Universally Invariant Learning in Equivariant GNNs
Authors: Jiacheng Cen
Anyi Li
Ning Lin
Tingyang Xu
Yu Rong
Deli Zhao
Zihe Wang
Wenbing Huang
Venue: NeurIPS 2025 poster ()
Keywords: Equivariant Graph Neural Networks
Geometric Isomorphism
Completeness
Primary area: deep_learning

Abstract: Equivariant Graph Neural Networks (GNNs) have demonstrated significant success across various applications. To achieve completeness---that is, the universal approximation property over the space of equivariant functions---the network must effectively capture the intricate multi-body interactions among different nodes. Prior methods attain this via deeper architectures, augmented body orders, or increased degrees of steerable features, often at high computational cost and without polynomial-time solutions. In this work, we present a theoretically grounded framework for constructing complete equivariant GNNs that is both efficient and practical. We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; and 2) a full-rank steerable basis set. Leveraging this finding, we propose an efficient algorithm for constructing complete equivariant GNNs based on two common models: EGNN and TFN. Empirical results demonstrate that our model demonstrates superior completeness and excellent performance with only a few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy.

Universally Invariant Learning in Equivariant GNNs
Jiacheng Cen 123 Anyi 2i ,Nig Lin 23, Tingyang Xu Yu Rong
Deli Zhao 45 Zihe Wang" 23 Wenbing Huang' 23+
Gaoling School of Artificial Intelligence, Renmin University of China
2 Beijing Key Laboratory of Research on Large Models and Intelligent Governance 3 Engineering Research Center of next-Generation Intelligent Search and Recommendation, MOE DAMO Academy, Alibaba Group, Hangzhou, China HHuaan Lab, Hangzhou, China (jiacc. en, 1i ___yn,,, ninglin00 (n00))ooulooo com; {xuty_00m yu rong) @ho thotmail.. com; zhaodeli i@gail com; wang zihe@ruc. edu, en; hwenbing@126. com
Abstract
Equivariant Graph Neural Networks (GNNs) have demonstrated significant suc- various applications. To achieve completeness- -hat is, the universal approximation property over the space of equivariant functions- the network must effectively capture the intricate multi-body interactions among different nodes. Prior methods attain this via deeper architectures, augmented body orders, or in- creased degrees of steerable features, often high computational cost and without polynomial solutions. In this work, we present a theoretically grounded framework called Uni-EGNN for constructing complete equivariant GNNs that is both efficient and practical. We prove that a complete equivariant GNN can be achieved through two key components: a complete scalar function, referred as the canonical form of the geometric graph; and 2) full-rank steerable basis set. Leveraging this finding, we propose an efficient algorithm for constructing complete equivariant GNNs based on two common models: EGNN and TFN. Empirical results demonstrate that our model demonstrates superior completeness and excellent performance with only few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy.
1 Introduction
Various types of scientific data, including chemical molecules, proteins, and other particle-based physical systems, are often represented as geometric graphs [1; 2]. This data structure only captures node characteristics and edge information but also includes a 3D vector (such as position, velocity, etc.) for each node. To effectively process geometric graphs, equivariant Graph Neural Networks (GNNs) have been developed that enable equivariant message passing over nodes while adhering to the E(3) or SE(3) symmetries inherent in physical laws. These models have achieved significant success in various scientific tasks, including physical dynamics simulation, molecular property prediction, and protein design [3-30].
In current literature, a key objective in the design of equivariant GNNs is to achieve com- pleteness, which refers the universal approximation property [31]. This concept
ini- tially explored by [32], who proved the universality of high-degree steerable models, specif- ically TFN [33]. Subsequent models, such as SEGNN [34] and MACE [35], can be simi- larly confirmed to be complete by establishing their relations with TFN. Regrettably, the the- ory proposed by [32] applies only to point clouds (i.e.. fully connected geometric graphs) and relies on sufficiently large values of the layer number, body order, and feature degree.
WWeniing Huang is the corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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More recently, the introduction of the GWL- test [36] and further researches [37-40] have provided a new perspective by connecting the completeness of invariant models to the geo- metric isomorphism problem through an exten- sion of the Weisfeiler-Lehman (WL) test [41]. Unfortunately, the GWL-test has only quasi- polynomial solutions [42], inheriting the same weakness as the original WL-test.
d (g) Σn (
( EL ie (0)
MLP
basis
[(o) = Imhn (..)
Minic Stiee
In this paper, we explore complete equivariant GNNs in more effective and operable way. To begin with, we reformulate existing equivari- ant GNNs as expansion of multi-body high- degree bases coupled with the
6: t 9 -
r(0) (()))
(a) Corrent Merhods
(b) Our Method
Figure I: Diffference between current methods and our Method. (a) Current methods can be under- stood as expansion of multi-body high-degree bases 6,4) coupled with the tensor product (Eq. (3)). The completeness is achieved with sufficiently large values of the body order and feature degree. (b) In contrast, our method proposes a novel expansion which does not necessarily require CG tensor product (Eq. (4)): the sum over a finite basis set with dynamic weights dependent on the input. The complete- ness is ensured if the weights (the scalar function I [(g)) are complete and the basis set (1) is full- rank.
(CG) tensor product (Eqs. (2) and (3)). This reformulation allows us to clearly identify the key limitations of current methods: they fail to achieve complete expansion when the degree and body order of CG tensor products are con- strained, or incur prohibitively high computa- tional costs otherwise. To overcome these weak- nesses, we propose a novel expansion of equiv- ariant GNNs (Eq. (4)), which does not necessar- ily require CG tensor product: the sum over a finite basis set with dynamic weights dependent on the input. The completeness is achieved if the weights (called the scalar function) are complete and the basis set is full-rank.
(CG)
Interestingly, constructing complete scalar function is equivalent to solving the geometric isomor- phism problem, which, unlike the traditional isomorphism task on topological graphs, can be resolved in polynomial time [42]. Actually, we have presented algorithm operating with complexity of (NN) where N is the number of nodes, based on the four-point positioning principle, and further turned it into canonical form for comprehensive embeddings of geometric graphs. Additionally, theoretically prove that full-rank basis set of any degree can always be constructed, if the input geometric graph is asymmetric. Building on this foundation, we propose more efficient algorithm with a complexity of O(N2) to construct the canonical form specifically for asymmetric graphs
Thanks to our theoretical findings, we modify existing equivariant GNNs (such as EGNN [43] and TFN [33]) into complete models. Specifically, we introduce two complete implementations of EGNN, termed EGNN/TFN,pt gllobbal and EG We conduct seven sets of experiments, validating our theoretical results and demonstrating the superior performance of our method compared to previous models. Our model achieves great performance with few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy².
2 Related Work
Equivariant GNNs. According to the operators used build the model, equivariant GNNs can be categorized into two main classes: scalarization-base models and censor-product-basee models. Scalarization- based models (e.g., EGNN [43] and PaiNN [44]) utilize scalars (e.g. distance and angles) as coefficients to linearly combine 3D vectors during node updates. In contrast, tensor- tensor-product-be product-based models (e.g., TFN [33] and MACE [35]) utilize spherical harmonics maintain the equivariance of message passing and realize interactions between steerable features of different degrees through CG tensor products. Historically, researchers maintained that tensor- tensor-product based models could provide richer information, despite their significantly higher computational resource demands [45- 471. However, the recent success of spherical-sciie scalarizzaaiinionn models such as S03KRАTES [48]. HEGNN [49], and GotenNet [50] has prompted reevaluation of this assumption, suggesting
?Code is available at https: h //github/ com/GLAD LLADDRUC/URS RUC/Uni-EENI

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potential shift towards scalar learning. Our research provides an important conclusion in this direction: a complete /th-degree steerable model can be constructed using two components a complete scalar model and full-rank basis set of /th-degree steerable features.
Completeness of Equivariant GNNs. Completeness in equivariant GNNs refers to their ability to approximate any continuous function with arbitrary precision on geometric graphs, thus sparking diverse research avenues. Initially, [32] demonstrated the universality of munso--product-basee models, notably TFN [33], in fully connected geometric graphs. Subsequently, building on the work of [51], later studies began examining the completeness of scalar models. For instance, [52] established the completeness of models such as DimeNet [53], GemNet [54], and SphereNet [55] under A- unsymmetry conditions. Meanwhile, frame- baane- models [56-59] emerged, innovatively separating geometric structures into group operations and orbits for separate processing. However, this approach can disrupt permutation invariance symmetric graphs. Furthermore, the GWL. test framework, inspired by the WI.-test on topological graphs, offers an upper bound on the expressive power of equivariant GNNs over sparse graphs, thereby influencing further researches [37-40]. Nevertheless, it is worth noting that many of these methods rely on message passing and require multiple iterations, akin to multi-layer networks. Our method utilizes the canonical form to derive a complete scalar function, and when integrated with a basis set, facilitates the construction of fully equivariant neural network that can be effectively achieved with just a single-layer architecture.
Method
In this section, we first introduce necessary preliminaries in $ 3.1. Then in $ 3.2, we review existing equivariant GNNs from the perspective of basis construction. Subsequently, we present our framework in $ 3.3, which proposes to construct complete equivariant GNNs from the perspective based on output space. The two components to achieve completeness, including a canonical form of geometric graphs and full-rank steerable basis set, are presented in 3.4. Finally in 3.5, demonstrate how to modify typical equivariant GNNs to be complete based on our theoretical findings. Comprehensive theoretical analysis and proofs provided in Appendix A.
3.1 Preliminaries
Geometric graph. A geometric graph of N nodes is defined as G = (H, X; X A), where ==
(h; E RCH y and X == {æ, E R A E N is the adjacency matrix representing topological connections between different nodes and each edge could be assigned with an edge feature e,j if necessary.
ji=1 are node features and 3D coordinates, respectively;
Let X and Y be the input and output vector spaces, respectively. A function Φ X + Y is called equivariant with respect to group & if
Equivariance.
Vig 6, (px(())) p)(g)
where fx and py are the group representations in the input and output spaces, respectively. Since translation invariance can always be achieved by translating the center of all coordinates to the origin, we omit translation and focus solely on equivariance concerning 0(3), the group consisting of rotation and inversion. This implies 0(3) = SO(3) x C,, where SO(3) represents the rotation group, and C; = i denotes the inversion group, with e as the identity and i as the inversion. We define two additional groups: the Euclidean group E(3) that further involves translation into O(3), and permutation group SN acting on a set of N elements.
Irreducible representations and steerable features. For group element r E SO(3), its rep- resentation is typically achieved through irreducible representations such as the Wigner-D matrix (t) E (22++)))(11-1 where / denotes the degree. We call features ((1) as Ith-degree steer- able features they are transformable using D(), The most common steerable features can be obtained via spherical harmonics, expressed as ( = [X=! (#/|\=)))>>,, which satisfy the property y (R..t/ [R provide a complete basis set for SO(3) - functions on the sphere. Clebsch- (CG) tensor products are usually leveraged to facilitate interactions between steerable features of differ- ing degrees. Specifically, given features ((4)) and g(ta), new feature ((()) can be generated, for
= (e) ( According [60], spherical harmonics
141 - 12] < 1 < 4 + lz. This caleulation is abbreviated as (()) ((t.) Bog 5(tz), By including
3

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parity [61], the above concepts can be naturally extended to the orthogonal group O(3). A more detailed discussion is available in Appendix A.I.
3.2 Rethinking equivariant GNNs: a perspective based on basis construction
By definition, complete equivariant model is able express any equivariant function. For in- stance, spherical harmonics provide complete bases for (0(3))-quivaria) functions on the sphere: [[ enabling the representation of all such functions. In 3D space beyond the sphere. a com- plete function is achieved by combining spherical harmonics with a learnable radial basis function (([ However, deriving complete equivariant functions on geometric graphs, namely "0) (g), is more complex due the need consider multi-body interactions and permutation symmetry.
To begin with, let define function set F(O), which encompasses all -quare-integrabe /th-degree steerable functions on geometric graphs. Let B (1) C F(() denote the basis set that expands the function space of equivariant GNNs. The learning process of equivariant GNNs becomes an approximation:
お ( (g) = EB.0 EBID Earo , (g),
where a, E IR are weight coefficients independent of the input. Enhancing the expressiveness of equivariant GNNs involves enlarging TB(()) until it coincides with F()) When span(B()) = F()), the model is termed a complete Ith-degree steerable model, or simply complete model, and the basis set denoted as (²()) opl Specifically, complete Oth-degree steerable models are referred to as complete scalar models.
Below, we demonstrate that standard equivariant GNNs share a common basis set construction, differing primarily in their body-order formulation. For simplicity and without loss of generality, assume uniform node and edge features that can either be treated as identical or safely ignored. This abstraction allows us to focus purely on the geometric and topological aspects of the problem while maintaining the core theoretical insights.
Example 3.1 (Basis Set of Common Equivariant GNNs). The basis set for common equivariant GNNs, such EGNN [43], HEGNN [49], TEN [33], and MACE [35], can be unified into the following form:
B = =E, & (Tiss/ /| II)}
(3)
where v 2 1 denotes the body order, j = (Ji.....Jv) represents all chosen ordered neighbors, and 1), Eq. (3) forms the basis set for EGNN [43] and HEGNN [49], applicable to degrees / = and 2 1, respectively. In contrast, for multi-body interactions, Eq. (3) corresponds TFN [33] when the body order 1, or
Lije E; j2- When only single-body neighbor is considered (i.e.,
MACE [35] with higher body orders v 2 1.
Although these models adhere to the form of Eq. (3), their expressive power differs markedly. Multi- body models can create entirely new bases through tensor products and achieve completeness by either utilizing a sufficiently high body order [62] or stacking multiple layers [32], whereas single- body steerable models could not. However, multi-body models suffer from notable limitations: a) They may not encompass steerable features of all degrees, if input degrees are improperly chosen. b) The computational complexity increases sharply with higher degrees and body orders v, resulting in significant computational costs.
3.3 Reformulating equivariant GNNs: perspective based on output space
Expanding the basis set of a complete equivariant GNN theoretically appealing but practically challenging. In this section, we adopt an alternative approach by deriving the necessary model design from the characteristics of the desired output results. We identify a key insight: a complete /th-degree steerable model can be constructed using two components complete scalar model and full-rank basis set of /th-degree stecrable features.
$ For clarity, our main text focuses exclusively on graph-level functions, where o (0(( node permutation. Notably, our discussion is generalizable to both node- and level functions, which will be discussed in Appendix A.2
(g) remains invariant to

