. Therefore, assuming (H) E (v(Vi @ V2), the Michel Criterion gives us
dim V + dim Va2 < dim VHH + dim V.H
(24)
By taking H' Ho (I),),, we see that at least one of the following two conditions must be true:
dim V.H dim V.Ho
== 1.2.
(25)
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This implies that for all adjacent supergroups of H, at least of the following two conditions must hold:
dim V < dim V,",, i= 1.2.
(26)
Therefore, (H) is orbit type of either Ví or V/2
The above theorem shows that when we construct mustimultlllictty representation for which (H) is orbit type using high-multiplicity components, the only way is to find migh-lultiplicityy component that already contains (H) as an orbit type.
For 5O(3), all closed subgroups satisfy the bottleneck condition. Consequently, for high-multiplicity representations, we have (V UV2) = (& @ V2). This means that [V @ V2. H) will be the minimum of (V, H) and (V2, H). Furthermore, the set representations {V ] is sufficient to generate all closed subgroups orbit types, because for any closed subgroup, there always exists a high-multiplicity representation for which is an orbit type.
For (3),, the bottleneck condition does necessarily hold. For example, for H Coo: symmetry increase can result in either Doc or Coce, which shows that no bottleneck group exists. The fact that Co never appears as an orbit type in any representation por demonstrates this point precisely. 1=
This introduces subtle issue when we apply the guideline on $ 4.2: for certain orbit types, repre- sentation exhibiting the target orbit type cannot be constructed simply by including the component V associated with it. Fortunately, Coo is the sole instance of this phenomenon encountered when 1=
G = (3)). Regarding the construction of a ((3) representation containing Coo, following Prop. 4.2, it suffices to simultaneously select components v with both odd and even degrees to >> 0.
1=10
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D PROOFS OF DENSITY OF (ALMOST) ISOVARIANT MAPS ($ 5)
D.1 COUNTEREXAMPLE IN $ 5.1
Lemma D.1 (Borsuk- Ulam Theorem, Guillemin & Pollack (1974), Chap. 2. Sec. 6). For any continuous odd function g S + R", there exists a point x E S"h such that g(x) 0.
Lemma D.2 (Weak Borsuk- Borsuk-Ulam Theorem, Nagasaki (2003), Thm. A). Let M and N be G-spheres in representation space for compact group G. If there exists an equivariant map from M to N, then the following dimensional inequality holds:
<a(dim M dim MD S dim N dim N
(27)
where PG : IN- + N a non- non-decreasing function that diverges to infinity.
Counterexample D.3. For a compact Lie group G, consider a representation Y with no trivial component, iee. YG = {0}. Let Y = yor and X = y for some r, no > dim G. By Prop. 4.2, we have Ox(X)= OY(Y), yet isovariant map exists from the unit sphere in X to Y for a sufficiently large integer ne-
In particular, for G = Zz, if is the non-trivial irreducible representation, then for any X = yon and Y = B with T > T2, no isovariant map exists from the unit sphere of X to Y.
Proof. For any compact Lie group G. consider equivariant map f from G-sphere M in a vector space X to a G-representation Y. We assume that the multiplicities of the trivial representation components in both X and Y are zero, i.e., X {0} and y {0). Therefore, only the origin in X and Y has the orbit type (G), and the sphere M does not contain any point of orbit type (G). Consequently, if f is an isovariant map, must have no zeros.
From such zero-free map f. we can define a map f the G-sphere N in Y:
J:M-N, i(x) - (<)\/11/(=)lll
(28)
Since scaling in a vector space does not change the orbit type, f is also an isovariant map. Therefore, by Lem. D.2, we obtain the dimensional relation:
pa M - dim M9) = pa X - S dim N dim No = dim Y 1.
(29)
Since Pe is non-decreasing function that diverges infinity, there must exist integer no dim G such that (no) > dim 1. This implies that
(dim(y +)) 1) > dim Y - 1.
(30)
This shows that no isovariant map can exist from the unit G-sphere M in the space X = (1+++11) to the space Y. To complete the counterexample, let Y y where r >> dim G. By Prop. 4.2, we have Oo(Y)= Oa(X), (X), yet isovariant map exists between them.
For the special case of G Zz, let Y be the non-trivial irreducible representation. Let X = ere and Y = 0r with it is an odd function. Since the representations have no trivial components, an isovariant map is equivalent an odd function that has no zeros. However, by Lem. D. any odd map from sphere in a nigher-dimensione space to a lower-dimensional space must have zero. Thus, no such isovariant map exists.
r2. A map f from the unit sphere in X to Y is equivariant if and only
D.2 PROOF OF THM. 5.1
TFN is an ENN based tensor products of hidden features. The TFN architecture is composed of a feature lifting map followed by an equivariant pooling stage. The feature lifting map contains equivariant convolutional layers that update node features by aggregating information from neighbors. These convolutions employ filters built from learnable radial functions parameterized by a Multi-Layer Perceptron (MLP) and real spherical harmonics Yim-
Mathematically, the parameterized map fTEN E FTPN is defined as a sum over feature channels. Each term in the sum is composition f pooo 0 fteats consisting of feature lifting map followed by
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equivariant linear pooling. Let Z represent the hidden features associated with each node. The feature lifting map fieat takes the input point cloud to node features Z @ R This map is constructed as a composition of layers:
freat Tzaz o (f(l), id) o .... (fa), id) ext o C,
(31)
where a centering operation, ext is a constant extension map, and each layer f(() updates the node features 1)(k-1) to ((()) according to the rule:
U (R) = ilgma
(1-1)) ilyms
ma) + Eÿfz 2m J2.m2 (12,m)).(1 Fmz (x: - 23) )u f
(1-1)
jlims
(32)
Here, Vilym, corresponds to the feature of type 13 at node i, and Clls..ms)
(12,m2)) .m)) are the Clebsch- Gordan coefficients. The filter function is defined Fn (x)) (1(/=)] V (z// [2]1)) where the radial function hy Rg0 - R is parameterized by a Multi-Layer Perceptron (MLP). For G (3) x Sn, the parities of and l3 must also be considered. We denote by JTFN the resulting family of TFN filters under MLP-parameterizee radial functions. In contrast, when ht is chosen as polynomial parameterization, we denote the corresponding parameterization by TFN-
We use L(X, Y) to denote the vector space of linear maps Y, and P(X, Y) the vector space of polynomial maps X + Y. Given the G-actions on X and Y, we write La(X, X) C L(X, Y) and (X, Y) P(X, Y) for the subspaces of G-equivariant maps, i.e., f(g(x) g(f())) for all g E G and E X. If Y carries the trivial action, equivariance reduces to invariance.
Lemma D.4. Ler X, Y, Z be representations of group G. Consider a subset of G-equivariant polynomial maps S Pa(X, Z) that satisfies the spanning condition:
P(X, Y) C span(L(Z,Y) o 5) == span (A (ZZY)EE))
(33)
Then, it follows that P)(X,Y) (pan(Lc(Z,Y)o35 In the special case where G Hi1 H2, the spanning condition can be relaxed
PH, (X,Y) S (pan(L(Z,Y)) ( S).
(34)
Remark.
The case for G = SO(3) x S, is from Thm. 1 of Dym & Maron (2021). In this context, the action of Sn on the representations faithful, while we are concerned with features that are invariant under Sn-
Proof. Let f E Pa(X.Y). Since PX( S P(X,Y), by the spanning condition, f can be written linear combination of compositions:
ApoPi
(35)
1=1
where Pt E S and A; (ZZ Here, we have absorbed the expansion coefficients into the lincar maps A;.
Since f is G-equivariant, it a fixed point the group averaging operator. Applying this operator to both sides of the equation gives:
f(x) pr (rr ()((p(())()))) dg
(36)
(
N Σ EM(o(()aa)) )
dg
(37)
N
M (
(r dg pr(o)).
(38)
In the last step, used the fact that the maps ps S are themselves G-equivariant and moved the integral inward. Let us define the averaged linear maps as
(Ac): := 1 py pr Aspzz())
(39)
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By construction, each (Ac); is G-equivariant linear map, i... (Ac); Lz( The expression for f() can now be written as f(z) Σn= (Ac); o pz(x). This shows that any map in P'(X, Y) can be expressed as a linear combination of compositions of maps from S and equivariant linear maps from LG(Z, Y). Therefore,
Po(X,Y) C span(Lo((,)))
(40)
The reverse inclusion, (p o S) S Y), is true by definition. since the composition of two equivariant maps is equivariant. Thus, the equality Pa(X, Y) = =saan[La Y) o S) is established.
The proof for the relaxed condition when G = H2 x H2 follows the exact same logic, by taking an I Po(X,Y) S (X, Y) and averaging over the group Hi.
Lemma D.5. Consider the TFN model with polynomial parametric radial function, input space X, a lifted representation space Z, and a final outuut space Y. Suppose the final output space Y Y @ R", where Y is a representation of SO(3) or 0(3), and the symmetric group Sn acts on Y by permuting the coordinates in the IR factor. Then the family of polynomial feature maps, F S CPc(X, Z). used in TFN satisfies the relaxed spanning condition:
Ps. X) S span span(L(Z,Y)) ((Z,Y)
feat )
(41)
Remark. Lem. in Dym & Maron (2021) proves the case for G = SO(3) x Sn. The proof for 0(3) is similar that for SO(3) and is therefore not repeated here. It is worth noting that in the ((3) case, the spherical harmonics map can still be used to construct the component-wis map from representation Vi=1- z R to symmetric algebra Sym, (Vi=1),, which in turn is used to build the lifted representation.
Lemma D.6. In Ex. 2.1, function families F-oly contain all equivariant polynomial maps.
Proof. According to Lem. D.4, to prove that a family of equivariant maps contains all equivariant polynomials, it sufficient show that satisfies the (potentially relaxed) spanning condition. We verify this for Fpoly
By Lem. D.5, for Y = @ R", the family J-poly satisfies the spanning condition
Ps. (X,Y) S span span(L(Z,Y)) ((Z
Fpoly) )
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We now only need verify the following spanning condition:
Ps. (X, Y) C (pan([((,,)) foar
(43)
For any g E Ps,(( F), we construct an Sn, equivariant polynomial map to Y as g:X X + Y, j(n,,..., = g( for j == 1,..., n.
(44)
This function is clearly Sq-equvvarr polynomial because all components of its output identical. Thus, g E Ps. (X,Y). ). By the spanning condition, we can write
Σ A ps, where A; E L(Z,Y) and p; E Jpoly.
(45)
Applying the averaging operator to both sides yields
1
Σ
(((((=))oo)) = =)(x)= = - ((As))oo(()))
(46)
SSa oES,
where
(As..): = |Sa| oES.
1
Σ ne 0 Lo(Z,ý)
(47)
This establishes the spanning condition
Ps. (X, Span((oo
Teat ).
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We consider the higher-order approximation theorem of MLPs, and obtain C* Cx via higher- order approximations based on polynomial parameterization.
Lemma D.7 (Higher-Order Approximation Theorem for MLPs, Pinkus (1999), Thm. 4.1). For any
compact set K C R", any function I E ("²),, and any non ron-polynomial activation function 5 (IR), lef > Then there exists an MLP parameterized map go with activation function
such that
max 1 #EK 21 ... 0h
Jhh 21 ... ...0f.9000))) <E
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for all non-negative integers kn with K +...... kn < m.
Theorem 5.1. In Ex. 2.1, the function families FTEN with smooth activation function are C -dese
in the space of smooth equivariant maps C (XXXY That is, for any integer r 2 0, any map f E C TXGXX (, any compact set K C., and any >> O, there exists function g FTFN such that
maxsek || ( DD < Sr.
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Proof Thm.. 5.1. On the domain of analyticity of function, its Taylor polynomials and their derivatives converge locally uniformly to the function. Therefore, an analytic function and its derivatives can be uniformly approximated by polynomial and its derivatives on any compact set. This establishes the -* -density of polynomial functions in the set of analytic functions. Furthermore, by Chap. Thm. 5.1 in Hirsch (1976), the set of analytic functions is Ce--dense in the space of smooth functions. Consequently, polynomials are Cro-denss in the space of smooth functions. By applying the group averaging operator, equivariant polynomials are also goe-dense in the space of equivariant functions.
We use mathematical induction to prove the C coe-density of FIFN. Let
f()) (f(m), id) ..... (1 id) ext C.
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We prove that for the TFN, the MLP-based parameterized map fkipp can approximate the polynomial- based parameterized map Jpoly- After establishing the approximation property of the feature maps, and noting that the equivariant pooling maps coincide in both settings, we obtain the C approximation of JTPN by FTEN. The desired approximation result then follows directly from Lem. D.6.
For the base case n = 1, there is no difference in the derivatives with respect to the X component corresponding to the identity map. Therefore, we only need to discuss the output of f()), Since
(1) Figng (ZI, I.) = 0 igmms
= 1,2,.....
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only need to consider the approximation of the derivatives of A (1)
ilams Expanding A (1)
dama yields
Σ
clssma)) (1amm)
Σ (1) (11x xzlle) (an - xxy)/l>;; x2l12)) (52)
Li,mo 22.m2
The difference between MMP and poly lies in the parameterization of hıg When taking derivatives of any order of Allymn. each term the result is product of derivatives of of various orders and a fixed function, and the number of terms is finite. By Lem. D.7, these terms can be approximated to any precision on a compact set.
2(k)
Now, assume the conclusion holds for n k- - 1. We will prove that it also holds for k. Let
(k-1) / (x),..,,,,. In)) Vigms Jilgmg
(53)
The update rule is given by
7(k) Jigga (#1...., xx) Byggs + iamss
1)
(k)
i= 1,2,,......
(54)
By the inductive hypothesis, the first term can be approximated by MLP model to any precision. We observe the second term. Similarly, expanding A (k)
iama yields
A
(k) Hymg
Σ
ms) (12.ma)
Σ j4i
(K) h/ (1| 121 ((x+ 11)//12; -x,, -2z 23llz)o(t-1)
(k-1)
Ja.ma
(55)
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The derivative of this term is a finite sum of products, where each product involves derivatives of various orders of both (1 and ((k-l)
To show that the error of this term can be controlled, we only need to prove that for vector space X, if for any compact set K C X, we consider maps f E C(X) and 9 E C(X), and for any El there exists /i E R1, and for any 62 there exists 91 R2 such that
max =Ek
[l) f1(x)] < <1,
max ll9(x)) (1))l <2,
(56)
wek
then for any > 0, there exist fi R1, 91 E R2 such that
max maxllf(z)g(z)) - fi (x)) <c. =ek
(57)
We bound the term as follows:
(I(())(()) (1)))) (z)]
< (1a (x()
$ +lo
I
(x) I (x)]le 61
(58)
Since continuous function compact set attains its maximum, let L1 be the maximum of [+)]]] on K and L2 be the maximum of (z)]l on K. We get
max (If())g(()) (z)g)) s Lzc + L + c1
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æek
This shows that the error can be made arbitrarily small, completing the proof.
D.3 SOME RESULTS ON TOPOLOGY
Here we adopt the definition of (H) and (H, H') from $ C.I. and consider the twisted product between G-sets defined in Chap. 2. Sec. 2 of Bredon (1972). We denote XH == {x X G2 H}. from which we obtain the following lemma on topology.
Lemma D.8. When the action of a group G on G-manifold M is faithful, the following decomposi- tions hold:
M(r) Mir XNo(H')HH G/H
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and
M = MH XNo(H))H No(H,H')/HE!
(61)
Proof. The first identity is derived from Thm. 1.31 in Meinrenken (2003), which states the existence of homeomorphism
a : Ma XNo(H)/H' G/H' - M(n given by ((x, = g(x).
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We now prove that (MH XNoLH)/H (No(H, H)/H") M The second identity then follows by restricting this homeomorphism.
(C) Take an element [H' where Myr and g E No H') Its image under a is g(x). The isotropy subgroup is Gy(n)) = gG.g g Since g E No (H. H"), we have H S gH' g! This implics g(z) E "M and since its orbit type is (H'), we have g(x) E M(lrr
(2) Take any y E ( This means H Gy and (Gy) (H'). The latter implies there exists 9o G such that G, Ho 96 The condition H C Ho 9o implies that E N, H'), Let 9n (v). Then G₂ H' so MH. We can then write
y go (o (y)) = go(x) = o gofd
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Since x E Mr and 9n e No H H'), this shows that y is in the image of the restricted domain. The inclusion is thus proven.
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From the following proposition onward, we need to invoke stratification theory. For the definitions of Whitney conditions and stratifications, we refer to Chap. 3, Sec. of Field (2007). Going forward, unless otherwise specified. all function spaces in this paper are equipped with the (weak) C topology. For the topology on function spaces, we refer to Chap. 2, Sec. of Hirsch (1976).