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We denote the target Ith-degree steerable function as (00) E B Although it might be repre- sented with infinite bases in ( the output of (g) still vector in IR++1, Let (g)
(g). ..... "c (g) (2(+1))xCC define an Ith-degree steerable model with C-channel outputs of 1th dth-de steerable feature (C 2 21 + 1). We have the following theorem:
(1)
Theorem 3.2 (Dynamic Method). Given a geometric graph G, suppose there is a matrix V (G)
with C channels of (th-degree steerable features denoted as : (g) satisfying span( ( (g)
(G) = (G) f()) E } C 2 Then for any lth degree steerable function (()) E F()),
there always exists in (g) = [we (0)) (g) with C-channel output scalars, such that
(G) - E Wc (9) (g)o!
notable distinction from Eq. (2) is that the coefficient in front of the steerable features changes from ag which is independent of the input, to a scalar (function) that depends on the entire graph, denoted we (g) (g). A natural idea is that if we can get complete scalar model (capturing all possible scalars), then we can always satisfy the requirements of Theorem 3.2. Another crucial consideration is that the validity of Theorem 3.2 requires the basis matrix (G) to satisfy span span( (G)) = (g). We call such basis matrix that meets this criterion as (F( (C)-full-ankk When Fr()) (g) = R2 the concept of ((() -fulll-ran is equivalent to the traditional definition of full-rank. For simplicity, in contexts with no ambiguity, we refer as full-rank.
By comparing Eq. (4) with Eq. (2), our proposed dynamic method offers greater flcxibilityy and simplicity, embodying the concept of complexity transfer. Instead of constructing the difficult complete basis set [B:4]]
<pl we transform the process into obtaining a complete scalar function and then design the complete steerable model through Theorem 3.2. In the next section, we will detail how to acquire the complete scalar model and the full-rank basis set of Ith-degree steerable features needed for Theorem 3.2.
3.4 Reach completeness: canonical form and full-rank basis set
In this section, we introduce how to obtain the complete scalar function (:(o) (g) (referred to as the canonical form) and the full-rank basis set (G) required by Theorem 3.2.
Geometric isomorphism and canonical form. An equivalent problem to constructing a complete scalar function determining the isomorphism of geometric graphs. To differentiate this from traditional graph isomorphism problem for topological graphs, we specifically refer to our task involving geometric graphs as geometric isomorphism. This concept can be defined similarly to the GWL-test [36], which assesses the equivalence of geometric graphs based on their geometric structures and embeddings.
Definition 3.3 (Geometric Isomorphism). Two geometric graphs ( ( (9)) and H((X (H), (K)) are called geometrically isomorphic if they fulfill both of the following isomorphisms
1. Point Cloud Isomorphism: The two point clouds ( and XH0 are isomorphic, i.c. (HH)
3o E SN.g E E(33, Vi, F. x Fo(s) Here, all (a, g) make a nonempty set MiG, H).
2. Topological Isomorphism: The topological graphs associated with the point clouds are isomorphic, i.c., 3(, E M), Vi, Vj, (9) = [A (H) (1)()))
Moreover, we denote the geometric isomorphism between G and 7 as g = H.
The most significant difference from the isomorphism problem in topological graphs is that, to date, it remains unclear whether there exists a polynomial-tin
algorithm capable of determining whether two topological graphs are isomorphic [42; 63]. In contrast, when it comes to distinguishing whether two geometric graphs are geometrically isomorphic in accordance with Definition 3.3, the situation appears to be more favorable. There are indeed algorithms that can resolve this issue in polynomial time, as highlighted by [64]. For example, we present Algos. and 2, which based on the four-point positioning principle and operates with complexities of O(ND) and ((N)), respectively.
5

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In most applications, the objective of an equivariant GNN is not to distinguish two geometric graphs, but rather to obtain comprehensive embedding of geometric graph. Consequently, we define the canonical form similarly to the definition provided in [64] as follows:
Definition 3.4 (Canonical Form of Geometric Graph), canonical form of geometric graph is a graph-level scalar function I' (R-*x, RNXNN R satisfy G = H - I(G) = (H)).
Built upon Algos. and 2, we have derived a canonical form r(-) as presented in Algo. 3. summary, the process involves three steps: a) traversing all ordered four-point sets to serve as reference points for four-point positioning, with a time complexity of O () ); b) transforming the original coordinates into distance vectors measured from the reference points, and converting both point sets and edge sets into scalar sets, with a time complexity of (NN); c) mapping these scalar sets to the desired canonical form using DeepSet [65].
Theorem 3.5 (General Canonical Form). Given any geometric graph G. Algo. 3 provides a method create canonical form with time complexity (
remains impractical because of its high complexity. We now explore the feasibility of attaining more efficient canonical form by sacrificing some versatility. The crux of Algo. 3 lies in the choice of four non-coplanar points, which contributes to a quartic complexity in traversal. If we reformulate this as generative problem, by treating the non-coplanar points as learnable graph-level features, we may reduce this quartic complexity factor. Specifically, we consider applying E(3) equivariiat GNN to generate four non-coplanar nodes (called virtual nodes below). We regard them as reference points in the four-point positioning principle, avoiding the fourth-order complexity cost by the original traversal. Thus, we derive Algo. 4, yielding more efficiency.
However, Algo. 3
Theorem 3.6 (Faster Canonical Form). Given an -e3)-iqrvva function Ç that generates four non-coplanar points on G. Algo. 4 able to create canonical form with time complexity O(N2).
It should be noted that Algo. 3 can encode any geometric graph, while Algo. 4 requires that four non-coplanar reference nodes can be learned through an equivariant GNN. In past studies, such as FastEGNN [66], it is assumed that non-coplanar virtual nodes can always be found, but in the discussion later, we will see that this is not the case.
Full-rank basis set. As previously addressed, the validity of Eq. (4) critically depends on the (G)-lullrraa basis set, condition not easy to satisfy. The study in [49] has demonstrated that, for symmetric geometric graphs (defined in Definition A.5), (G) will degenerate to zero functions for certain values of degree I, implying that it is hard to design full-rank basis set in this case. In practice, most scenarios study the case of asymmetric graphs, so we give priority to discussing them here. Our key inquiry is whether full-rank basis sets can always be constructed for asymmetric geometric graphs. We first present a pivotal theorem that significantly aids in our analysis.
Theorem 3.7 (Coloring Asymmetric Graph). In asymmetric geometric graph, each point can be assigned a unique color. This implies the existence of an E(3) invariant function that maps the features of the ith node distinct values h, Similarly, for directed edges ij, their features are mapped to distinct values eij-
We propose two node coloring methods: a) Distance to the center, denoted as @; b) Tensor product, denoted as Ø. We denote the uncolored model by ø and illustrate the coloring procedure in Fig. 2, taking the + method an example. A detailed description is provided in Table 9. This theorem is fundamental and forms the core of our subsequent analyses. It decouples the roles of different nodes and edges. simplifying basis functions by selecting appropriate weight functions, or even to focus on a particular basis function by setting the weights of all other basis functions to zero. Given that the node coordinates of the entire geometric graph are non-coplanar, the coordinate differences {Ø.) form a full-rank
discussion. This framework enables to enhance or suppress specific
matrix, which allows us to achieve the desired ( (G). Additionally, spherical harmonic functions facilitate the mapping of first-degree steerable features to higher-degree representations. We obtain the following theorem.
Theorem 3.8 (Existence of Full-Rank Basis Set). For any given asymmetric graph G. an BQ)
full-rank V (G) can always be constructed for any degree l.

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(a) Original
(b) Center
(c) Colored
Figure 2: Different vectors are decoupled by coloring, thereby expanding the space of virtual nodes. The yellow area
figure represents the out space of Ist-degree functions. Fig. 2(a): All nodes have the same features, and the output can only appear in straight line in one-dimensional space. Fig. 2(b): Calculate the distance to the center ||ã to update each node feature so that each node has different color (feature). Fig. 2(c): After coloring, all vectors are decoupled, so the output can be generated in the entire 3D space.
For symmetric graphs, the situation is more complex, making the construction of full-rank basis set considerably difficult. An potentially feasible approach could using a tensor maasor-product-basee model, such TFN [33] for basis set construction, which will be detailed in the appendix Appendix A.1.3.
3.5
Implementation of Complete Models
The last subsection demonstrated how to achieve theoretical completeness. Here, we present practical implementation of complete models, using EGNN [43] as an example due to its simplicity and universality. Example 3.1, we point out that EGNN could be incomplete since its basis set (()) is possibly reduced to the sum of all edges { {Zng} }. However, the edge itself {æ.y} full-rank and could be consider as the full-rank basis set (1) in Theorem 3.2. To make EGNN complete, the remaining issue is how to implement the complete scalar function wh (0) Inspired by Algo. 4, we derive the E(3)-equivariant function Ç in Theorem 3.6 using FastEGNN [66; 67], along with the pre-coloring process and several other modifications.
Essentially, we first color the geometric graph and then construct the reference virtual nodes through FasEEGNN in the following form:
Z 1, + 1/2 Em Pz(()) (#. -E), M, 14 2,
where Ž E R4x3 denotes the coordinates of four reference virtual nodes, M; E R computes coordinate difference between node i and all virtual nodes, and E coordinate center. In Algo. 4, the key point is to convert the coordinate information into a distance vector the reference virtual node. To further refine the information, we employ the inner product form from GMN [68], as an alternative to distance vectors in Algo. 4. The node features are updated follows:
Σ E. signifies the
(6) Afterward, global operations (e.g., pooling) aggregate the information from all points and edges, mimicking DeepSet [65] caleulations to obtain the desired canonical form. Strietly speaking, the virtual nodes forming the tetrahedron should also be included Algo. 4, but we omit this term as degrades performance in experiments.
h = h. + 4P% (m)),
m. = M, M//MM M.le..
we consider single-layer EGNN, becomes the following form:
EGNN epl (
Σ
Z Z. + Eii,es Pa Tig.
where the message mij.cpl pm (h; hj, || 11 eij) has been a complete scalar function since h; already contains the information in the canonical form. Through Theorem 3,2, we verify that such single-layer EGNN is a complete model.
The above approach can also be extended to tensor sesoo--roduct-basee models, and we have implemented corresponding improvements in TEN [33]. Since virtual nodes (i.c. Z) are generated globally across
7

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Table 1: The Completeness Test.
Table 2: The Chirality Test.
The Completeness Test
The Chirulity Test
2-body
3-body
Color # #TP
Fig. 4(a)
Fig. 4(b)
Fig. 4())
GNN Layer
(Table, GWL)
(Fig. 2(b) in IASR)
50.0
20.0
50.0
SchNetr.
50.0 =00
50.0
100.0 100
100.0
50.0 =00
50.0
50.0
Basic
100.0
100.0 100
100.0
GVP-GNN5aoy
100.0 200
$0.0
75.0
95.0
100.0
8.0
50.0
100.0 #00
100.0 2.0
100.0 :.0
M-CEs-oody
100.0
50.0 100
100.0
100.0
100.0
MACE
100.0 2.0
100.0 2.0
1
50.0
50.0
50.0 2:00
Basican
100.0 a.
100.0 8.0
100.0 8.8
100.0 aJ
50.0 0.00
SchNete
100.0
100.0
EGNN
100.0
100.0
100,0
EGNN.
100,0 0.0
100.0 8.
100.0 4.0
100.0 1.0
100.0 6.0
GVP-GNN,Gt
100.0 0000
100.0 8.0
100.0 8:0
100.0 2.0
100.0 8.0
TF
100.0
100.0
100.0
100.0
100.0
Table 3: MSE loss for predicting the coordinates of the Monge point and twelve-point center and
incenter of tetrahedron. Models with superscript indicate that they utilize only half of the hidden dimension of the standard node embeddings. The optimal value is denoted in bold, while the suboptimal value is indicated with an underline.
Monge Point
Twehee-oint Center (x10-1)
Incenter (x10-")
I-layer
2-layer
3-layer
4-layer
I-layer
2-layer
3-layer
4-layer
I-layer
2-layer
3-layer
4-layer
BONN
1.188
0.374
0.185
0.131
1.320
0.413
0.265
0.161
32.54
0.527
0.141
0.036
FasEENN
1.187
0.374
0.149
0.979
1.320
0.417
0.194
0.107
32.40
0.421
0.188
0.055
HEGNN
1.188
0.356
0,157
0.110
1.320
0.317
0.192
0.159
32.54
0.386
0.078
0,058
TFN
1,188
0.467
0.289
0.221
1.320
0.519
0.315
0.246
32.54
2.715
0.151
0.060
MACE
1.188
0.468
0.288
0.223
1.320
0,521
0.316
0.246
32.54
2.707
0.153
0,090
Equiformer
0.416
0.296
0.099
0.116
0.456
0.187
0.122
0.147
0.736
0.077
0.055
0.054
EGNN, globbal
0.108
0,086
0.097
0.104
0,112
0.097
0.099
0,088
0,025
0.018
0.015
0.012
BONN global
0.175
0.116
0.148
0.111
0.155
0.144
0.174
0.152
0.046
0.025
0.059
0.026
EGNN
0,460
0.210
0.083
0.052
0.510
0,232
0.095
0.067
0.388
0.258
0.031
0.029
SGNN-.-ooc
0.460
0.252
0.138
0.091
0.508
0,250
0.182
0.102
0.395
0,320
0.084
0.090
glooall
0.468
0.162
0,233
0.084
8,983
0.398
0.077
0.051
TEN; Gr global
0.473
0.291
0.158
0.096
0.525
0.317
0.181
0.105
7,858
0.354
0.085
0.054
TFN rlcall
0,460
0.210
0.089
0.088
U.518
0.237
0.097
0.078
0.676
0.048
0.029
0,021
TEN -lcca
0.473
0.244
0.082
0.073
0.519
0.271
0.122
0.085
0.693
0.056
0.039
0.024
the entire graph, we refer to these models as EGNN/TFN.prgglobbe To better capture local information,
virtual nodes can also be generated for individual nodes, resulting in the EGNN/TFNN.p-loo -lcal variants. Detailed model architectures are provided Tables 12 and 13.
Experiment
In this section, we conduct two categories of experiments, totaling seven in number, to validate our theory and methods: a) three toy datasets in $ 4.1 to assess the expressivity of our models: b) the remaining four in $ 4.2 evaluate the actual performance of our models. Here, we provide brief overview the experiments. For further details, please refer Appendix
4.1 Expressivity
Dataset setup. The completeness test is derived from the GWL-test [32] and the IASR-test [69], with the objective of distinguishing between two geometrically non-isomorphic graphs. We here focus on the k-chain test and 2-body/3-body tasks for testing. The chirality test is an experiment designed by ourselves, which includes three tasks as illustrated in Fig. 4. This study aims to address two objectives: a) assess the model's ability to distinguish geometrie graphs under reflection transformations; (2) investigate whether distinct coloring strategies generate different virtual node coordinates, as validated by determinant calculations serving as mathematical indicators for reflection determination.
Models. For the completeness test, we selected several baseline models, including SchNet [70]. EGNN [43], GVP-GNN [71], TFN [33], and MACE [35]. Furthermore, models utilizing our canonical form are denoted as Modelept, with Basicepl specifically using the canonical form as the graph embedding directly. For the chirality test, we provide two options to identify reflections: a) Determinant det([ 23, =1] 11133) b) Tensor product, as described in Example A.
8

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Table 4: MSE loss on 5-body system.
i Tet. Incenter
MSE Loss 5-body system
(x10-")
1-layer
2-layer
3-layer
4-layer
EGNN
4.214
0.780
0.710
0.712
FasIBGNN
3,983
0.705
0.509
HEGNN
4.114
0.801
0.561
0.489
TFN
2,411
1.758
1.758
1.739
MACE
2,403
1.754
1.746
Equiformer
0.805
0.682
0.465
0.657
(i0) 5- 5-body
EGNN gr
0,943
0,546
0.530
0.492
EGNN
0.985
0,554
0.533
0.498
EGNN
0.768
0.537
0.481
0.458
BGNN gar-ocall local
0.703
0.513
0.455
0.450
2.144
0.934
0.916
0.708
TFN:
2.163
0.910
0.825
0.734
(a) OG Model
(b) TFN Model
-locallccal
0.988
0.766
0.738
0.761
TFNG TFNgg-locul
0.883
0,792
0.734
0.746
Figure 3: Visualization MSE loss.
Table 100-body dataset.
Table 6: Prediction error (x10-2 MD17 dataset (3 runs).
MSE Loss (x100")
Agpirin
Bessene
Eihanol
Malonal.
Naph.
Salieytic
Tohoce
Uraeil
EGNN
1.36
EGNN
14.41.0.
62.40iam
4.64.oa
13.64
0.47 ooz
1.02
11.78
0.64
FasEGGNN
FISEGNN
9.8
60:84
12.82çauo
0
1.05amx
10.88sse
01566am
TFN
TFN:ca
12.37
58 58.481200
208
13.62sas
2.48
1,03aaxe
10.89
0.84
MACEigo
MACEsea
1044-o0
39.71.mm
4.83.om
13.78
0.44.mm
0,94vam
10.20.aa.
0.74
Eggiformetec
8.90
Equiformetee
33.28
1.86
1.50
0.57
HEGNNI
1.13
HEGNN.g1
4.63
0.90.aus
10.56:a1
HEGNN.
8.97
HEGNNN
10.0g
80lass
2.6
12.85gax
0.30
0.912ar
10.56asm
HEGNN..
0.94
HEGNNiia
10.20
628224u
12.85
0.37 mm
0.94gaa
10.55
0.52
HRCINNias
0.96
HEONNica
9.94gom
59.93ga5
2.62
12.85gam
0.37am
0.88gare
10.36
0. 0.54gom
0.98
SGiNNar-globe
9.00.mm
58.24
4.64-0m
12.85
a.39
0.95
10.374sm
0156vaxx
EGNN, local
-local
9.52gm
44 90
12,00ga
0.94gars
10.21
0.572mm
TFN globaloge
9.49
58.3 58.251m42
4.63
12.82
0.33.ox
0,80
10.24:00
1.53
1PNgg-ooaalaa
48.77
2.64gmm
12.85aai
0.
10.95amm
Results. Results are presented in Table (the 2-body/3-body of completeness test), Table (the chirality test) and Table 10 (the k-chain test of completeness test). As shown in Table 1, all models with canonical form, i.e. Modelept reach an 100% accuracy, no matter whether the original model could distinguish the geometric graphs. As shown in Table 2, the model utilizing the determinant highly dependent on the coloring scheme. In the special case illustrated in Fig. 4(c), only the model employing the tensor product can effectively distinguish between the geometric graphs, which demonstrates the importance of choosing an appropriate staining strategy.
4.2 Performance on Physical Systems
Dataset setup. We conducted four experiments evaluate our model's performance across different scenarios: a) tetrahedron center prediction, which involves predicting the Monge point, twelve- point center, and incenter of tetrahedron based on its four points (with identical node features): b) 5-body system [72], c) 100-body system [66]. d) MD17 dataset [68], and e) Water-3D mini [66; 67], where predictions are made based on initial coordinates and velocities. Specifically, for the 5-body, 100-body, and Water-3D mini systems, we predict the future positions of charged particles or fluid particles, while for the MD17 dataset, we predict future atomic trajectories within eight different molecules.
These experiments provide comprehensive evaluation of our model. We assess its performance on graph-level targets (a) and node-level targets (b-e), explore the influence of model architecture layers (a-b), and test its scalability large systems (c.c) as well as its applicability real-world datasets (d-e). Detailed information on each dataset can be found in Table 11.
Models. In both experiments, we selected the following baseline models: EGNN [43], FastEGNN [66; 67], HEGNN [49], TEN [33], MACE [35], and Equiformer [73]. We compared these baseline models with our proposed models, specifically EGNN/TFN,g--ooce and EGNN/TFN, global. More detailed information is provided in Tables 14 and 15.
We evaluated the performance of each model across varying numbers of layers in the first two experiment. For specific values, please refer to Table and Table 4. The experimental results are visualized in Fig. 3 to facilitate comparison. Our model demonstrates superior performance across all
Results.

---

Table Results on Water-3D-mini.
MSE Loss on Water-3D-mini
(x10
1-layer
2-layer
3-layer
4-layer
EGNN
4.904
4.323
3.649
3.338
FastEGNN
4.885
4.332
3.782
3,259
HEGNN
4.885
4.138
3.519
3.287
BGNN epNNeptgglobal
4.368
3.711
3.294
3.248
eGNNep--local
3.611
3.320
2.803
2.495
tasks, often achieving results comparable to those of other models with multiple layers- sometiies even with just a single layer. In addition, our improved model has also achieved significant improve- ments on lcage-scale datasets and real-world datasets, Tables 7, where our model is in the leading position in the former and in the latter, where we achieve performance leadership in more than half of the tasks.
Additional findings. Furthermore, the baselines in Fig. 3 demonstrate that losses are equal when single layer is used, confirming the degradation predicted in Example A.16 and highlighting the necessity of node coloring. While TFN and MACE outperform EGNN and HEGNN with a single more layers. This is likely because EGNN/HEGNN
layer, their advantage decreases with may initially lack necessary basis functions, whereas tensor products used in TFN/MACE in deeper layers produce redundant terms. The introduction of the dynamic method significantly enhances the performance of both EGNN and TFN, showing its ability to construct the missing basis initially and suppress redundant terms in tensor products, thus achieving adaptive basis construction. From this perspective, exploring how to encode invariant features with more powerful architectures (e.g.. advancing from GNNs Graph Transformers [74]) represents promising direction for future research. This perspective also helps explain the performance gains observed in models such as GotenNet [50] compared with HEGNN [49].
5 Conclusion
We propose novel framework called Uni-EGNN, which is dynamic method for constructing complete equivariant GNNs. In contrast to previous models, which required stacking multiple layers, increasing body order, and enhancing the degree of steerable features to achieve completeness, our dynamic method simplifies this process by requiring only two essential components: a) A canonical form (a complete scalar function) of geometric graph; b) A full-rank steerable basis set. Additionally, we have developed an efficient implementation of dynamic method leveraging a polynomial algorithm for geometric isomorphism problems. Experimental results demonstrate that dynamic method excels both expressiveness and practical performance.
Limitations
For symmetric graphs, constructing full-rank basis set remains open problem. To date, it has only been established that tensor-product-basee models (e.g., TFN [33]) can construct such a basis, while for scalar methods (e.g., HEGNN [49]) this remains uncertain-ann we even conjecture that it may not be feasible. Another limitation of this work lies in the experimental evaluation: due to the lack of tasks with objective functions of degree / > effectiveness of the proposed method in predicting high-degree steerable objective functions.
unable to empirically validate the
Acknowledgement
This work was jointly supported by the following projects and institutions: the National Natu- ral Science Foundation of China (No. 62376276, No. 62172422); Beijing Nova Program (No. 20230484278); the Fundamental Research Funds for the Central Universities; the Research Funds of Renmin University of China (23XNKJ19); Public Computing Cloud, Renmin University of China; Damo Academy (Hupan Laboratory) through Damo Academy (Hupan Laboratory) Innovative Research Program and Damo Academy Research Intern Program.
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Contents of Appendix
Theoretical Analysis and Proofs
19
A.I Detailed Preliminaries
19
A.1.1
Equivariance
19
A.1.2
Clebsch-Gordan (CG) tensor product
19
A.1.3
Discussion on Symmetric Graphs
20
A.2
Connection between graph-level and subgraph- leveel functions
21
A.3
The full form of dynamic method
21
A.4
The construction of canonical form
22
A.4.1
An algorithm for determining point cloud isomorphism
22
A.4.2
An algorithm for determining geometric graph isomorphism
23
A.4.3
A general canonical form
23
A.4.4
A faster method to construct canonical form
24
A.5
Construction of full-rank basis set
26
A.6
Comparison of different coloring methods
27
Experiment Details
28
B.1
Expressivity Tests
28
................................
B.1.1
Completeness Test
................................
28
B.1.2
Chirality Test
30
B.2
Performance Tests
30
B.2.1
Tetrahedron Center Prediction and N-body System
................................
31
B.2.2
Implementation Basclines
B.2.3
Implementation of Our Models.
31
C Rethinking the Equivariance and General Functions
33
C.I
Rethinking equivariance: A perspective from Fourier expansion.
33
C.2 Rethinking General Functions: the Equivariant Decomposition
34
18

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A Theoretical Analysis and Proofs
A.1 Detailed Preliminaries
This section introduces the equivariant representation and the CG tensor product with inversion, along with the symmetric graph structure and its related properties.
A.1.1 Equivariance
Let X and Y be the input and output vector spaces, respectively. A function ø X - Y is called equivariant with respect to group if
Vg B (px(g))) (r
where px and py are the group representations in the input and output spaces, respectively,
Since we can permit translation invariance by simply translating the center of all coordinates to the origin, we only discuss equivariance with respect to 0(3) in our paper. Specifically, the group ((3) consists of rotation and inversion, implying O(3) SO(3) x C, where SO(3) is the rotation group and C; fe. i} denotes the inversion group with representing the identity and i representing the inversion. the current literature, researchers typically utilize irreducible representations (i.e. Wigner-D matrix (t) (t) ((2(41)x(21+1) with / denoting the degree) represent SO(3) and parities E } to represent the inversion group C; as follows:
p (()) (()) (m)
c. When considering only rotations, (2l + 1) -iddmeensional variable that can be rotated by the /th-degree Wigner-D matrices is referred to as an (th-degree steerable feature. When further involving the concept of inversion, we adopt the notation presented in e3nn [61]. Specifically,
where (P) p i, while o (p) (m) =
-ivariannnnvaiiann (th-degree steerable feature (i.e., p 1) is referred to as an le-type, while an inversion-equivarian feature (i.e., p = 1) is designated lo-type. These designations correspond to even parity and odd parity, respectively.
Example A.1 (Spherical Harmonics). Spherical harmonics (z) = [Y (T) are Ith-degree
steerable features, i.e. spherical harmonics are lo-type when / is odd and le-type when is even. According to [60], spherical harmonics offer complete set of function bases of (3q--uqvvvar functions.
D (") (æ), and the parity is given as p = (-1)'. It means
Another example about common physical quantities is presented in Example A.2.
Example A.2 (Common Physical Quantities). Scalars (e.g. mass, charge and energy) are Oe-type (invariant to both rotation and inversion). Pseudo-scae scalar (e.g. magnetic charge and magnetic flux) are Oo-type (invariant to rotation but change signs under inversion). Vectors (e.g. coordinate and velocity) 1o-type, while pseudo-vectors (e.g. torque and angular momentum) are 1e-type.
Example A.3 (Common Invariant features). Due to the various advantages of scalars, there are large number of construction methods, which we briefly list here:
1. Purely mathematical representation: tensor product (spherical-scallrizatioo [48-50], topo- logical characteristics [75-77], cluster expansion basis [78; 79], subgraph blocks [80-82], frames [56; 83; 84), normalization operators [85; 86), L.LM-based information [87];
2. Features based physical and biochemical prior knowledge: distance matrix [40; 88 92], chemical bond length, angle and dihedral angle [93-97], force [98; 99], fractional coordinates [21; 100; 101], canonical ordering [102; 103].
A.1.2 Clebsch-Gordan (CG) tensor product
Tensor product +)) is a common operator to describe interaction between steerable features. Given an 4 th-degree steerable feature ((1)) and an l2th-degree steerable feature ((tz) calculating their tensor product yields a new feature (44) & ((?2) with its group representation as ( & ( (T). The Clebsch-Gordas rule reveals that any tensor product of irreducible representations can be represented as the direct sum of the irreducible representations, Let w(41) and (122) denote the irreducible
19