Proposition D.9. Consider smooth manifolds M and N. Let (Sa; Sg) be a pair of submanifolds Sa at point I E f (S.), then there exists a neighborhood U of f in C N) and a neighborhood V of x in M, such that for any map g E U and any point y V, g is transverse both Sa and Sg at y-
in N that satisfies Whimey's 's condition (a). If map f : M + N is transverse
The proof is adapted from Prop. 1.3 of Trotman (1976). We proceed by contradiction. Assume the conclusion false. Then for any neighhorhood V of x and any neighborhood U of S, there exists map g E U for which the condition g Sa, S3 on V does not hold.
Proof.
For a neighborhood Vi of æ with radius less than é1, we can construct a sequence of maps ( converging to f such that transversality to S. S3 fails on V1. Let the sequence of non-transverse points be {yh () }. We can similarly construct, for neighborhoods V2 of x with radius less than 6t. sequences of maps (9h²) converging to f and corresponding sequences of con-transverse points {yn ( }.
Now, consider the diagonal sequence of maps {gh")) } and the corresponding sequence of non- transverse points {um ( }. We have gn + f and (n) + x. We can partition the sequence (u (")) into a subsequence lying in Sa and another lying in Sg. At least one of these two subsequences must be infinite. is therefore sufficient to negate the following proposition: There exists a sequence of maps {gn} + and sequence of points {yn) for each n, In is not transverse to S. or S3 at yn-
with either {y^} C Sa or {v.} C Sg, such that
This is impossible. Note that for sufficiently small 61, the closure of the neighborhood V1 is compact by local compactness of the manifold, and we can consider the control of the neighborhood U over functions this compact closure in the C topology of functioin space. Also note that d(gn)s (Ty, M) converges to dfx(T, For the case of Sa. by the smoothness of the manifold. (yn( Sa converges to T(r Sa. For the case of Sg. by Whitney's condition (a), the limit of T ) S3 contains Sa However, f is transverse to S., at meaning
d So N.
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Therefore, due to convergence, the transversality condition for S, or S3 must hold for In at yn for sufficiently large n, which is a contradiction.
Corollary D.10. Under the conditions of the previous proposition, assume that the stratification
S = (S,) satisfies Whitney's 's condition (a). If a map I is transverse to a stratum So at a point (S.), then there exists a neighborhood Va of x and a neighborhood U2 of f in O (M, N) such that for any E Uz and any E Vz, is transverse to the entire stratification S in the neighborhood Vz.
Proof. Let x be a point where f is transverse to Sa. We consider all strata S3 that satisfies SpnSa * Ø. Since a stratification is locally finite, there are only a finite number of such adjacent strata. For each such adjacent stratum Sg, the pair (S) satisfies Whimey's condition (a). By Prop. D.9, there exist neighborhoods U.(a.3) of f and v((a.)) of x where transversality to both S. and Sg holds. We can then construct the desired neighborhoods by taking the finite intersection of these individual neighborhoods labelled by 3:
Up =
n SAnS.
U(a.9) and
V2 ==
((a.8)
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SANS
Since the intersection is finite, Un and V, are still open neighborhoods of f and x, respectively. For any g E U, and y E V,, 9 is transverse S, and all its adjacent strata Sy y. Therefore, it is transverse to the entire stratification S within the local neighborhood Vn..
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Lemma D.11. For G-manifold M and G-representation Y, the collection of submanifolds
{M x XXur))n)(ara
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forms Whitey stratification of (M x ())-
Proof. By Prop. 3.7.1 of Field (2007), we obtain that these point sets in the collection are smooth G-submanifolds. By Lem. D.8, have
(M(n x Y(m))a =
(M x Y(m)) XNol G/H
(67)
(M(H) x Y(Hr))) (No(H)/H G/H
(68)
= -Mii)) x ((H))) XNe(u)/B G/Hl
(69)
By local trivialization, similar to the proof of Prop- 3.9.2 of Field (2007), is sufficient to prove that the pair ( V(k)) satisfies Whitney's 's condition (b). By Prop. 3.9.2 of Field (2007), the orbit type stratification for representation space a normal orbit type stratification, and therefore the pair of strata (Yar), Y(K)) for (H')= (K) satisfies Whitney's conditions. We now show that ((H)) (K)) also satisfies Whitney's conditions.
The condition holds for the pair (YHr), Y(K)). Therefore, for a point x E ((K) n Y(HY) and for sequences [yn} C Y(K)) + x and {rn) Y(H) - x such that the secant lines Tnli converge to line 1 C. T.Y and the tangent spaces Ty Y(K) converge
linear subspace E T,Y, we have I S E. By restricting the point x to be in (K) n (Hr) and the sequences {z,, and {yr} to be in Y.H. (Ho) and Y respectively, the conclusion C E still holds. Therefore, the pair (X Y also (Hry (K)
(K)
satisfies Whitney's s condition (b).
Lemma D.12. Let M and N be topological spaces, with M being compact. Consider continuous map f : :M > N. Let C be a closed set in N, and suppose there exists neighborhood V of the preimage (C)
Then there exists a neighhorhood of f in C(M, N) with the C topology (compact-open topology) such that for any map 9 E U, we have g(M (V) n Ø.
Proof. Note that M \ V is a compact set, and N \ C is an open set. By the definition of the compact-open topology, e.g. Definition 43.1 of Willard (1970), the set
U := (h E C(M,N) h(M CN)
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an open set.
Since V is neighborhood of f we have C V. which implies Therefore, the map f itself is an element of U, meaning U is an open neighborhood of f. For any map g E U, its definition directly implies that g \ V) N \ C, which is equivalent to g(M VVnn = Ø. Thus, this set is the desired neighborhood.
D.4 GENERIC EQUIVARIANT MAPPINGS
Proposition D.13 (Equivariant Smooth Extension). Let M be a smooth G- G-manifold and let S C M be smooth and compact G-submanifold. For any smooth equivariant map J E C. (S, Y) defined on S that maps into a representation space Y, there exists smooth equivariant extension f E C2 (M,Y) such that restriction to S is f. i... fls f
Proof. The proof strategy is the same that for the Tietze- Gleason Theorem (see Chap. 1, Thm. 2.3 of Bredon (1972)).
First, view the map f as collection of dim Y smooth real-valued functions defined on S. By the standard smooth extension theorem, c.g., Lem. 5.34 in Lee (2012), there exists a smooth (but not necessarily equivariant) extension yo E C (M, Y) such that ypls =
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We then construct equivariant map from yp using the group averaging operator. Define the map & : M + YYby by
(xx) = 1 px(xx))llexxx) dg.
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Due to the linearity of the integral and the properties of the Haar measure, this map is equivariant. For any h E G:
I pr
v(((x))
dg
(72)
[ ("((ogh")-"))<00 dg'
(73)
= =pr(h) /. pr pr (x) =r (h))()()).
(74)
(75)
Next, we verify that the restriction of d to the submanifold is equal to the original map f. For any
E S:
(s))
pr(gg') ()((((s)) dg
(76)
pr pr (()(())
(77)
G
pr(g 1)py (s)
(78)
f(s) f(s) / 1dg =
ddg- f(s).
(79)
Finally, the smoothness of &i follows from the smoothness of and the properties of integration over a compact group. This can be verified by a local coordinate analysis. Thus, tp is the desired smooth equivariant extension f.
Corollary D.14. Consider maps from G-manifold X to representation space Y, where X contains compact, smooth G-submanifold M. Let S be subset of CG (M, Y) that contains an open dense set. Then the set of maps f C (X, Y) whose restriction M lies in S (i.e., flM S) also contains an open dense subset of Co (X,Y).
Proof. We recall that a set contains an open dense subset if and only for any non-empty open set, its intersection with the contains non-empty open subset. Let A (s Cg (X, Y) I'M S). Our goal to show that for any f E C (X, Y) and any of its open neighborhoods U, there exists non-empty open set V CUA.
Consider the restriction map RESM (X, + Cg (M,Y), (, by Prop. D.13, this map is surjective. Furthermore, this map is continuous and open.
Let U be an open neighborhood of an arbitrary map f- Since Res is an open map, its image, Res (U), is an open neighborhood of fla. By our hypothesis on S, its intersection with the open set ResM (U)) must contain a non-empty open subset. Let us call this non-empty open set VL, We have
V' C ResM (U) N.s.
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Now, let's consider the preimage : Res (V"). Since Res is continuous and V is open, V is an open set. Since ResM is surjective, the preimage V must also be non-empty. Therefore U n V is non-empty open subset of U ( A, which proves the corollary.