---

representation space of 11th- and l2th-degree, respectively. Then, we have the following relation:
v(ti & v(l) - V()
(10)
This further demonstrates that the group representation matrix (t) @ (t) is similar to the
matrix D/1t// -1= (t). In general, the /th-degree component from the tensor product ū(1) @ ((()))
can be expressed as
l
((lm))
(a) (1.mm))(l2.ma)) mi
(11)
m
mh 1 12==112
where (1 (.).02m))m))))) is called the Clebsch-Gorlan coefficients. To simplify writing, when specifying the degree of the input and output, we directly abbreviate the above formula to 5(() = e(h)) 8eg 5(1a), We introduce the decomposition of the moment of inertia in Example A.4 to enhance understanding.
It is straightforward to extend the tensor product to incorporate inversion. When performing the tensor product of two steerable features one exhibiting odd parity and the other exhibiting even parity-the the resulting feature will necessarily exhibit odd parity. Conversely, if both features share the same parity, the result will be of even parity. For simplicity, we will omit discussions related to inversion in the subsequent analyses.
Example A.4 (Decomposition of Reducible Representations). For the moment of inertia veo( E) (1 1. E & EE,) R The first item is scalar (0e) of channels denoted as 9x0e, while the corresponding group representation of the second item >. के. 8 H is R. @ Rg (the inversion are canceled). According the Clebsch-Gordan rule, this representation is similar to Di (t)... which can be transformed through a specific coefficient matrix, and Σ, ã @ z can be decomposed into irreducible features of three types: Oe, 1e, 2e.*
A.1.3 Discussion on Symmetric Graphs
A symmetric graph is special kind of geometric graph. [49] points out that in symmetric graph, steerable features of a particular degree will degenerate, so we list this case separately for analysis.
Definition A.5 (Symmetric Graph). geometric gruph G called symmetric graph, if there exists finite and nontrivial subgroup f) S 0(3), & {c], satisfying that Vh E S, h G G. All subgroups making G symmetric yields a set HG),, and all geometric graphs that are symmetric w.r.t. H) constitute a set denoted G()). Moreover, all other graphs are referred to as asymmetric (geometric) graphs.
It is straightforward to conclude that asymmetric geometric graph must contain non-coplanar nodes; otherwise, if all the nodes lie in the same plane, the graph would coincide with itself after reflection across that plane. Therefore, the presence of non-coplanar nodes is necessary condition for the asymmetry of the geometric graph.
Lemma A.6 (Image Space of Functions on Symmetric Geometric Graphs). Given a symmetric graph G, (G) E Aseag) [2+1 - p() (5)]*, where (() = 1 averaging.
[5| Enes (() denotes the group
Here, we strengthen the conclusion of [49]. Below, we give an example that makes the steerable features confined to a subspace instead of degenerating into zero vector.
Example A.7. Consider a cone with a square base, defined by the set G
{(E1,0, 0,1,, (0, -1), 4)). For = 1, is straightforward to find that (g) = ((0,0,))) +)] C R33 and constructing a (F)) (G)-full-rann) function is relatively easy. However, for cases where 1 > 1. while the analysis can still be performed using the orthogonal complement of the group average, constructing a (G)-luull-ran function becomes significantly more challenging. We propose using tensor- cittor-rrducttbbee based models, such as TFN [33], to construct a basis set. These models, despite their computational complexity, have been proven to be complete. Once constructed, this basis set can be applied with our method to achieve the desired result.
"Not that since it is symmetric tensor, the components of 1e are actually zero.
20

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A.2 Connection between graph-level and subgraph-level functions
In practice, the focus of many problems lies in some local features, such as node-level or edge-level features, which we collectively refer to as subgraph-level. In this section, we will show the connection between graph-level functions and subgraph- level functions.
In fact, the local features also require the message from the whole geometric graph G. thus for a subgraph sab C 9, the Ith-degree steerable feature could be denoted as fl 9). Moreover, we denote F()) (g G) == { (a 9)If(() E } C R Note that the whole graph is also a subgraph, (() (g) and (G) are actually the simplified version of (G) 9) and F()) (9,9) g).
Lemma A.8. Given a geometric graph G. where Goub C 9 is subgraph (c.g. node, edge), then we have (1)) (g) C B(O) (Gsu::)) 9).
Proof. Note that for the subgraph- levell function (g 9), let the function ignore the previous input Gub. Therefore, for any graph-level function, there is always subgraph-level function that is equal to it.
Lemma A.9. Given a geometric graph G, where Gaub C G is a subgraph (e.g. node, edge). We have (( 9) = 2.2+1, (g)) 211+1
Proof. It is obvious that ((GSub. 9) C 2 And we have TR F()) C F(gutb 9), thus FG 9) F(G) 2
Proposition A.10. Given a geometric graph 9, where gmub C. is subgraph (e.g. node, edge), suppose there is matrix w (g 9) with C channels of lth--d steerable features denoted as de (Gsub, 9) satisfying span (Guub, Siii,,9 9) = TF(1) (Gruts 9). Then for any lth-degree steerable function (!)) E F(), there exists T (0) (gmu 9 = [w)] (0) (Grub, 9) with C-channel output scalars, such that
(G 9) = > C
c=1 w (0) (Guur, "
(grub:))
(12)
Proof. Note that the proof of Theorem 3.2 does not depend on the input form of the function, so the generalization from Theorem 3.2 to this proposition still holds.
Proposition A.11. Given a geometric graph G. where Guub C G is a subgraph (e.g. node, edge) and F(G.u 9 F()). Suppose there is a matrix (() with C channels of lth-degree steerable features denoted as "! (G) satisfying span (g) = 1F() (g). Then any Ith-degree steerable
function (()) F(O) there exists w (0) (Grub. 9) == [w. (Asub, 9) with C-channel output scalars, such that
E(0) (Gubs, 9) DE (: (Amb. g)
( (g).
(13)
Proof. Note that 8!"(() E (G) C FF 9), this proposition still holds.
Note that for invariant functions, if we can obtain the canonical form at the graph-level, then we can naturally obtain the canonical form at the subgraph-level. Therefore, the discussion of subgraph-level and graph-level is consistent. To simplify the writing, only the graph-level case will be discussed in the following.
A.3 The full form of dynamic method
Before giving a complete theoretical analysis, let us review Theorem 3.2 as follows.
Theorem 3.2 (Dynamic Method). Given a geometric graph G, suppose there is a matrix ())(g)
with C channels of ith-ee steerable features denoted as (i(!) (g) satisfying span( (())) =
(()()) = (G) f()) E } C 1 Then for any lth-degree steerable function 300) E F()),
y(y)
there always exists
(wt! (0) (9) with C-channel output scalars, such that
(G) Ee w( (G)oo' (g).
(4)
21

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This form decomposes the problem of constructing a complete equivariant function into two compo- nents. However, determining the explicit forms of ( (g) and ( remains a highly nontrivial task. To provide insight into this construction, we present two corollaries of Theorem 3.2, corresponding
two special cases of the equívariant space (G): one where it reduces the trivial set consisting
solely of the origin, 0 {02+1), and the other is the opposite, i.e. dim F (g) 0.
Proposition A.12. Given a geometric graph 9 with (G) = 0, let w(O) C F(O) denote all invariant
functions that can be expressed by model, any lth-degree steerable function could be written into the form of Theorem 3.2, if ()) is not empty.
Proof. Since ( (g) = 0, any equivariant output of degree must be the zero vector. In particular, both the target output (g) and each basis component ! (g) in the decomposition of Theorem 3.2 are identically zero.
As a result, the decomposition in Theorem 3.2 holds trivially for any choice of invariant scalar function ( E vo((), because both sides of the equation reduce to the zero vector. Hence, any dth-ree steerable function admits the desired representation, as long as W(() is non-empty. O
Proposition A.13. Given a geometric graph G with dim (G) > O, let Wi() C F(()) denote all invariant functions that can expressed by model, any lth-degree steerable function could be written into the form of Theorem 3.2, (1(0) (p(0), i.e. the invariant model is complete.
Proof. The sufficiency is evident. By Theorem 3.2, there exists an invariant function of the required form, and such a function is clearly contained in the space FF()).
To prove necessity, assume by contradiction that w/() Ç [F()) but that the decomposition in Theo- rem 3.2 still holds for all steerable functions. Let (G) be set of linearly independent equivariant vectors that spans ((t) (G). i.e., basis of (Tt) (g). Now, consider any w(()) E [F(0) \ w(()) Then the function (g) Ee w (G)o," (!) (g) is valid steerable function in F(O) (g), as it results from linear combination of basis elements with coefficients in IR(0), However, since (i()) is not expressible by the model, the decomposition cannot be achieved using elements in w700, contradicting the assumption that the decomposition is always possible.
Therefore, the assumption must be false, and we conclude that W [F() is necessary for the representation to hold for all stecrable functions.
A.4 The construction of canonical form
In order to losslessly encode the information of a geometric graph, we define the concept of canonical forms in this section. Then, we give method construct canonical forms in polynomial time. Here, we assume that the geometric graph is fully connected and use double-loop set comparison method to estimate the worst time complexity of the algorithm.
A.4.1 An algorithm for determining point cloud isomorphism
Here, we give Algo. for determining whether two geometric point clouds are isomorphic and prove its correctness. is noteworthy that this algorithm resembles that of [104], centered on the principle of four-point positioning in dimensiimnii space. Our subsequent designs will leverage this principle to adapt it for use in geometric graph scenarios and neural network applications.
For two given point clouds, represented as X are isomorphic, there must exist local substructure (composed of at least four points, i.e. U and V3) that is also isomorphic. It is worth noting that this is the isomorphism judgment between two point clouds of certain size, so is always of constant complexity.
(o) and X (H), we recognize that if these point clouds
According to the four-point positioning principle, the coordinates of any point य, ( or T. E X (H) can be expressed as distance vectors, i.e. (9) or df relative to these four foundational points. Consequently, sets of coordinates (G) X (P)) can be transformed into sets of distance vectors (9))) ((H), When there is set of isomorphic substructures with matching distance vector
22

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Algorithm 1: Check point clouds isomorphic with time complexity O(N)).
Data: Two point clouds ( and X (H),, each containing N nodes.
Result: True or False, indicating whether the two point clouds isomorphic.
// O(N4), the worst case requires traversing all permutations. 1 Find an ordered tuple containing four non-coplanar points Ua f (wta )== in 9; elements from set size N.
// O(N"), the permutations
for all ordered subsets of X (H) containing four elements donated by V ( /=1 do
(H)
// O(1), comparison of two ordered sets of certain size.
3
if U. and VB is not point clouds isomorphic according to matching (ūa, 0g
((H) then
continue;
end
for
ū. E ( do // //N,,,, get 4-channel scalar vector.
( (l Ua i,; Tagll [ū; Uag ll, || Uas II):
8
end
for E (H0 do
// ONN), get a 4-channel scalar vector. d,tt d + (1 ū. ll. 10 T..11. 11, Tg lI |6 - ūz, II):
10
end
// O(N), change the node set into scalar set.
D(()), (HH) + set(Id set(ld
12
// O(N3), the complexity of comparing two sets varies depending on the choice, such double loop comparison.
if (C) (H) then
14
return True; end
15
end
return False;
sets, the point clouds are necessarily isomorphic. Conversely, if no isomorphic substructure can be identified or if the distance vector sets cannot be rendered equivalent for any potential isomorphic substructure, then the point clouds must not be isomorphic.
A.4.2 An algorithm for determining geometric graph isomorphism
Furthermore, we discuss how design an algorithm distinguish the isomorphism of two geometric graphs. Note that at this point, all node coordinates have been converted to distance vectors, so is also easy to transform the edge set as follows.
A.4.3 A general canonical form
Now, we further give a method to construct the canonical form of geometric graph. From Algos. and 2, it is not difficult see that the core of the algorithm is transform a set of 3D coordinates into a set of distance vectors. Here we give Algo. 3 as follows.
To encode the set, employ DeepSet [65] for encoding, which losslessly encodes data. Here, ignore the complexity of the neural network (because this is a function of the hidden layer dimension H). and only consider the complexity of the size. It difficult see from Eq. (14) that for set of size M, the complexity of DeepSet is (MM).
Lemma A.14 (DeepSet). A set function could be always written into such form:
f(1z,) = o(xx)).
(14)
23

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Algorithm 2: Check geometric graph isomorphism with time complexity ((N7).
Data: Two geometric graphs 9 and H. each containing N nodes.
Result: True or False, indicating whether the two point clouds are isomorphic.
// O(N4), the worst case requires traversing all permutations.
1 Find an ordered tuple containing four non-coplanar points Ua f- (wa: )== (ua))) in 9;
// O(N)), the permutations
elements from a set of size N.
2 for all ordered subsets ((H) containing four elements donated by V (0, )==1 do (H)
// O(1), comparison of two ordered sets of certain size
if a and V3 is not point clouds isomorphic according to matching (uo, 03
(H) then
4
continue:
end
6
for
ū7 E ( do 1/ O(N), get a 4-channel scalar vector.
(.9) (1iū Tai ll, ūl Woalh | Tas l/;; Tas I1);
8
end
for
E x do // (NN, get a c-cannell scalar vector. de00 + ([]ū, U. ll. 10 u..ll. [0 uن I. |6 - 03,11);;
10
end
// O(N + N2), change the node set and edge set into scalar set. (M0)) D(H) set((d [==), set(ld H=:):
R(0), B(H) t set((d))) (9) d eei (1)e)))) (et((d) (N) (M) (1) i (e))))))
(9)
13
// O(N2 Nº), the complexity of comparing four sets (node-node,
edge-edge) varies depending on the choice, such as double loop comparison.
( = (H) and ((() = B(H) then
14
15
return True;
16
end
end
i8 return False;
Proof. This proof is a simplification of Theorem 7 in Deepset [65]. The necessity, that is, that this form satisfies the permutation invariance of sets is obvious, and its sufficiency is proved below. Consider an Nith degree polynomial P()) ΣN Ei= att', which satisfies the set J.N of all the N roots of Ptt) 0. Thus, the set can be mapped to ordered polynomial coefficients. Vieta's theorem and Newton's identity show that the space of polynomial coefficients is consistent with the space of sum-of- power functions, so the universal approximation theorem of MLP can be used to prove that such a mapping can be found.
Theorem 3.5 (General Canonical Form). Given any geometric graph G, Algo. 3 provides a method to create canonical form with time complexity O
Proof. Due to the completeness of Deepset's encoding, Algo. 3 can naturally distinguish arbitrary geometric graphs. For detailed time complexity analysis, see the comments in the algorithm block.
L
A.4.4 A faster method to construct canonical form
In practical applications, it is impractical to traverse all ordered four- four-point sets because the complexity too high. In order to reduce complexity, a simple and intuitive way is to reduce the number of four-point sets. Here, a very clever change of mind is that we change the ordered four-point set from being selected from a geometric graph being generated from geometric graph. We call these generated ordered four-point sets virtual nodes. In order to ensure the correctness of the algorithm,

---

Algorithm 3: A canonical form of geometric graphs.
Data: A geometric graph 9, and Wnode; Vedge; Vgraph DeepSet models.
Result: The canonical form I' E IR of point clouds g.
// O(N4), traverse all permutations. T Ø;
2 for any ordered set containing four non-coplanar points Ua + {Ta, liz1 in 9 do
for ú; E x9) do
// O(N), get a 4-channel scalar vector.
d, < (Iliz tai ll: ii, Vag [ uas ll, || Uau II): end
5
// O(N + +22,, convert point sets and edge sets into scalar sets.
TD <- set([d) E + set((d,, dj. eijla,ee:;
1/ 0(1), decentralization of the four reference points. Ua (14x4 Uaa
1/ O(N + V/N++NN' +22,, get the embedding based on current four points.
Ta + concat(U..) Mmonte (D), teddge (E));
T TU(Ta):
end
r
Veraph (T);
return
need use B(3)-equivariant method to implement the generation process. We call this generating function Ç, thereby simplifying Algo. 3 to Algo. 4.
Algorithm 4: faster method to construct canonical form.
Data: A geometric graph G, and Undde teedge are DeepSet models.
Result: The canonical form I' e RR' of point clouds g.
// Get four coplaaaa reference points via generation.
1 V + (());
for ū; (9) do
// O(N), get a 4-channel scalar vector.
d, < (llū, l.ll प. 8211 l|ū Ball. i -;,l);; end
// O(N + N convert point sets and edge sets into scalar sets. DD + bet(ld)))) E + set(Id, d,. enlo,ee)):
// O(1,, decentralization of the four reference points.
6 V (Ixx V;
// O(N + N2), get the embedding based on current four points.
7 e concat V V. Phods (D), Vldgg (E)):
8 return
is worth noting that Algo. 4 is applicable only when the four virtual nodes generated by Ç are not coplanar. We summarize it as:
Theorem 3.6 (Faster Canonical Form). Given an E(3) -quivvarna function Ç that generates four non-coplanar points on G, Algo. 4 is able to create canonical form with time complexity O(N2).
Proof. Let I be the function of Algo. 4, we only need to prove that G _ H + r(g) T(H). We first prove sufficiency, let G H with
(O, i) E E(3) be group element. Since is have (9) (()) ((g H) 0V + (m)) After ( In addition, the encoding of nodes and edges depends on the relative distance to the virtual node and is therefore
an E(3) -equiivavaa function, decentralization, we have (g) (H)) = (V
25

---

invariant, i.e. ID()) = D) E(C) = E(H), Thus, we get [()) = I(H). Next, we prove the necessity.
Note that when we take the reference four points as (()) and ((H), we can directly get 9 = H. in Algo. 2, thus completing the proof.
Table 8: Comparison of time complexity.
General Methed (Algo. 3)
Faeter Method
Algoritiun Deuils
Time Complexity
Algorithm Detiil
Time Complexity
Poosible chosen of VNs
Traverse all possible quadruplets ((2)
O(N))
Generate directly.
((1)
types).
Coustract VNa
Use real nodes vistual nodes.
(1(1)
Aggregate by all modes.
((()
Get distance vector
Calculate mudes possible VN+
OLM4-N NN)
Calculate nodes possible VNs.
Oi1 NN)
distunce
Each chosen peodive
(1(A+))
Ench chosen prodive
((1)
Embed sot wish Deepset
O( (VV NP))
node edge
OX1- X9))
Tocal
A.5
Construction of full-rank basis set
In this section, we discuss the full-rank basis set, including its existence and construction method on asymmetric graphs.
Theorem 3.7 (Coloring on Asymmetric Graph). In an asymmetric geometric graph, each point can be assigned a unique color. This implies the existence of an E(3) invariant function that maps the features of the ith node to distinct values h. Similarly, for directed edges ij, their features are mapped to distinct values Eij.
Proof. The theorem can be described in the following formal language, given an asymmetric point cloud X R3 and all function here discussed are node-level scalar function. This theorem means
3f X y E N. vi + j. yi + yj.
(15)
First, we prove that it can be expressed as an equivalent proposition, which is
Vi j X y E N Y4 # yj-
(16)
According to the logical relationship. it is obvious that Eq. (15) leads to Eq. (16). We will prove below that Eq. (16) can also lead to Eq. (15). Let füj X + y denote function let y # v then consider map 9: X + Y == B We can find that Y, # Yj, and notice each fd only uses finite colors, which means we can further assign a new color to each possible output of g, which is the mapping we want in Eq. (16).
Then we prove by contradiction Eq. (16), assuming that the following proposition is true:
3i + j.vf X y N y
(17)
We now construct a contradiction, hoping to prove that Eq. (17) will lead to point cloud symmetry. Let the permutation matrix P E SS represent the exchange between node and node j. so we can get:
f(PX) Py f=
(18)
Since / is an E(3)-invarine function, we guess PX will fall in set F(3). X. And now we prove that PX must fall in such set. And the canonical form I is obvious B(3)-invarinal function and could distinguish whether two point clouds are isomorphic, which means PX must be point cloud isomorphic with X. And notice that f is node-level function, which means
- = -- ب.
= g. T..-
And we have assumed there are no overlapping points, i.e. I. * ē.. so * e. And it shows that such a point cloud is symmetric, which constitutes a contradiction and the original proposition is proved.
Theorem 3.8 (Existence of Full-Rank Basis Set). For any given asymmetric graph G. an (G)- full-rank (G) can always be constructed for any degree l.
26

---

Proof. We give the proof via a constructive method. According to Theorem 3.7, there exists mapping let node be colored differently, and without loss of generality, we denote the color of node i as Now we construct function:
(x(X) Ok.c Σ; Tk
(19)
It is obviously that such a funtion is E(3)-rqunariant, and we and further combine fu of different k which can construct a full-rank 7(1) since the input point cloud is non-coplanar.
We just give existence proof here, which is still affected by the structure of the model itself and the form of message passing. We give an example in Example A.15.
A.6 Comparison of different coloring methods
Node coloring necessitates updating node features. Here, propose alternative solutions:
1. Distance to Center: A simple preliminary coloring can be achieved by calculating the distance || - Tell from each node to the coordinate center. While this method is straight- forward, it lacks practical significance and may not perform well in real-world problems. Additionally, there are counterexamples, such as Example A.15, where nodes cannot be distinguished.
2. Tensor Product: This method extends the previous approach. First, we construct a global feature Σ (α, æz), which is then used to create the coloring through moments. While this method more complex and incurs higher computational costs, enhances expressive power.
Table 9: Different Color Methods. For the tensor product method, the set = ..,..., L}} denotes all degrees and the learnable weights of tensor product are omitted.
Nothing (2)
Distance Center
Tensor Product
Caleulation Formila
h (b) aoeaa
EM or [frll]]) yol (s7.e)
h; +- h; a(I[E,)))
h. Rignal Bay Blg (2 (su))
Scenarios Avoid
Spanse Graphs
Coumterexample like Fig. 4(c)
Founded
Below, we present two examples to illustrate potential issues without coloring.
Example A.15 (Fig. 4(c)). We use the center pooling method to construct such an example. First, we ensure that the center of gravity of the image the origin, and secondly, make it asymmetric. The construction here requires basic point cloud:
X. (-1,0,0), 3 3
2
0
(
1
})}
00)
All nodes Xo are in the unit circle, which means the distance between each node and the center of mass is the same. And the final point could is
X (X)
where sample some random rotation matrix R. And then it obviously only using global pooling will lead all virtual node in the center of mass if we do not use any edge to color this nodes.
However, it could also be solve by high-degree steerable feature. The global steerable features can construct different virtual nodes with tensor product method in Table 9.
Example A.16 (Degeneration in Uncolored Models). There are specific scenarios in which mes- sage passing uncolored geometric graphs can result in degenerate phenomena. A particularly illustrative example found in geometric graphs where node features are identical and all edges bidirectional. Considering single-layer EGNN example, it may degenerate because the
27

---

condition h; = h implies maj = Mjj. Consequently, we have EdWee Pa(m:y) Biy 0g. As result, Eq. (7) can only yield the coordinate center T.. This phenomenon can be observed in the tetrahedron center experiment in $ 4, where besides EGNN, advanced models such as HEGNN [49], TFN [33], and MACE [35] also degenerate.
B Experiment Details
Our code is available https://ggihabb.co///AA-AUUU
B.1 Expressivity Tests
B.1.1 Completeness Test
In this section, we provide additional details that were omitted in 4.1, focusing on the following points:
1. The 4-body non-chiral counterexample implemented by the GWL-test (Fig. 2(f) in LASR- test) and the 4-body chiral counterexample (Fig. 2(c) in IASR-test) exhibit a significant issue:
the geometric graphs presented by [32] are geometrically isomorphic.
2. Details regarding data construction and model design for the chirality test, which primarily include the coloring strategy and specifics of the construction of Oo-type features.
Metric. Referring the settings of the GWL-test [32], utilize the average accuracy across tests as our evaluation metric. The detailed experimental design is based on the implementation of the GWL-test [32], available in the code repository'. In this setup, single GNN layer is employed to encode the geometric graph, followed by simple classifier that predicts the label. If GNN fails distinguish between two geometric graphs, the output embeddings will be identical, resulting in one graph being classified correctly while the other is misclassified, leading to an accuracy of 50%. Conversely, an accuracy rate exceeding 50% indicates that the GNN can generate distinct embeddings and successfully differentiate between the two geometric graphs
Additionally, the accuracy of the Basic model in Table 2 does not reach 100% due to limited number of training epochs for the classifier (set to 100). Increasing the number of epochs (e.g.. to 200) achieve 100% classification accuracy. Notably, the EGNN model, with the same settings, also achieves 100% accuracy. We speculate that EGNN improves the embedding of geometric graphs during the message passing process.
Results of k-chain test. This experiment comes from GWL-test. The canonical form introduced by model can easily obtain global information, so only one layer is needed to identify the geometric graphs differentiation task that other models require multiple layers.
Table 10: k-chain Test.
(k: 4-chains)
Number of layers
GNN Layer
k
2 +
6 +
k
EGNN
50,0 =0.0
50.0 0.0
50.0 = 0.0
50.0 0.0
100.0 0.0
GVP-GNN
50.0
100.0 0.0
100.0 0.0
100.0
100.0 0.0
TEN
50.0 a.0
50.0 0.0
50.0 0.0
100.0 = 0.0
100.0 0.0
MACE
50.0 2.0
100.0
100.0 0.0
100.0
100.0 0.0
Basicepl
100.0
100.0 0.0
100.0
100.0 0.0
100.0 0.0
SchNet.pl
100.0 0.0
100.0 0.0
100.0 = 0.0
100.0 0.0
100.0 0.0
EGNN
100.0 0.0
100.0 0.0
100.0 = 100.0 = 0.0
100.0 0
100.0
GVP-GNNGp
100.0 = 0.0
100.0 0.0
100.0 0.0
100.0 0.0
TFN
100.0 = 0.0
100.0 0.0
100.0 0.0
100.0
100.0 0.0
Geometrical isomorphism of 4-body counterexample. In the open source notebook provided by GWL-test [32], four tasks are presented, the first two which are the 2-body and 3-body tasks tested
'https://gitan http://hw....ccaaaiaaaeeeoaaaeee hwtp://gggeeetieeeeeeeeeeoeecece www.eetttrrnnnddorroeeeee goj-d
28