For C" manifolds X and Y, C*(X) is the set of C" real-valued functions X, and C* (X) is the set of C" maps from X to Y. For manifolds with a C G-action, (X Y) is the set of C" equivariant maps from X to Y. We assume these function spaces endowed with the C" topology; in our proofs, we always consider the C topology. For a map f E C* (X, Y) and submanifold A C Y. dh A denotes that is transverse to A. For G-manifolds, f the A denotes that f is in
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equivariant general position with respect the G-summanifold A defined in Bierstone (1977). For E CX Y), the jet map j" X - J" Y) maps a point z to the equivalence class of the first derivatives of f at x. We obtain the following results.
Proposition D.15 (Bierstone (1977), Thm. 1.3). Let M and N be smooth G-manifolds. Ps closed Gssummnnifold of N and K is a compact subset of M, then the set of maps f (M, satisfying Mgg P K forms open subset of C (M, N) (in the Whiteyy C topology).
Proposition D.16 (Bierstone (1977), Thm. 1.4). Let M, N be smooth G-manifolds and P be G-submanifold of N. The set of maps f E C (M, N) satisfying f the P forms residual subset of CE (M, ) i.e., countable intersection of open dense sets (in the Whitmey C topology).
Remark. For compact M, note that the Whitney C topology coincides with the C topology-
Proposition D.17 (Stratumwise Transversality Theorem). For any orbit type (H) E Oc(M), map f E C (M, N) with f mi P satisfies the stratumwise transversality property:
flaa Mu +> N fflm m pH
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Alternatively, this can be expressed in the language of jets as
1e Pflma: MiH) + (May x N) f flaan di (M(H) x P)(
(82)
Remark. The fact that f the P implies stratumwise transversality on the fixed-point sets is a funda- mental conclusion derived from Prop. 6.4 of Bierstone (1977). Our proof below is straightforward corollary of this.
Proof. From the discussion in Prop. 6.4 of Bierstone (1977), have
(f M(n (M x N)),, {f ti (M P)(n-
(83)
Note that the image of the map (f M(n)) is contained within M(H) N. The transversality condition an equality of tangent spaces:
((f"( (T, + +T,((M x P)(n) = T,M x M)))
(84)
Intersecting both sides of this equation with [y((M(H) x ())) yields
+ (MM(( x P)(n) = TTM(((H) x N)))).
(85)
Therefore, we have
(f M(H) + (Min x N)m,, { ti (Min) x P)
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L
Proposition D.18 (Bierstone (1977), Sec. 7). Consider smooth G-manifolds M and N. Let P be a
smooth G-submnnifold of N. If an equivariant map f satisfies f the P at point x E f-)(P,, then there exists a neighborhood U of f in Cg (M, N) and a G-imvariant neighborhood V of the orbit G(r) such that for any map g U and any point y E V, it holds that g meg P at y-
Remark. The proposition above can also be derived from the properties of stratifications given in Prop. D.9 by the definition of equivariant general position in Bierstone (1977).
Proposition D.19. Let G be a compact Lie group, M be a compuct, smooth G-manifold, and Y be G-representation space. The set of smooth equivariant maps E C& (M,Y) that satisfy the transversality condition
j (fllM()) dh (M(H) x Y(H)))))
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for all pairs of orbit types (H) and (H') such that (H) is present in M and (H') 2 (H), contains an open dense subset of C (M,Y)
Remark. The proof of density is straightforward. However, the openness of this property does not generally hold. A counterexample can be constructed following Ex, 2.1 of Bierstone (1977).
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Proof. Since M is compact, by Proposition 3.7.2 of Field (2007), the orbit types are finite. For any given symmetry type (H), by Prop. D.16 the set of maps satisfying f the ((r) is an intersection of a finite number of residual sets, and is therefore itself a residual set. By Prop. D.17 the set of maps satisfying the transversality condition stated in the theorem is dense.
Instead of directly proving openness property, we prove related proposition: for a map that satisfies f the Y(H for all orbit types (H), there exists a neighborhood U of f such that for any g E U.
( ) th (Mar x (()))))) for all (H') 2 (H).
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The proof proceeds by induction on the dimension of the strata in the orbit type stratification of Y. ordered from lowest to highest (or equivalently, from highest to lowest symmetry). We discuss the fixed orbit type (Hf) in M. Then we can construct neighborhoods that hold for all (H) by taking the intersection of the neighborhoods corresponding to each orbit type (H).
We start with the lowest divees-dimeiioonal strata. Let Y(m), be a stratum of minimal dimension. Such stratum is unique which corresponds to maximal symmetry type (G) and is closed G-submanifold Y By Prop. D.15, the set of maps in general position to Y(H.) is open in C (M, Y). Thus, there exists neighborhood U1 of f such that for any g E U1. g the Y(..). By Prop. D.17, this implies that for all (H).
(g)))))) m (M(r) x Y(H,))())
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Assume the proposition holds for all strata of dimension up to k - 1. Let Uk-1 be the neighborhood of f found from the inductive hypothesis. For any map g E Ux-1 and for any stratum Y(H.) with dim Y(H,) I k 1, we have
( m (Mas) x (())) for all (H.) 2 (H).
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Note that by Lem. D.11 the collection S(F) ((Mr) Y(Hr) (H)) a Whitney strat- ification and satisfies the frontier condition. Therefore, by Cor. D. for any E M(H) with f(x) Y( there exists neighborhood Vg of and neighborhood U2 of (f such that for any map in Uz, it is transverse to the stratification S(m)
Next, we need to obtain an open set in Cg (M, Y). We use the following sequence of maps between function spaces:
ReaMon)
(M,MY)
(M Y)
(91)
ca (MB) (MB) ())))
(92)
(M (Min xan)).
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The maps between these function spaces are continuous in C topology. For example, the 0-jet map is a restriction of the continuous 0-jet map on the general function space, which can be obtained from the continuity of jet map in the Whitney O topology established in Chap. 2, Prop. 3.4 of Golubitsky & Guillemin (1973). Since the topology on the equivariant function space is induced from the general function space, the restricted map is also continuous. Similarly, ResM, and the inclusion are continuous. From this analysis, we can pull back the neighborhood U, to obtain neighborhood U² of f in (M, Y) where the transversality holds.
We deal with the global result. For any i with dim Y(H,) S k 1, the collection of neighborhoods (VaJeef (((,,)) for all with dim Y(H.) < k-1 1 forms open cover of (Y By the frontier condition, any stratum that intersects the closure Y(H.) has strietly smaller dimension than Y(H.)- Hence U, Y(n,)) is closed. Since M is compact and f is continuous, the preimage f (U, Y(n)) is compact, so there exists a finite subcover { (Vz, J of the preimage. We construct the neighborhood Vx == - Va,, of the preimage in M, and the neighborhood U(() = - Uk, of f in the function space.
.
In the Co topology, by Lem. D.12, there exists neighborhood U of f in C(M,Y) such that for any y U(2).
g(M) \ Vi) vi) U = =.
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By pulling this back through the inclusions (M,)) - C (M, + Y), we obtain neighborhood ( in Că YM, The neighborhood is open in the C topology, so also open in the C topology. Let U% Un-1 n nuth n (
For any map g E U% we have
for E V { [ m S(n)
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o(()) & Y(H)
f UnzEMAVA
for all i with dim Y(H)) 1k
Now we show by contradiction that there exists a neighborhood Uj. C Ug of f where the transversality condition holds for strata of dimension k. By the induction hypothesis, the transversality condition holds for all strata of dimension at most l: - 1. Therefore, it also holds for all strata of dimension most k. The proof idea is from the proof of openness of equivariant general position in Sec. 7 of Bierstone (1977), where the closedness of P is replaced by the condition that f does not intersect low-dimensional Y(H,)) outside of Vi-
Assume openness does not hold. Then there exists sequence {g.} - f in U% such that each In fails the stratumwise transversality condition for some stratum of dimension k. Since the stratumwise transversality condition fails, it follows from Prop. D.17 17 that In Mg Y(H,) for all j with dim Y(u,) = l also does not hold. The points of non-transversality for these maps, {yn), can only occur in M Vx and (UY) E Ug Y(H)). Since M \ V& is compact, there is a convergent subsequence {ym, J with limit x. Then 9. + f(x). By construction of U/ f(M \ V&)) does not intersect Y(H.) for any dim ((H,) Sh-1 Thus, we must have f(x) E Y(H,) for some
We claim that 9n. (Um.) E +(HH,) ) for all sufficiently large i. Otherwise, there exists a further sub- sequence {9m., (Um.,)) }} such that In., (Um..) ) E Y(HH,) for some j' * and all a, while still In:. (Un:,) (xx) E Y(H,). Hence Y(H,) Y(H))) j Ø. By the frontier condition of stratification this implies dim Y(H,) < dim Y(H,,). ). However, Y(Hr) and Y(H,,) have the same dimension, yielding a contradiction. Therefore, for i sufficiently large, In: (yn,) ) Y(H,)-
We now show that this contradicts the assumption that f the Y(H,) at If were to satisfy the Y(u,) at x, then by Prop. D.18, there would exist a neighborhood U of f and a G-invariant neighborhood V of G(x) such that any g E U satisfies the Y(H,) at any y E V. This contradicts the existence of sequence {gn} and points {yn} where transversality fails. This completes the proof. D
Proposition D.20. Let F be a C -dense family of smooth parameterized maps in (X, and {Mj} be finite collection of compact, connected and smooth G-submanifolds of X. Let
(()-(1(((f)) fo E (G2) (H), (G(()) = (H')}
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There is a -ense subset G F. g E G, the set (HH) is disjoint union of smooth G-submanifolds of X, and its dimension satisfies
dim (H)la dii(m
(dim V dim )(Hr))
(97)
if the right-hand side of the equation is not smaller than dim G dim H. Otherwise, the set ((H) M,) is empty.