---

in Table 1. The other two tasks are termed the 4-body non-chiral counterexample and the 4-body chiral counterexanple, both originating from IASR-test [69]. According to the experimental results provided in GWI.-test, only MACEs body successfully passed the 4-body non-chiral counterexample; moreover, the results for the 4-body chiral counterexample were not presented.
We successfully reproduced and cited the experimental results of the 2-body and 3-body from GWL- test [32]. However, we were unable reproduce the results reported in the article despite multiple lesting attempts, consistently achieving an accuracy of only 50%, indicating that MACEs-tody failed this task. Furthermore, recent literature [77] also reported similar failures in the test. Upon thorough investigation, we discovered that the geometric graphs used in the two tests were geometrically isomorphic. We suspect that there may be a minor mistake in the GWL-test.
Specifically, the two graphs 91, 92 of the 4-body non-chiral counterexample are constructed as follows:
1. Consider three sub-graphs: Hi =
-33, H2
{(3, -6), (0, 22 5). (-3, -2, [2444 and H3 ()) Ry is random matrix for rotation around the Oy-axis. Let M,, M, denote the reflection about yOz-plane and =Ox-plane.
2. 91 H1 U (R. H2) U H3 and G2 H1 J (R, UM, Hz).
Now we show 91 = H2, notice there are serval relation:
Internal symmetry of geometric graph: H, = M Hi(i = 1,2), H3 = Ryy
Symmetry between geometric graphs: H1 = MM, H2i
Properties between geometric transformations: M = M2 = I. M2 = R. My R.
Then we have:
G1 H1 (R. H2)
= (M, H2) (R, H4) U (M, Ha) = (Ry H1) U UM, H2) (M, Hz)
-,, UN)((M U UM, Hz)
-(M,R,) ) .H4 U.U.R. R2 U RyHss
-,M,R,) ).R.)..P) - U R,H2UHH
-(M, M.) (M,HI M, M,,,H,H H2 U M2
-MM M,) (H U U, U M,HHs)
R,, M2) [H1 UR M,
R,, M2) -2
-(M,R,,,,) G2.
Moreover, the 4-body chiral counterexample also contains two geometric graphs G3, Gr, where G3 _ G1:
1. Consider two sub-graphs: H4 = {33.0, 0, -2) 4), (-300,), -4)) and Hs = (0,,5, 5, (.)). Ma, My denote the reflection about yO2-plane and 2Ox-plane.
2. G3 H4, L H. and G1 H. U (M,
Now we show G3 = H3, notice there are serval relation:
Internal symmetry of geometric graph: H3 = -M H3. H.3 = H3 and H4 = M.
Properties between geometric transformations: M = M2 =
Then we have:
G3 = H4 U Hs. (M, [H4 U M,H.s (Ma G1.
29

---

B.1.2 Chirality Test
Details of the chirality test.
1. Fig. 4(a): Choose 91 from the 4-body non-chiral counterexample, and another geometric graph is -
2. Fig. 4(b): Since G3. G4 are symmetric, we modify to Gs = {(3, 0, 4), R.
(0, 0, 5), (-3,0, -0-4 (0, 5, 0)}. where Rg is random matirx for rotation around the Oy- axis.
3. Fig. 4(c): Similar to Example A.15.
(a) Inversion
(b) Mirror
(c) Counterexample
Figure 4: Examples to test whether the model recognize chirality. Figs. 4(a) and 4(b) modified from GWL-test's implementation [36] of LASR-test [69], while Fig. 4(c) is designed by ourselves.
Fig. 4(a): Test whether models could distinguish this geometric graph with itself after inversion. Fig. 4(b): Test whether models could distinguish this geometric graph with itself after mirror about the =12neee Fig. 4(c): The task is same Fig. 4(a), while all nodes
the unit sphere, and the center point coincides with the center of the sphere, Such counterexample cannot produce separable virtual nodes by simple center distance coloring.
Details of construction of Co-type features. There are two ways to construct Oo-type features:
1. Determinant: Calculate det((02 31, 03 31 t -1));
2. Tensor Product:
We still begin by constructing Σ ( - Zz), and employing moments, specifically least 3rd-order moments, derive Oo-type features. This approach clearly serves as an extension of the previously established method.
B.2 Performance Tests
We conducted four experiments:
1. Tetrahedron center prediction, which entails predicting graph-level targets, i.e. Monge points, twelve-point centers, and incenters, given the coordinates of four points (with the same node features) of given tetrahedron.
2. 5-body system [72] involves predicting node-level targets, i.e. the coordinates of each charged particle at a specific moment in the future, given its initial coordinates and velocity.
3. 100-hody system [66], generalization of the body system, is used to test the performance of the model large-scale systems.
4. MD17 dataset [68], containing eight molecules, is used to predict node-level targets, i.e. the future coordinates of each atom in the molecular trajectory. This dataset be used to test the performance of the model on real datasets.
5. Water-3D mini dataset [66; 67], is used to test the performance of the model in large-scale systems and real datasets. Due to time constraints, we only evaluate substantial subset (Water-3D-e 3.000/300/300 for train/val/test).
30

---

Table 11: Details of each dataset.
Datuset
Target
Type
Asample (truin vaali/
Node Number
Edge connection
Teurahedron
graph-level
Syuthetic Dataset
500/2,000/ 2,000
fully_ccnneee
5 hody
mode level
Symhhttic Dataset
5.000//2.000 2,000
5
fully. comeecedd
100-body
mode-level
Syuthetic Dataset
5,000/ 2,000
fully_concected connected
MD17
mode
world Daluset
500 2,000/ 2,000
5-13
distance cutaff
Water-3D mini
node-level
Real-world Dataset
3,000 300 300
~8,000
ciatance_cutoff
B.2.1
Tetrahedron Center Prediction and N-body System
We conducted experiments on a single NVIDIA H20 GPU. In both experiments, we configured the batch size to 100, the learning rate to 5 x 10m², and the weight decay to 10 For the tetrahedron center prediction task, the maximum number of training epochs was set to 300. In the case of the N-body system task, we implemented early stopping with a patience of 100 steps.
Metric. We Mean Squared Error (MSE) measure the accuracy of the prediction results in both experiments.
Tetrahedron Center Prediction. We generated our dataset using the calculation formulas for the Monge point H as. twelve-point center Ty and incenter H, from [105]. Given a tetrahedron defined
by four points žo, A,, B, #c. with the vectors = A æo, ēn Ho, C = #c Fo, we can compute the desired points as follows:
ā (b
c) x C) b. (c (ē a) + (ā (a x)
EM Zo
2ā xbx
_T IM + 1oo HM),
(20)
6 x ā
ē 0 b ā b
b ह| + e a +
ā x bl
+ b xi-xxxxxx bl|
We utilize 500/2000/2000 samples for training, validation, and testing, respectively. e 10 was added to the denominator to avoid division by zero. In addition, to avoid numerical problems caused by the center point being extremely large, we generated the cireumseribed sphere of the tetrahedron when generating the tetrahedron, ensuring that the radius of the sphere is less than or equal to
N-body system. N-body system [72] is dataset generated from simulations. In our simulations, each system consists
charged partieles with random charges e; E [+1), whose movements are governed by Coulomb forces. The task is to estimate, node-level target, i.e. the coordinates of the N particles after 1000 timesteps. The initial node feature set as the kinetic energy, while the edge features include the distance and the product of the charges. We use 5000/2000/2000 samples for training, validation, and testing like the setting in HEGNN [49]. Since the particles have different initial velocities, they initially have different characteristics (coloring).
B.2.2 Implementation of Baselines
We undertook further development based on the codes of EGNN [43], TEN [33], and MACE [35] provided in the GWL-test repository. Following testing, we eliminated the Gate modules from both TFN and MACE, as their inclusion resulted in significantly degraded performance. Additionally, for HEGNN [49], we utilized the code from the original paper and opted for the inner product normalizati in option.
Each model utilizes at most 4 layers, and the dimension of the hidden embedding is fixed at 64. The last three models employ at most 2nd-degree steerable features. Both TFN and MACE utilize channels for steerable features, with the correlation parameter for MACE set to 4.
B.2.3 Implementation of Our Models
We provide two complete implementations of EGNN [43] and TFN [33] follows.
Unlike FasEEGNN [66], the virtual nodes in this approach are not updated following their initialization during the message-passing layers. We define the normalization coefficient as a; := 1/))(()) For
'nttps://githin http:::://gttuu..com//A.0-
31

---

the ith node, we denote its features as h, and its coordinates as क. Here, C represents the channels of the virtual nodes, and Z E R denotes all coordinates. Specifically, for -llobal we employ the canonical form to build a new feature w, for each node, thereby rescaling the norm of steerable features in manner analogous to HEGNN [49], which is consistent with Theorem 3.2.
Table 12: Complete global models.
EGNNorllobal
TESeoglobal
Node Coloring
h. Color(Ay, distance, to. center)
A Color(h dissance, center)
Generate VNs
2 lett + nz S (2z(h)) E
2 1 - E Va (2 (6 -B.
Create Canonical Form
M. l. Miaas MJ M./IMTMMM
M, 1 misst M, M,/1 NYT ab,lle
Update Node Feature
h. h, Sillm.pp))h h,
91
w. h." AAp (m..)))
my Vmlls, by mg. ||2g (2), myy (a(mn)) dy
1 1, 0i (oo(F,)
Message Passing (x L)
mi nz Ejeni Myy 1i, oo Ejenio Mo
R R Eyerto may
h h, Palh,, mi), d R. ris
A Fax (6)") un) R.).
yPu (Mn, A.)
Readout (node-level)
b Rall)) R. 4.
h, Aaalh 4. R, a3nn a3 a3. I....anarr (1(1))
Readout (graph-level)
hg Spall 4 Yoi- h.), Ro + S 2,
hg - #pmil 2 h.). Zo- + E. 4.
To enhance the model's performance, we permit the virtual nodes be updated layer by layer, similar to the approach taken in FastEGNN [66]. We denote the normalization coefficient a; == 1/ ( We define the normalization coefficient ag N/)(- Notably, we allocate C' virtual nodes to each node, denoted as Z, E RCx3, and these C virtual nodes share a scalar feature represented by 8, Table 13: Complete local models.
TFNgellocal
my Pm.(h..hp by. eyy [| [P)
my Pm (o) ..) e.y. e.g. (?) Valiiaa (6(2)) my)
Generate VNs
8, (alyaa((h)) my)
2. Lod, E (2)(h))) By
2. = loz, + Exevio (a(h)) -E,y
my = Pmils, by. eu. [iay my = palmy) d.
m 1 oue (()())))
m. oz Seaar may mi ou Dijeere my
6. h Eyerty mi
h. = h. palh, ma), di d.
4. = aax (m)) R,
Message Passing (x L)
My tedy Z Myye My, Molli Ms
M, loū, - 2n masae Mr. / M,, Mylle
(real virtal)
maj Smaqu (8., myga))
Musat Smasi(s). h mo.a))
h h2 Phias (mi) 81 maal
youqe (Imgge))
#. 2, E Eaarto if (mmnad) My
5. 5. Ejenos 20 (moud) Mu
2. 2, ExeNoo) 22 (00 (m)naa)) My)
Z. ž, + au Eienin F2 (masapp) ( (a.)
Readout (ndde-lovc))
vaalh)), Z,= smalt ElL Ao),, do Ei
(halh & e3mn, 03 (inear(h
Readout (graph-lovel)
Powil + EN, It). EZi,
Parameter Quantities. Here, we present the number of parameters for each model in the tasks shown in & 4.2. Since the inputs for the two tasks differ, the number of parameters for models with the same number of layers also varies.
Table 14: Tetrahedron dataset.
Table 15: N-body dataset.
Total Parameters
Total Parameters
1-layer
2-layer
3-layer
4-layer
1-layer
2-layer
3-layer
4-layer
EGNN
33.7k
67.1k
100.6k
134.1k
EGNN
33.5k
66.9k
100.4k
133.8k
FasIEGGNN
67.4k
134.4k
201.7k
268.8k
FaSLEGNN
67.5k
134.6k
201.4k
268.4k
HEGNN
46.9k
84.9k
122.9k
160.9k
HEGNN
46.7k
84.6k
122.6k
160.5k
TEN
46.5k
93.0k
139.4k
185.9k
TFN
46.4k
92.8k
139.2k
185.6k
MACE
48.9k
97.8k
126.6k
195.5k
MACE
47.2k
94.3k
141.5k
188.6k
Equiformer
313.5k
340.5k
367.5k
349.4k
Equifonner
313.6k
340.6k
367.5k
394.5k
EGNN globbal
181.7k
215.2k
248.7k
282.1k
RGNN of global
181.3k
214.7k
248.1K 1k
281 1.5
EGNN Se global
46.8k
55.4k
63.9k
72.5k
EGNN gloobal
46.6k
55.1k
63.6k
72.1k
EGNN -local
107.5k
180.7k
253.8k
327.0k
EGNN.--oock
107.3k
180.4k
253.5k
326.66 6k
EGNN ge local
28.2k
47.3k
66.5k
85.7k
EGNN qp local
28k
47.2k
66.4k
85.5k
TFN gglobal
215.2k
291.1k
267.
443.1k
TFN.gglloba
215.6k
291.6k
367.7k
443.7k
TEN globba
70.3k
104.6k
138.9k
173.3k
TIN: TIN rgglbbll
70.5k
104.8k
139.2k
173.6k
TFN TFNga-locul
138.0k
241.7k
345.5k
449.2k
TEN TNNa--loca
138.2k
242.0k
345.8k
449.7K. 7k
TEN -local
51.0k
93.1k
135.2k
177.3k
TEN -locall
51.1k
93.3k
135.4k
177.5k
32

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C Rethinking the Equivariance and General Functions
In this section, we give Fourier series-like approach understanding equivariant GNNs.
C.1 Rethinking equivariance: A perspective from Fourier expansion.
According to [106], the Fourier expansion of multivariate function necessitates the simultaneous consideration of the basis functions associated with all variables, as demonstrated below:
f(1) Σ fa exp(-27rjn) s(t) Σ exp( 2mjn (nnTtt n
f(t)
Σ nEZ4
fm II
k=1
(21)
represents multi-dimensional variable. In fact, any function defined on the unit sphere can be expressed as a summation of spherical harmonics. Let & denote a point on the unit sphere.
00
00 (a (())).
f(i) = Σ
1 (2)
(22)
1=0 m=-
This formulation offers two representations. The fully expanded form aligns more closely with the Fourier series, while the inner product formulation facilitates connection equivariance. At this juneture, we introduce second point ý on the unit sphere and consider the expansion of the function with two inputs, & and y.
f( 9)
(1a2)) (44 (+)) mm. Mii
(23)
12 m==-12
We observe that the product ylll) ( (1)) (
(j) constructs (21, + 1) (212 11 basis functions,
iee. elements in ( @ (()) E (22+11)) Let ((.0)) E (22+++))-( be the coefficients. Consequently, the expansion can also be expressed in an inner product form as follows:
f( 0)
(((0))))) (()) (())
(24)
According to the notation of tensor product, we denote the Ith-degree steerable feature derived from the tensor product (+))(()) @ (12)) (g) as b111812-1) (+. 9). which can be expressed as follows:
(11 11) (z, 3) = ((11))). (x 1 (b)
(25)
1=
22l
Let us define
114 11=hh-4
will (11112-11) (Qi(1882)) (11881l)) This leads us to the following expres-
sion:
f(a. ú)
(w0( (1110
(26)
1=0 14 shihh
where a Ith-degree steerable function ( 1) incorporates only the Ith-degree components in Eq. (26), and can be expressed as follows:
(2,
Σ
(102-+1) f0h042 (á. 9).
(27)
141 11
Eq. (27) offers more concrete understanding of the framework. It is important to note, however, the formula in Eq. (3) is invariant to permutations of the inputs, while the imputs in Eq. (27) are presented with specific order. Nevertheless, this specificity does not preclude us from conducting an analysis based on this formulation:
Since I and la can take any values, there will be infinite basis functions in B()) when the geometric graph accommodates interactions involving more than two bodies.
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2. Not all bases in Eq. (27) will be meaningful; that is, some bases may not emerge in the objective function, although they satisfy the equivariance constraints.
From this perspective, the advantages and disadvantages of the equivariant model become evident. By incorporating an equivariance prior into the model design, effectively eliminates components that cannot exist within the framework. However, this constraint necessitates limited selection of operators in the model design. the current setup, only the tensor product exhibits strict equivariance (while others can be transformed and expressed using tensor products, e.g- tensor decomposition in frame averaging [56; 84]), whereas other operators, such as group convolution, require approximations under current floating-poini operation systems and can achieve only partial equivariance. This limitation, turn, constrains the model's capacity for learning.
C.2 Rethinking General Functions: the Equivariant Decomposition
Generally, the imput of a geometric graph could be denoted as {ē; J;= without loss of generality, we assume that these coordinates are decentralized. Although in practice, people generally use edges as
input (i.e. {zt H ) 1), for the sake of convenience we still discuss the vectors of points here, and it is easy to verify that they are equivalent, Then normal function in l-degree problem could be rewritten as:
3R3xN R2++1 want
(g)
Σ
(g) + S
(28)
6 (G))B()
To completely give the expansion formula of such ((), we need to analyze each dimension separately. And we denote the mth- (-1 < m I 1) of f(() as f(t.m) (similarly, ((t.m) represents the
mth dimension in b (()), which is:
fll.m) (g)
Σ
(1) (l.m)
(9)
Σ
Σ
w
(o.1) (0.0)
10.00100000 (g).
(29)
6 (B))
==0
B
(G)
And it could find that he aut-dimension
fels
(L.m) of
(()) could be donated as:
20
(l.m)))
(9)
Σ
Σ
(..0)) (g).
(30)
8=0
(G)
Here and B are dummy index for enumeration and both wa and w (l.m) function about all radials ()) Notice that the same basis may appear in both two items in the equation, ensure uniqueness, set
are constant (or only the
w argmin |1 ( (2) argmin !))
Σ
(31)
w
(G)ER()
And with definition of equivariance error in Spherical CNNs [107], we could find that
Legu =
(1) (g) ( ( 9)
(g) 1 (g) f1 (g 9)
(32)
<R (0())) f (g)ll + f (g 9)
<
[1 (9)
Traditionally, practitioners have opted for data-driven methods, such as data augmentation, to achieve equivariance. This approach aims constrain the norm || by adjusting the parameters wa (a.() throughout training, with the goal of minimizing their values. In contrast, models derived from geometric learning directly incorporate the equivariance constraint into the model architecture, effectively achieving ( 0. A fundamental aspect of these models is the construction of the
34

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basis set B(O), allowing them to systematically remove irrelevant or non-existen components from the learning process.
Moreover, for a (g), obtaining (9) or calculating | (9) caa intractable. However, plausible approach is to utilize another general model whose output is bounded by constant M to learn the residue. This leads us to the relationship [| (g)) & ( = M. We introduce this concept here, hoping it will inspire future research.
Additionally, it important emphasize that only require the inference model to be equivariant. In practice, we allow the use of non-equivariant methods for training our model. For instance, noise injection VaVeN [108] and the Huber loss in MEAN [102] are examples of such methods. These approaches modify the coefficients Wa. but they do not violate the equivariance constraint.
35