Proof. By Prop. D.19, for each My. there exists open dense set in C (M,, Y) such that for any map in this set,
((l((M))))) th ((M)) Y(H))( for all (H') > (H).
(98)
Therefore, writing codimx := dim X dim Y for the codimension of Y in X, the dimension theorem for transverse maps (see, e.g., Sec. 2.3 of Amold al. (2012)) yields
codim(M,) S(H)+(H)) = codim((M)))s) xY) ((M)) x Y(H))
(99)
Furthermore, from the proof of Lem. D. we have
(()))(H) Y(H))) ((M)) vHH XNo(H)/H G/H.
(100)
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Thus, obtain
(M ((P) = codimy (H')
(101)
Moreover, since the dimension of orbit with orbit type (H) in a G-manifold is dim G' - dim H, the dimension of (HP) calculated above is less than dim dim H, then (( is an empty set.
For each Mj we take the corresponding open dense set Uj C C (M,, Y) on which the maps satisfy the dimension theorem. Then, by Cor. D.14, the set of functions f CG (X.Y) that satisfy flM; E Uj contains an open dense set U². Since [M,} is a finite collection, the intersection U= n= UJ is also an open dense set. Therefore, the intersection of U with the -ense set F is a dense set G. Thus, this intersection the set of maps F that we sought to construct, which is dense and whose elements satisfy the dimension theorem.
D.5 PROOF OF TIM. 5.2
Theorem 5.2. Let F be a equivariant parametrization with C approximation capability. If for every (H) E (a(M) we have (py H)) E (Y)).. then for any finite union of compact, smooth G-submanifolds M C X, any I E C9 (X,Y) Y), any integer 2 0, and any >> 0, there exists map 9 E such that
(11) and glas is almost isovariant relative to Y. Furthermore, if the feature space Y contains representation yer for integer > max, (dim M2}, where Y itself satisfies the condition (H)) E (&).Y, then the approximating map as can be chosen be isovariant relative to Y.
max D* f(x) D* < S,
Proof of Thm. 5.2. By Lem. D.8, we consider the relation
dim No(H, H²) dim (H) = aH,, H²) dim Y dim Y
(102)
Regarding the G-representation as a faithful representation of G G/kerpy, for orbit types satisfying (py (H))) > IHH, Y). the dense and open property of the minimal orbit type by Prop. B.4 in the fixed-point space implies that for each Mj.
dim H dim Y (Hr) = dim y dim H (I, 0.
(103)
By Prop. D.20, implies
dim (Mz) > dim ((H)
(104)
With respect to the Hausdorff measure Hd since the dimension of S( (9)M, is strictly less than the dimension of the manifold it lies in, its measure is zero. Then, by the finiteness of the set of orbit types for compact manifold and the finiteness of the collection { (M,) ). it follows that g) M is an almost isovariant map relative to Y.
Furthermore, if we require g to be isovariant relative to Y, we need
S(H) (gl (gl M2)) Ø.
(105)
The condition for this set to be empty related to its codimension. Since the inequality
dim y dim Y H' al H²) 2
(106)
scales with multiplicity r to become
r(dim y dim y it is sufficient to choose the multiplicity for the representation fr such that r max, (dim M,
2
(107)
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E TABLES
All closed subgroups of (()) are denoted using Schoenflies notation, where should be noted that K S0)3) and K₁ = 0(3). When interpreting the tables, care should be taken recognize the low-dimensional equivalences: C. Cia = Civ. Dj C2 Dıa Czv. and Dia C2n In the tables of symmetry infimum, we list representative subgroup from each conjugacy class and omit the class notation (e.g., C2 instead of (C2)) for clarity.
E.I MINIMAL PROPER SUPERGROUPS IN SO(3) OR O(3)
Minimal proper supergroups table of SO(3) or (3). Some of the results can be found from Fig. 3.2.1.6 1.6 of Aroyo (2016). We only present the results for (()). The diseussion for SO(3) can be obtained by removing the subgroups that are not subgroups of SO(3) (i.e., the subgroups the first kind), namely Ck, Dx: T,O, O, Coo, Dxe, and K.
Table 6: Table of minimal proper supergroups H axial closed subgroups of 0(3). In the supergroup notation, p denotes prime number and p" denotes an odd prime number.
G
H for General k
H for Special k
Ck
Cpk. S2k, Ckty Cku DE
Dr- (k 2); (k =3)
S2x
S2 C2k.hi Dkd
Tn (k: = 3)
Ckh
Cpk, Dich
Cpu (k: 11; Dyr: (k: 2)
Cku (k > 1)
Opki Diasy Did
Ta (k 3); (k ===] 2)
DR >
Dpks Dkt: Did
T (k 2): (k 4); = =35)
Dan (k > 1)
Dpkc
Th (k 2n:0n (k =4)
DEd (k>>
Dprk,d Dzh
Ta = 2); On (k: 3); Jn (k 335)
Table 7: Table of minimal proper supergroups H/ of other closed subgroups 0(3).
G
H
T
Ta,
0,1
Ta
On
T.
On, Ib
O
On, K
On
Kn.
1
In, K
I
Kn.
Coo
Cooks Coo,vy Doo
Coch
Do0,
C
Doc.
D
Doc,h,
D Doch
K
K
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E.2 DIMENSIONS OF FIXED-POINT SUBSPACES FOR SUBGROUPS OF SO(3) 0(3)
Dimensions table of fixed-point subspaces for subgroups of 50(3) or O(3). Some of the results can be found from Table B.1 and Table B.2 of Linehan & Stedman (2001). We only present the results for (3). The discussion for 50(3) can be obtained directly from the table.
Table Dimensions of fixed-point subspaces for closed subgroups of (3) acting on the irreducible representations V m where 0 (7) [ and (1) = [2 + k)/ (2k)
ln
Subgroup
lo even
to odd
to even
to odd
Ck
20x((o) +
even
(6,,(4
24z (l)) +
S2x
A odd
0
2ah (la) + 1
even
2ap (ln) + 1
Cuu
odd
2bx((0)
2azz (lo) +
Ckv
(x(1
a (1o) + +1
ax +
a& (4))
Dy
ax + 1
a (40)
ax(1o) +
a& (2)
even
(x:41) +
ax (x))))
DEn
odd
by (4o)
(2)
02k (lu)
h even
by (7o)
a2 (1o) +
02k (la)
DRa
odd
0x (l)) +
ag (4o)
T
200 (lo) + a2(1))
lo
Th
2az(4n) + az a2(4)) - lo +
Ta
as(1o) + b2(4)) + by (lo) - to a. (1() + as((1)
a1(4o) + az ay(4)) + a2(1)) - lu +
O
+ a2((l) Lg +
O
(4(4) 0z(4)) + a2(1)) - lo
(5(4)) + a3(4))
+ a2())) - 10 +1
In
as(4)) + az(46) + o2(4)) 4o +
C
Coch
0
Coon
0
Doo
Doch
0
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E.3 SYMMETRY INFIMUM FOR SUBGROUPS OF SO(3)
Symmetry infimum table for subgroups of SO(3). In the following tables, we use a color-coding scheme classify the types of symmetry increase. An increase the full group is termed full degeneration and is marked in red An increase a supergroup
strictly higher dimension is termed continuous degeneration and is marked in blue An increase to supergroup of the same dimension is termed discrete degeneration, and marked in yellow No increase is termed no degeneration, and is marked in green All subgroups increase to to K when / = 0.
Table 9: Symmetry infimum of general axis subgroup of SO(3) on Ver T > 3, 40 0. /=
to
<
lo 2 k
even
lo odd
Ck
Doc
Coo
C₁
Dx(k > 2)
Dos
K
Dk
Table 10: Symmetry infimum of special axis subgroup of SO(3) 1 r> 3, lo
4
3
lo 2
4
D2
K
D2
T
D2
Table 11: Symmetry infimum of polyhedral subgroup of S0(3) on V r >> 3, lo > 0. The "Caption" in the table takes the value lo
6, 10, 12, 15, 16, 18, 20
22, 24 28.
6, 9, 10
40 = 3,7
4,8
4 2 12
other
T
T
T
0
T
K
lo 4,6,8,9,18 10
4 2 12
other
O
0
K
Caption
lo
30
other
I
I
1
K
Table 12: Symmetry infimum of infinite subgroup of 50(3) on 1=b, > 3, 3, >
40 even
4o odd
C.o
Dooh
Coch
D
Doch
K
K
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E.4 SYMMETRY INFIMUM FOR SUBGROUPS OF ((3)
Symmetry Infimum table for Subgroups of ((3). The color scheme for full, continuous, and degeneration the same as for (3)3). For cases of discrete degeneration of lo = 1=,, we distinguish between (a) light Green which indicates the predictable increase to (=((H)) for the projection map ((3) 5O(3), and (b) yellow which indicates all other exceptional cases of discrete degeneration. In the following table, the first type of subgroups degenerate K and the others to Kn. when l = 0- All subgroups increase to to K₁ when lo = 0+
Table 13: Symmetry infimum of general axis subgroup of 0(3) on V 1=1
r > 3, la > 0.
< lo <
k
S lo
<
2k
10 2 2k
Subgroup
tu even
odd
lo even
Lo odd
la even
lo odd
C₁
Doc
Coce
C
C
C
Ck
k even i odd
Kh
Kh
S2N
S2x
Szk
S2u
S2k
K₁
Kh
Kn
Kn
Kh
Kn
i even
K₂
Kn
Kn
K
Kh
Kn
k odd
K₁
Kn
Ckn
Cku
Cku
k even
K₁
Coco
DRd
Cku
Cku
Ckv
Cku > 2)
k odd
Kh
Coce
DRh
Ckv
Cke
Cku
k even
Doo
Kh
Dx
Dka
Dy
D& > 2)
k odd
Doc
Kh
DR
DRa
DE
DR
k even
Kn
Kn
Kn
Kh
Kh
Kh
(ktck 2)
k odd
Kn
Kn
Dun
Dun
Duh
DEn
k even
K₁
Kh.
DRa
DEa
D&d
Dwa
Dea(k > 2)
k odd
Kh.
Kh
Kh.
Kh.
Kh.
Table 14: Symmetry infimum of general axis subgroup of 0(3) on fhr 1=1
r > 3, lo > 0.
0 < lo < k
k S lo
< 2k
to 2 2k
Subgroup
to even
odd
to even
lo odd
lo even
lo odd
l even
Dock
Coch
Ckh
Ckh
Ckh
Ckh
Ck
& odd
D Doch
Coch
S2k
S2x
Szu
S2x
i even
Doch
Coch
Dooh
Coch
Czuh
C2uh
S2s
k odd
Dock
Cxch
S2k
S2x
S2x
S2x
k even
Doch
Coch
Okk
Ckh
Cku
Cku
Okk
i odd
Doch
Coch
Dooh
Coch
Czuh
Czkh
k even
Doch
K₁
Dkn.
Dkk
Deh
DR
Cki(k>>> 2)
I odd
Doch
Kn
Dka
Dka
Ded
D
i even
Doch
Kh
DRh
DEh
DEh
DEh
D&(k > 2)
k odd
Doch
K₁
DRa
DRa
Dea
DEa
& even
Doch
Kn
Dun
Dun
Duh
DEN
Dan (k > 2)
i odd
Doch
Dooh
Kh
Dzkh
Dzkh
k even
Doch
Kh.
Doch
Kh.
D22h
D2Rh
Dka(k 2)
k odd
Doch
Kn
Dka
Dka
DEd
Ded
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Table 15: Symmetry infimum of special axis subgroup of 0(3) on Ver
> 3, > 0.
1=1,"
lo 2
4
L0 =
t0
l0 = 3
lu even
lo odd
Czu
Coca
D2d
Czu
Czu
C2v
D2
Kλ
D2
Ta
D2
D2
D2h
KA
Kh
Kn
Kn
Kn
D2d
Kn
D28
Tg
D2a
D2a
Table 16: Symmetry infimum of special axis subgroup of 0(3) on Ver
3,
1=1
2 4
lo
to
3
to even
lo odd
Czu
K
D2h
Th
D2h
D2n
D2
K^
D2h
Th
D2h
D2h
D2h
Kh
D2h
Th
D2R
D2h
D2a
K
Doch
K₁
Dak
Dah
Table 17: Symmetry infimum of polyhedral subgroup of (3)) on /r
r >
3, 1o
> 0. The
|=1,
"Caption" in the table takes the value lo = 6, 10. 12, 15, 16, 18,20
20
22, 24 28.
lo 6, 9, 10
10 = 3,7
1o = 4,8 1o 2 12
other
T
Ta
0
T
K₁
1o
3, 7
lo 2 9
other
Ta
Ta
Ta
K₁.
all lo
Tp
K₁.
to 4, 4,6,8,
lo 2 12
other
0
O
K₁
all lo
O
Kh.
Caption
lg 2 30
other
7
I
Jn
K₁
all lo
In
K₁
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Table 18: Symmetry infimum of polyhedral subgroup of ((3) on V
>>>> 3, 4 > 0.The "Caption"
1=1
the table takes the value 4 = 6, 10, 12, 15, 16, 18, 20
22, 24
28.
T 40 3,6, 7,9, 10
4 = 4.8
4 2 12
other
O₂
K₁
lo
4, 4, 6,6,8800 9, 10
2 12
other
Ta
Oh
On
K₂
lo = 3,6,7 7, 9, 10
lo = 4,8 8
to 2
12
other
Th
Th
On
Tp
K
1 4, 6, 10
2 12
other
O
O
K₁
lo 4,6, 16,8,90 9, 10
lo 2 12
other
On
O
On
К₁
Caption
2 12
other
Tn
K₁
Caption
4 2 12
other
I
In
In
K₁
Table 19: Symmetry infimum of infinite subgroup of O(3) V
3, to 0.
1-17
even
lp odd
Coo
Doo
Coco
Cook = S200
Ky
Ky
Coor
Kh
Coce
Doo
Doo
K
Doch
Dood
Kh.
Kj
Kh
K₁
Kn
Kn.
K₁
Table 20: Symmetry infimum of infinite subgroup of O(3) on /rr
1= 1=
> 3, 40 > 0.
to even
lo odd
C.o
Dzoh
Coch
Coch S200
Dooh
Coch
Coor
Doch
Doo
Dzoh
Kh
D Doch =Doh=DDoo ood
Doch
Kh
K
Kh
Kh.
K₁.
K₁
K₁
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F DETAILED EXPERIMENT
Experiments in $ 6.1 and 6.2 are conducted with randomly initialized TFN (Thomas et al., 2018) and HEGNN (Cen al., 2024) architectures as the underlying equivariant models, whereas the experiments in $ 6.3 first pretrain a HEGNN encoder to obtain molecular embeddings and then fine-tune separate MLP heads for the final prediction task under different settings.
For TFN (Thomas et al., 2018), we use the implementation from GWL-ttst (Joshi et al., 2023), which relies on irreducible- rerpesentonnnnn features with alternating parity. For HEGNN, we extend the original design to multi-channel variant. In particular, the spherical scalarized features computed for nodes j and are defined as
1C) 3/.
1/VO
(6
(108)
where uhe and ij j.c denote the 1-th degree steerable features of channel e among a total of C channels. To obtain the degree-lo graph-level representation, we apply global mean pooling for each graph 8(V,8):
Gg.c = 1//2 E,
(109)
We will detail, in this section, how each experimental task processes these graph-level features.
F.1 VISUALIZATION OF REPRESENTATION SPACE
We a TFN with single-layer calculate the embedding. After that, extract features of degree la. we append an o3. Linear layer from e3nn (Geiger & Smidt, 2022), denoting this setup as TFN:=1, in our experiments.
F.2 EXPRESSIVITY ON SYMMETRIC GRAPHS
This section first introduces the detailed settings of 6.2 in $ F.2.1, and then introduces the reproduc- tion of the original experiment of GWL-test (Joshi et al., 2023) in F.2.2.
F.2.1 EMBEDDING DIFFERENCE NORM EXPERIMENT
We employed both TFN (Thomas et al., 2018) and HEGNN (Cen et al., 2024) to compute the norm of the embedding difference across 12 configurations for each model, varying the number of channels (I, 4, 16) and layers (1, 2, 3, 4). The degree-l discrepancy between two graphs Go and G1 is defined as
A 1/ICI Σ
Go.c
(110)
These choices give rise to 2 x 3 x 4 24 distinct A In Fig. 5, we report the maximum norm across all A Since every norm strictly positive, a maximal value below 10-6 indicates that all corresponding norms fall below 10-6, .eaninng none of them can distinguish Go from G1.
F.2.2 ORIGINAL GWL-TEST ON SYMMETRIC GRAPHS
Dataset. Same as the setting in (Joshi al., 2023), we construct four symmetric k--fold structures (k E {2,3. 3, 4, 6)), each centered at the origin. For each structure Go we apply random rotation to produce 9i which ensures Gi does not coincide with the original Go. The goal is to evaluate whether different equivariant neural network architectures can distinguish Go from G1. To validate distinct aspects of our theory, we consider rotations separately in 2D and 3D; in the 3D experiments we additionally ensure that G1 is not coplanar with Go.
Embeddings. The extracted lo do-dgree embeddings from TFN are fed into a vanilla classifier for the classification task. The model trained for 300 epochs to ensure the classifier had sufficient capacity to discriminate the classes.
Results. The detailed experimental results are presented in Table 21, which can be observed that the color blocks this table are completely consistent with Fig. 5. It demonstrates that our theoretical predictions in Table are in complete agreement with the empirical findings obtained from the model.
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Such remarkable consistency not only confirms the correctness of our theoretical analysis, but also highlights the importance of constructing mappings with appropriately chosen features.
Table 21: Results of distinguishing k-fold structures rotated in 2D/3D space.
Rotational Symmetry
3D Rotational Symmetry fold
GNN Layer
fold
fold
fold
6 fold
2 fold
3 fold
6 fold
=F
a00
50.0 =0.
20.
50.0 :0
50.0
a0.0
=0.0
50.0
TEN
50.0
50.0 =00
a0.
50.0
50.0
50.0
=0.0
50.0
TFN:
100.0
50.0
50.0
50.0 =00
100.0
100,0
100.0
100.0
TEN
50.0
100.0 200
50.0
50.0 :00
50.0
100,0
50.0
20.0
TEN:
100.0 =
50.0 0.0
100.0 a..
50.0 2.0
100.0 =0
100.0 0.0
100.0 n.
100.0 =
TFN
50.0 0.0
100.0 nu
50.0
50.0 00
50.0 0.00
100.0 a.n
50.0 000
50.0 =0.0
TFN
100.0 0.0
100.0 0.0
100.0 0.0
100,0 1000
100.0 =0.
100.0 0.0
100,0 #0
100.0 =00
TEN
50.0
100.0
50.0
50.0
50.0
100.0
50.0
50.0
TEN
100.0 :
100.0 8.
100.0 2.0
100,0
100.0
100,0 0.0
100.0
100.0 =0.
TFN
50.0 0.0
100.0 a.
50.0 00.0
50.0 0.00
50.0 0.0
100.0 6.0
50.0 0.0
50.0 +00
TEN:=10
100.0
100.0 0.0
100.0 0.0
100.0 2.0
100.0 1.0
100.0 0.0
100.0 8.0
100.0 0.0
TEN
50.0 0.0
100.0 8.0
50.0 0.0
50.0 0.0
50.0 0.0
100.0 0.0
50.0 0.0
50.0 =.0
F.3 MOLECULE PROPERTY PREDICTION WITH PRETRAINED EQUIVARIANT FEATURES
F.3.1 DETAILED EXPERIMENTAL SETUP
We employ HEGNN as the encoder in our experiments. TFN is computationally prohibitive in this setting: its tensor-product operator has a complexity of (LL), and when the maximum degree is set to L = 11 (the upper limit supported by e3nn), training a four-layer TFN with only four irreducible- repebbee-trrestttiioonnn channels on QM9 requires roughly 10 A100 GPU-hours per epoch. This far exceeds any practical budget. In contrast, HEGNN adopts spherical scalarization, where interactions across different degrees are mediated solely through scalars, reducing the complexity to (L2)) For this reason, we choose HEGNN our encoder.
Concretely, we use a four-layer HEGNN with 16 irreducible-epreee reprriittccrrrreeenttttiiono channels and a hidden dimension of 64. The resulting features passed through scalarization layer and then into two-layer MLP to predict the molecular isotropic polarizability a on QM9. During fine-tuning, we freeze the encoder, apply mask selectively remove information, and train separate two-layer MLP for each setting. Specifically, our designed mask multiplies the features of the unselected degrees by 0, followed by scalarization calculation, that
( HEGNN(S), 8 (1) vec(( 000 D M0xxx 1...,, 11,
(111)
(112)
â MLP E uinetune (7) (1) (11)),
(113)
Following the standard protocol, we train on the first 10K molecules for 300 epochs and fine-tune for an additional 30 epochs. Notably, although the final visualizations are produced by running the trained models on the entire QM9 dataset', this does not affect our theoretical conclusions, as our analysis relies solely on horizontal comparisons rather than absolute predictive performance.
F.3.2 CASE STUDIES
To further validate our theory, we analyze three prominent point groups C2a. Cas. and Ta (see Fig. 7). Each exhibiting a representative pattern of symmetry increase. These groups display a symmetry increase to the full group 0(3) under specific conditions: for Cz. when lo is odd, for Css when lo 1, and for Ta when lo 2.5. This increase corresponds to full degeneration, making the features non-discriminative The impact of this degeneration is evident in our experiments. Due to the full degeneration performance for molecules with C2n, Cas, and Ta symmetry. Following the introduction of 2-degree features, a marked improvement in performance observed for C2 and C'... Conversely, for Ta. performance again decreases, result directly attributable to the fact that its 2-degree features also undergo full degeneration.
1-degree features, introducing them paradoxically decreases predictive
Minor discrepancies between = 0 in Fig. 6 and S 0 in Fig. 7 result from not fixing the random seed and do not affect the conclusions.
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camplng
(17.784 samples)
samples)
camples)
KMC 1002
CDv (67 samples)
C2h (48 samples)
(15 samples]
DSh 015 samples)
D2d samples)
axemple))
Din samples))
C-v samples)
MAC (og
samples)
samples)
samples)
samples)
MA4
Nlods
Mopel Deggnn
Model Degres
Made Degies
Figure 7: MAE loss (in units of a)) for isotropic polarizability prediction with degree S to across molecules from the top-16 point groups by molecular count. Each boxplot shows the distribution of errors at given degree, while diamond markers denote the corresponding mean MAE.
The experiment shows strong dependence between model performance and feature choice on real-world dataset validates our discussion on symmetry increase and feature space dimension. It demonstrates a critical model design principle: not only should one avoid building model entirely from fully degenerate features, but one should also avoid including individual feature components that undergo full degeneration, as they can actively detrimental to predictive performance.
Table 22: Symmetry infimum of point group symmetry the QM9 dataset.
1
2
4
5
7
10
C₁
K
C1
C,
C1
C,
C1
C,
C1
C,
C1
C,
C₁
C,
Ka
C,
Czu
C,
Czu
C.
Cza
C.
Czu
C.
Czu
C,
C2
K
Crev
Czu
C2
Cau
C2
Czu
C2
Czu
C2
Cza
C²2
Czu
Ky
Covv
Dzn
Cae
Dan
Ca.
Dak
Ca.
D2n
Ca.
D2h
C2e
Ca
K₂
Dach
Coen
Can
Dza
Ca.
Dsa
Can
Dza
Ca
Dsa
Can
Oz
K₂
K₁
Ozu
K₁
Czu
K₁
Czu
K₁
Czu
K₁
Czu
K₁.
C
S2
K₁
C:
K₁.
C.
Kh.
Kh.
C:
Kn.
C.
K₁.
Dah
Ky
K₁
Dock
Dah
Doch
Dah
Den
Dah
Doh
Dab
Deh
Dah
Dza
Ky
K₁
Dook
T'a
Dah
D2a
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47


Reviews:
[{"review_id": "i54EoryoTt", "reviewer": "Reviewer_Xv3y", "summary": "The paper studies symmetry increase in equivariant neural networks. When symmetric inputs are passed through a G-equivariant map, they can acquire more symmetry in their outputs than what the input possesses, which should degrade expressivity. There has been some work on the issue but a unified formulation and treatment has been lacking thus far. The paper proposes (a) a symmetry infimum determined by the feature space, (b) a computable procedure via orbit types to predict that infimum, and (c) under manifold-type regularity and sufficient approximation capacity, it is shown that most equivariant maps achieve this infimum (i.e. can take care of unwanted increases).\n\nThe paper motivates the problem with degradation with k-fold symmetric structures: rotated copies can collapse in feature space despite not being equivalent physically. Several empirical works (Joshi et al. 2023) and some theory works (Cen et al. 2024) are cited, and then the problem is framed via orbit types and Curie's principle (which in this case would amount to saying that equivariant maps can not decrease symmetry). To summarize the main ideas and their development. If the input orbit type $G_X$ does not occur in $Y$ (not in $O_{G(Y)}$, then the equality $G_X = G_{f(x)}$ is impossible. This means that strict increase (degeneration) is forced. These are illustrated by three flavours (full, axial, half), qualitatively shown in figure 2. Then a partial order on orbit types is defined by subgroup inclusion (a standard move), then in a fixed point space $X^H$ the minimal orbit type exists is unique (this is stated in theorem 3.1). The symmetry infimum $I_{G(X,H)}$ is that minimal orbit type. The symmetry increase is said to be \"unexpected\" if $G_{f(x)}$ exceeds the infimum. Isovariance is $G_X = G_{f(x)}$, and the necessary condition for it is $O_{G(X)} \\subseteq O_{G(Y)}$. Equivalently, $I_{G(Y,H) = (H)} $ for all $(H)$ in $O_{G(X)}$. If the kernel rho_Y is not equal to the identity, every isotropy in $Y$ contains the kernel, so some increase is forced. Then a projection operator $p_{Y(H)}$ (closure under the kernel) is introduced and the relative isovariance defined. The necessary condition becomes $I_{G(Y,H)} = (p_Y(H))$. Using decomposition of representations (and for high multiplicity using Michel's criterion), a union or min of infima type behaviour is obtained for direct sums (corollary 4.3). Later the details are provided for k-fold geometry (D_kh symmetry), the minimal supergroups among candidates identifies and summarized in table 1. The analysis then discusses three degenerations that matches the visualization figures. Other results at the end work out the \" most equivariant maps achieve this infimum\" bit.", "questions": "There seem to be a lot of typos in the paper that should be correct. For example, there are several just in the abstract (\"enpowerred\", \"scienctific\", \"indepth\"). \n\nCan the authors add pseudo-code for the orbit-type test and infimum selection in the main text, and a small worked example? This should make the theory easy to follow. \n\nIt would be good to have ablations for r, for different architecture families and depth. It would also be good to include one or two real, even if simple, domain-based sanity checks (perhaps from molecular or crystal contexts).", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I think the core contribution is original and provides a crisp, representation-aware lower bound that predicts degeneration modes and unifies disparate observations (collapse to zero is seen as a special full case). \n\nMain theorems look good and provide useful machinery and results, which formalize quite a few intuitions that had been floating around in the literature. \n\nThe paper is also generally clear, and fairly easy to follow. The k-fold example and three degeneration regimes are intuitive.\n\nThe main message of the paper can also lead to actionable design guidance: choose feature spaces whose orbit-type sets include the required (p_Y(H)); increase multiplicity to guarantee realizability; expect almost-isovariance generically.", "weaknesses": "There is a heavy focus on SO(3)/O(3) and dihedral cases, it would be good to add a discussion for other groups (like SE(3), E(2)), products groups. \n\nThe synthetic experiments are fine and align with the theory, but see some suggestions in 'questions.' Despite the theoretical nature of the paper, I think it would be strengthened if the authors include some more experimental setups.", "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": "1bGjSL1u1b", "reviewer": "Reviewer_cFwq", "summary": "The paper presents a rigorous theoretical framework for analyzing symmetry increase in equivariant neural networks, when outputs become more symmetric than inputs, reducing expressivity. It introduces the notion of a symmetry infimum, a computable lower bound determined by the feature space, and provides an algorithm for computing it. Experiments on synthetic symmetric structures validate the theoretical predictions. However, the work remains highly abstract and focuses mainly on characterizing rather than reducing symmetry increase, offering limited practical guidance for model design.", "questions": "The title and framing suggest that the paper provides methods for reducing symmetry increase, yet most of the work focuses on characterizing it. Could the authors clarify what concrete design principles or architectural adjustments actually help reduce symmetry increase in practice?\n\nCould the authors provide more intuition or visual explanation of the symmetry infimum concept to make the theoretical framework more accessible to practitioners? What about simpler examples? (Consider a 2D shape such as an equilateral triangle. The triangle itself has 3-fold rotational symmetry, it looks the same every 120° rotation. Now, imagine an equivariant network that processes this triangle and produces a feature representation...)\n\nBeyond synthetic examples, do the authors foresee domains (e.g., molecular modeling, materials science) where symmetry increase significantly impacts performance and where their theory could guide practical model improvements?\n\nThe phenomenon of symmetry increase seems loosely analogous to aliasing in the Nyquist–Shannon theorem, where insufficient sampling along the translation group causes distinct frequencies to become indistinguishable. In the present context, a restricted feature space (e.g., limited degrees of irreducible representations) similarly prevents the network from distinguishing between different orientations, leading to a kind of symmetry aliasing over the rotation group. Could the authors comment on whether this analogy is meaningful, and whether the symmetry infimum can be seen as a group-theoretic counterpart to the Nyquist limit for translational symmetry?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper addresses an underexplored but conceptually important issue in equivariant learning, the tendency of models to exhibit unwanted symmetry increase. It introduces an original and rigorous theoretical framework based on the notion of a symmetry infimum, supported by solid mathematical analysis and consistent experimental validation. Although abstract, it represents a meaningful theoretical advance in understanding the limitations of equivariant neural networks.", "weaknesses": "While the paper offers a rigorous theoretical framework for characterizing symmetry increase, it lacks a clear and consolidated discussion of how to mitigate it in practice. The ideas for reduction, such as adjusting representation degree or multiplicity, are scattered across sections and never formulated as concrete design guidelines, making the work hard to apply in real settings. The paper also feels rather abstract and narrowly focused; the topic is niche within equivariant learning, and the presentation does little to connect theory to practice. Moreover, the title’s promise of “reducing” symmetry increase is not fully realized, as the paper mainly analyzes rather than mitigates the phenomenon.", "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": "TN59JxUQrE", "reviewer": "Reviewer_gQcX", "summary": "The paper investigates why equivariant models can unintentionally increase symmetry in their outputs, causing distinct inputs to become indistinguishable. It introduces a practical notion of a symmetry lower bound determined solely by the model feature space: if the feature space cannot realize the input symmetry, the output is guaranteed to gain extra symmetry. Building on this idea, the authors provide a computable test that predicts when symmetry increase will happen and how severe it will be. They also show that, under the manifold hypothesis, generic equivariant models can avoid unnecessary symmetry increase. Experiments on controlled setups closely match the predictions.", "questions": "1. Can you discuss the practical feasibility of using higher-dimensional irreducible representations in the output representation, or of increasing their multiplicities? Does this lead to computational or memory bottlenecks during training or inference?\n2. Is it possible to adapt Theorem 5.2 to a tubular-neighborhood hypothesis, namely that the data distribution is supported in a tubular neighborhood of, rather than exactly in, a smooth low-dimensional submanifold?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "- The paper is well structured and uses clear notation.\n- Although I did not check the proofs in depth, the results appear sound and internally consistent.\n- The paper positions itself as a relevant contribution to the literature on symmetry-breaking inputs in equivariant machine learning. It formalizes the notion of a symmetry infimum, providing a simple, checkable target for assessing symmetry increase.\n- The authors provide all the necessary machinery to derive this object.", "weaknesses": "- Despite the paper being well organized, the exposition is overly compact and hard to follow, as strong backgrounds in differential geometry and Lie groups are uncommon in the ML community. While the paper need not be self-contained, a more explicit notation guide in the appendix would help. In the eventual camera-ready, the extra page could be used to introduce key concepts in plain language with brief examples.\n- The (almost)-isovariance results rely on the manifold hypothesis and smooth parameterizations. Non-smooth and distributional regimes, common in practice, are not addressed. Consider clarifying the scope of the claims and discussing whether any results extend beyond the smooth setting.", "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)