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We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; 2) a full-rank steerable basis set.
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Reviews:
[{"review_id": "ZMDiWb7AOs", "reviewer": "Reviewer_yiPB", "summary": "This paper presents a theoretically grounded and efficient framework for constructing complete equivariant graph neural networks (GNNs). While existing methods often rely on deep architectures, high-order body interactions, or high-dimensional steerable features to achieve universal approximation capability, they typically incur high computational costs. The authors show that completeness can be achieved through two key components: (1) a complete scalar function, referred to as the canonical form of the geometric graph, and (2) a full-rank steerable basis set. Based on this theoretical analysis, they propose an efficient algorithm for building complete equivariant GNNs using EGNN and TFN as base models. Experimental results demonstrate that the proposed approach achieves superior completeness and strong performance with only a few layers, significantly reducing computational overhead while maintaining practical effectiveness.", "questions": "See Weaknesses.", "limitations": "YES", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- 1. This paper systematically summarizes several tensor-product modules.\n\n- 2. The paper is clearly written and well-organized.\n\n- 3. The canonical form of scalar functions may help replace certain equivariant features, offering a new perspective on model design.\n\n**Weaknesses**\n\n- 1. The theorems in this paper mainly focus on proving completeness, but lack theoretical analysis regarding expressive power, similar to Theorem 2 in [1]. I would like to know whether this canonical form can only substitute for simple (low-order) equivariant functions in practical applications.\n\n- 2. Constructing GNNs using high-quality invariant features is also a common practice. In my opinion, the authors should compare their method with existing invariant representations, such as FPFH in point cloud processing or the embedding layer of GotenNet [2].\n\n- 3. The authors could try more complex real-world systems, such as QM9.\n\n- 4. Theorem 3.5 is more like a definition.\n\n[1] Nadav Dym and Haggai Maron. On the universality of rotation equivariant point cloud networks\n\n[2] Sarp Aykent and Tian Xia. GotenNet: Rethinking Efficient 3D Equivariant Graph Neural Networks", "was_revised": true, "final_justification": "The theory presented in this paper does not have obvious flaws. In the rebuttal, the authors addressed some of my major concerns, particularly whether such invariant features can truly be effective in practical applications. I believe the paper's completeness meets the acceptance criteria. However, I cannot rate this paper higher than a 4, because the practical applications provided by the authors are quite limited, using non-mainstream datasets, and thus no comparison with state-of-the-art equivariant models is included. The real-world utility of the proposed approach still needs to be validated over time.", "rebuttal": "Thank you for your comments and questions, and we salute your hard work and time in the review process. We summarize and respond to your questions as follows:\n\n> **R1 (W1). Discussion of Expressive Power** \n\nThank you for highlighting this important aspect. Expressive power refers to a model's ability to approximate functions. In general, the more functions a model can accurately approximate, the greater its expressive power. Completeness, actually, denotes the strongest level of expressive power: it is the capacity of a model to approximate any equivariant function to arbitrary precision. \n\nBoth of Thm. 3.2 in our paper and Thm. 2 in D-spanning [1] ultimately establish completeness, although their derivation processes differ substantially. Specifically, whereas D-spanning [1] relies on constructing equivariant polynomials via sufficient layers of tensor-product operations, our approach requires a set of complete invariant scalar functions along with a full-rank steerable basis set. \n\nFurthermore, our canonical form can substitute for complex and high-order equivariant functions in practical applications (not just simple and low-order equivariant functions), as our porposed model is a complete model  of arbitrary degree $l$.\n\n> **R2 (W2). Comparison with Existing Invariant Representations**\n\nWe are grateful for your insightful suggestion regarding comparisons with common invariant representations such as FPFH and GotenNet. In response, we have conducted additional experiments as follows:\n\n1. Inv$_\\text{FPFH}$: We construct additional node features using the FPFH descriptor and introduce them into the EGNN framework.\n2. Inv$_\\text{GotenNet}$: We similarly inject node features generated by the embedding layer of GotenNet.\n\nAll methods are evaluated on the N-body dataset, with the results presented below (Table S1):\n\n**Table S1. Results on the N-body Dataset ($\\times 10^{-2}$).**\n\n||1-layer|2-layer|3-layer|4-layer|\n|-|-|-|-|-|\n|EGNN|4.214|0.780|0.710|0.712|\n|HEGNN|4.114|0.801|0.561|0.489|\n|EGNN+inv$_\\text{FPFH}$|3.800|0.701|0.487|0.674|\n|EGNN+inv$_\\text{GotenNet}$|2.307|1.922|2.083|1.030|\n|EGNN$_{\\text{cpl}}^*$-global|0.985|0.554|0.533|0.498|\n|EGNN$_{\\text{cpl}}^*$-local|**0.703**|**0.513**|**0.455**|**0.450**|\n\nWe observe that, FPFH-derived features can enhance EGNN performance, and GotenNet embeddings only outperform EGNN at shallower depths. In contrast, our model consistently achieves leading or comparable results, especially as depth increases. We will discuss these insights more thoroughly in the revised manuscript.\n\n> **R3 (W3). Additional experiments on Complex Real-world Systems** \n\nThank you for encouraging us to investigate more challenging datasets. As QM9 is primarily an invariant prediction task, but our work aims to advance the performance of equivariant functions, we instead select the large-scale Water-3D dataset from [A] for evaluation. This dataset involves an equivariant task to predict future coordinates for fluid particles (average >8K nodes/system). Due to time constraints, we only evaluate a substantial subset (Water-3D-mini: 3,000/300/300 for train/val/test) using up to 1,000 training epochs.\n\n**Table S2. Results on Water-3D-mini ($\\times 10^{-4}$).**\n\n||1-layer|2-layer|3-layer|4-layer|\n|-|-|-|-|-|\n|EGNN| 4.904|4.323|3.649|3.338|\n|FastEGNN| 4.885|4.332|3.782|3.259|\n|HEGNN|4.885|4.138|3.519|3.287|\n|EGNN$_\\text{cpl}$-global|4.368|3.711|3.294|3.248|\n|EGNN$_\\text{cpl}$-local|**3.611**|**3.320**|**2.803**|**2.495**|\n\nOur approach (EGNN$_\\text{cpl}$-global/local) consistently outperforms the baselines at every depth, and achieves superior results with fewer layers. These findings further support both the scalability and efficiency of our method of handling large, real-world systems.\n\n> **R4 (W4). Improved the Presentation of Original Thm. 3.5**\n\nWe apologize for our previous confusing presentation. Our intention was to present a theorem regarding the time complexity of Algo. 3. We will clearly state this theorem and provide a detailed proof to ensure the correctness and completeness of our claims in the revised manuscript.\n\n***\n\nThank you once again for your constructive feedback, which has directly contributed to improving the clarity, empirical evaluation, and theoretical discussion of our work. We will incorporate all clarifications and additional results into the revised version of the manuscript. We sincerely hope that our responses and revisions will address your concerns, and we would be grateful if you might consider these improvements in your evaluation.\n\n> **Reference**\n\n- [A] Zhang Y, Cen J, Han J, et al. Improving equivariant graph neural networks on large geometric graphs via virtual nodes learning. ICML 2024."}, {"review_id": "oDlx0kGZtW", "reviewer": "Reviewer_Y7Ue", "summary": "This paper introduces a novel dynamic method for constructing complete equivariant graph neural networks (GNNs) for geometric graphs. Unlike previous methods that use infinite basis to achieve completeness, the proposed method only needs a complete scalar function (of the geometric graph) and a full rank steerable basis, which are finite. This way, the architecture doesn't need to use higher-order representations or stacking multiple layers to achieve completeness. The authors introduce an efficient implementation of the method via geometric isomorphism for asymmetric graphs, and empirically demonstrate its expressiveness and performance through synthetic and real-world datasets.", "questions": "1. As mentioned in weakness, can you explain the definition of \"steerable function\" in line 146, and explain the Thm 3.2 in more detail, such as what is  $F^{(l)}(G)$ , how does it depends on G, how to construct it (and $V^{(l)}(G) )?\n\n2. For simpler dataset like tetrahedron center prediction, 5-body, 100-body system, the proposed dymaic method significantly improves over the baselines, but for MD17, the improvement becomes much less, with some even underperformed the baselines. Can you explain why?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**Strengths**:\n1. Novelty and Significance. The author proposes a new framework for constructing complete equivariant maps for geometric graphs, which is fundamentally different from previous approaches and potentially expands the perspective on this important question. Plus, the proposed framework is clear, rigorous, and straightforward to implement.\n2. Experiment. The experiments and results are solid, covering different tasks, a strong set of baselines and insightful analysis. And the performance improvements brought by the newly proposed method are significant.\n\n**Weakness**:\n1. Clarity. The explanation of some concepts can be improved. For instance, in Thm 3.2, I'm not quite sure how does the $F^(l)(G)$ depends on G, I didn't see definition of it previously. Also, it's better to explain what a geometric graph and equivariance for geometric graph is in the introduction section. As I mainly work with topological graphs, I was initially confused by the description of completenss of equivariant GNNs. Another example is in line 145-146, I'm not quite sure what is \"steerable function\", is it just \"equivariant function\" w.r.t. O(3)?", "was_revised": true, "final_justification": "All the issues I mentioned are resolved. And as per our discussion, I'll maintain my score of \"5 Accept\".", "rebuttal": "Thank you very much for your valuable and constructive feedback. We greatly appreciate your careful reading, positive remarks, and thoughtful questions. Please find our detailed responses below:\n\n> **R1 (W1 & Q1). Clarification of Concepts**\n\nThank you for highlighting areas where further clarification and improved definitions are needed. We acknowledge that terms such as \"geometric graph\" and \"steerable function\" may not be universally familiar to all readers, especially those who primarily work with topological graphs. We will address these points and strengthen the introductory section to include more context and definitions. \n\n**Geometric Graph and Equivariance:** We agree that a clearer distinction between geometric and topological graphs would benefit readers. In our context, a \"geometric graph\" refers to a graph not only with nodes and edges, but with associated coordinates in Euclidean space. Equivariance for geometric graphs then refers to functions whose outputs are transformed in a specific way—consistent with group actions (e.g., $\\mathrm{O}(3)$ rotations and reflections)—whenever the graph’s geometric data is transformed. You may refer to review [A] for a more detailed introduction.\n\n**Steerable Function:** Indeed, a \"steerable function\" is synonymous with an \"equivariant function\" under the action of a group, particularly $\\mathrm{O}(3)$ in our case. The term \"steerable\" is often used in signal processing and geometric deep learning communities to denote equivariant mappings of higher degree ($l>1$).\n\n**Clarification of why $\\mathbb{F}^{(l)}(\\mathcal{G})$ depends on $\\mathcal{G}$ in Thm. 3.2:**  Here, $\\mathbb{F}^{(l)}(\\mathcal{G}) \\subset \\mathbb{R}^{2l+1}$ represents the set of all $l$th-degree equivariant function values on the geometric graph $\\mathcal{G}$. Each such function outputs a vector in $\\mathbb{R}^{2l+1}$, corresponding to the degrees of freedom for the representation of degree $l$ under $\\mathrm{O}(3)$ transformations. Importantly, space $\\mathbb{F}^{(l)}(\\mathcal{G})$ may not span the entire $\\mathbb{R}^{2l+1}$, depending on the structure of the graph $\\mathcal{G}$ and its geometric realization. As discussed in [B] and outlined in Appendix A.1.3, in some configurations (i.e. symmetric graphs), the equivariant functions are restricted to a subspace. Thus, the expression and universality results must take these distinctions into account.\n\n**Construction of $\\tilde{\\boldsymbol{V}}^{(l)}(\\mathcal{G})$:** The construction of the basis $\\tilde{\\boldsymbol{V}}^{(l)}(\\mathcal{G})$ is two fold: 1. We formally prove (Thm. 3.8 and Appendix A.5) that such a basis always exists for the space of $l$th-degree equivariant functions for any $\\mathcal{G}$; 2. In Section 3.5, we provide a concrete example: the set of coordinate differences $\\\\{\\vec{\\boldsymbol{x}}_{ij}\\\\}$ used in EGNNs can be leveraged to construct $\\tilde{\\boldsymbol{V}}^{(l)}(\\mathcal{G})$ in practice.\n\nWe will update the main text to make the definitions, dependencies, and construction procedures more transparent, and will provide these explanations in the main manuscript rather than only in appendices.\n\n> **R2 (Q2). Discussion of** **Performance** **Improvement Difference** **over Baselines**\n\nThank you for your insightful question regarding the performance on the MD17 dataset. Indeed, we notice that the improvements of our dynamic method are less pronounced on MD17, with some metrics even underperforming compared to the baselines. We believe this is due to the following reasons:\n\n- Baselines such as Equiformer and HEGNN directly utilize high-degree steerable features, which are closely related to spherical harmonics and can effectively represent complex patterns (such as electron cloud shapes) essential for molecular dynamics tasks. While their architectural advantages may contribute significantly to performance in this setting, it is important to note that computing high-degree steerable features also introduces substantial additional computational overhead.\n- In contrast, our approach does not rely on the high-degree steerable features. Nonetheless, our method achieves superior performance on 5 out of 8 tasks and demonstrates better generalizability. \n\nIn addition, we supplement our work with a challenging experiment on the large-scale Water-3D dataset [C], which involves predicting future coordinates for fluid particles in systems averaging over 8,000 nodes each. Due to time constraints, our evaluation is conducted on a substantial subset (Water-3D-mini: 3,000/300/300 for train/validation/test) with training capped at 1,000 epochs.\n\n**Table S1. Results on Water-3D-mini ($\\times 10^{-4}$).**\n\n||1-layer|2-layer|3-layer|4-layer|\n|-|-|-|-|-|\n|EGNN| 4.904|4.323|3.649|3.338|\n|FastEGNN| 4.885|4.332|3.782|3.259|\n|HEGNN|4.885|4.138|3.519|3.287|\n|EGNN$_\\text{cpl}$-global|4.368|3.711|3.294|3.248|\n|EGNN$_\\text{cpl}$-local|**3.611**|**3.320**|**2.803**|**2.495**|\n\nOur approach (EGNN$_\\text{cpl}$-global/local) consistently outperforms all baselines at every depth, and achieves superior results with fewer layers. These findings further support both the scalability and efficiency of our method of handling large, real-world systems.\n\n***\n\nThank you again for your generous recognition of our work; your encouragement is truly appreciated. We will address these considerations and limitations in the revised manuscript to provide a more balanced perspective on the strengths and applicable scenarios of our method.\n\n> **Reference**\n\n- [A] Han J, Cen J, Wu L, et al. A survey of geometric graph neural networks: Data structures, models and applications. Frontiers of Computer Science 2025.\n- [B] Cen J, Li A, Lin N, et al. Are high-degree representations really unnecessary in equivariant graph neural networks? NeurIPS 2024.\n- [C] Zhang Y, Cen J, Han J, et al. Improving equivariant graph neural networks on large geometric graphs via virtual nodes learning. ICML 2024."}, {"review_id": "2WpkesOD0M", "reviewer": "Reviewer_Qhpc", "summary": "The paper describes an advanced, theoretical grounded method for obtaining equivariant predictions from point clouds. The derivation is such that the authors can guarantee any equivariant function can be parametrized through an canonicalization procedure (complete scalar function) and a corresponding complete steerable basis set. The paper supports such claims via 5 theorems and experiments in which geometric isomorphism tests, in which the model succeeds in all cases, where other models (not using the completeness procedure) fail.", "questions": "See numbered comments/questions above. Thank you.", "limitations": "Are adequately addressed. Checklist is included.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "# Strengths\n\n1.\tApart from some challenges to get into this rather technical paper (see comments), the exposition of the main method (the algorithm) and the derivation of a *complete* equivariant model is clear and the results of Theorem 3.2, 3.5-8 are an important contribution to the field of equivariant deep learning. The derivation seems sound, I only thoroughly went through the proof of 3.2, but seem intuitively correct and I trust the proofs given in the appendix based on my understanding of canonicalization methods and steerable/tensor field methods.\n2.\tThe method is thoroughly validated with controlled experiments that confirm the theory.\n3.\tThe code is provided as supplementary materials. It looks clean and the paper seems as such very reproducible (I didn’t not try running the code though).\n4. The paper is quite extensive in technical details, but could still benefit from some more intuitive clarifications\n\n# Weaknesses\n1.\tAlthough theoretically very strong, and with experiments supporting the theoretical claims, the practical impact is at this stage hard to assess as the tasks other than completeness-type tests still seem a bit artificial. The paper would be more impactful if demonstrates something on atomic point clouds, where equivariant models typically make most impact and where there is a strong need for efficient models.\n2.\tAlthough generally complete in discussion of literature on equivariance (and well written!), it misses in my opinion a clear connection between the scalarization methods and tensor-product based methods. They are unified through the group convolution viewpoint (cf [Vadgama et al 2024]) in which the scalar class can be converted into the other via point-wise SO(3)-Fourier transforms and working with CG products instead of otherwise constraind linear layers. As such, the distinction into two classes is a bit arbitrary and often comes down to an implementation preference. Related is the exposition of the method relative to canonicalization methods (here categorized as frame-based methods) can be made more explicit. It seems that the presented paper is a form of canonicalization which simultaneously expands the predictions in a particular basis. Could the authors clarify how the work should be considered relative to canonicalization methods (e.g. [31])?\n3.\tThe paper has many forward references to equations and algorithms which makes the reading sometimes a bit difficult. A striking example is “Actually, we have presented an algorithm …” which confuses me and I starting to doubt myself, have I missed something, or did the authors forget to put in a citation, or a reference to a section in the paper.\n\n# Minor comments\n1.\tIn 3.2 the authors talk about an “equivariant function”. I understand how an *operator* (or a map, or layer, or network) can be equivariant, namely via eq (1). But how is a function equivariant. E.g. the text states that spherical harmonics provide a compete basis for SO(3) equivariant functions, but I am not sure if I follow. The SHs provide a complete basis for functions over S^2, but not for SO(3), that would require all Wigner-D functions. Or are we talking about a basis for linear operators (like tensor products)? I don’t think so as the next line talks about “SE(3) equivariant functions over the sphere”. But isn’t any function over the sphere (regardless of basis) equiriant via the regular reprsentation phi(n) -> phi(R.n)? Or is this function to be considered vector valued, for example a convolution kernel? (cf Lang and Weiler, 2020 “A Wigner-Eckart Theorem for group Equivariant Convolution Kernels”, https://arxiv.org/abs/2010.10952 ). I am probably misinterpreting the statements here, but perhaps this can be clarified. It would help to make precise in what sense a spherical function is called equivariant. \n2.\tThe claim of “without loss of generality” on line 154 should be further justified, I would recommend doing this in an appendix, together with a response to the following comment:\n3.\tCould you please include a detailed derivation of Example 3.1? I’m worried that something is not correct, namely that TFN is of order \\nu=2 and EGNN of order \\nu=1. Aren’t both of order \\nu=2? TFN can be written in group convolution form, and only involves pairwise interactions (albeit with l>0). Same for Schnet (albeit with l=0). EGNN is not of convolution form due to the position update, but it’s still pairwise interactions. I don’t understand how TFN is different to EGN in this regard. The tensor products used only involve two quantities. Apart from this concern, it would be nice to see where the equation comes form because it seems quite fundamental to the rest of the exposition.\n4.\tIn definition 3.3 I don’t think doublestruck M is defined.\n5.\tWhen forward referencing to the algorithms, please mention where we can fined them (in the appendix)\n\n# References\n\nVadgama et al, 2024 “Probing Equivariance and Symmetry Breaking in Convolutional Networks” https://arxiv.org/abs/2501.01999", "was_revised": true, "final_justification": "My final justification follows my last comment:\n\nI have no further comments/questions. I appreciate the extra work done, and I agree that the reference [A] provides all the context needed regarding the g-conv paradigm. I am sure the paper will be much improved with the mentioned updates, and considering the paper was already of high quality I continue to support for acceptance and as such maintain my score \"accept\".", "rebuttal": "Thank you for your thorough and insightful review. We are grateful for your positive comments on both the theoretical contributions and the literature discussion, as well as for your helpful suggestions to further strengthen the manuscript. \n\n> **R1 (W1). Additional Experiment on Atomic Dataset**\n\nThanks for your suggestion. Since there are few equivariant tasks on atomic point clouds, we have now included results for the large-scale Water-3D dataset [A], which requires predicting future coordinates for fluid particles (average >8K nodes/system). Due to time constraints, we only evaluate a substantial subset (Water-3D-mini: 3,000/300/300 for train/val/test) using up to 1,000 training epochs.\n\n**Table S1. Results on Water-3D-mini ($\\times 10^{-4}$).**\n\n||1-layer|2-layer|3-layer|4-layer|\n|-|-|-|-|-|\n|EGNN| 4.904|4.323|3.649|3.338|\n|FastEGNN| 4.885|4.332|3.782|3.259|\n|HEGNN|4.885|4.138|3.519|3.287|\n|EGNN$_\\text{cpl}$-global|4.368|3.711|3.294|3.248|\n|EGNN$_\\text{cpl}$-local|**3.611**|**3.320**|**2.803**|**2.495**|\n\nOur approach (EGNN$_\\text{cpl}$-global/local) consistently outperforms all baselines at every depth, and achieves superior results with fewer layers. These findings further support both the scalability and efficiency of our method of handling large, real-world systems.\n\n> **R2 (W2 & MC1). Theoretical Discussion**\n\nThank you very much for your thoughtful and insightful comments, which have greatly enriched our understanding of the theoretical landscape. \n\n**Connotation of Model Methodology:** As you astutely observe, there are indeed two main perspectives on unifying scalarization-based and tensor-product-based methodologies. The first approach, which we employ in our work, relies on message passing consisting of message computation and aggregation and it achieves equivariant constraints by leveraging known equivariant variables. The second approach, based on group convolution and decomposition of the convolution kernel (representation matrix), as you have perceptively pointed out, proceeds from the opposite direction—deducing equivariant variables directly from established equivariant constraints. Historically, the second perspective predates the first, and can be traced back to Eqs. (10), (11), and (16) in the 3D Steerable CNN paper [A]. The two additional works you referenced are, in fact, further developments based on [A]. This method offers profound insight into the underlying algebraic structures; however, it is often less accessible to researchers without a strong background in group representation theory. As a consequence, the second approach is most prevalent in research concerning equivariance in physical domains, while the first approach tends to be favored within the literature on equivariant graph neural networks. We will include a dedicated section in the appendix of our revised manuscript to introduce this alternative methodology and will update the main text to include appropriate citations. \n\n**Spherical Harmonics and Bases:** We apologize for the confusion. Our intention was to convey that \"for functions on the sphere that are equivariant under $\\mathrm{SO}(3)$, the spherical harmonics form a basis,\" rather than to imply that \"the spherical harmonics are a basis for $\\mathrm{SO}(3)$\" itself. We will clarify this statement in the revised manuscript to prevent further misunderstanding.\n\n**From Sphere to Euclidean Space:** This can be addressed by incorporating an RBF function, as described in Lines 141–143 of the original manuscript. This design is also inspired by [A] (see after Eq. (16) in Section 4.2). We sincerely appreciate your thoughtful suggestions, which have helped clarify these important distinctions.\n\n**Model Interpretation from Canonicalization:** Very nice comment! As you have pointed out, the essence of Frame-Averaging [31] lies in eliminating the influence of group transformations on the input by constructing a suitable frame, thereby enabling compatibility with any downstream architecture. From this perspective, we agree that the canonical form utilized in our method is conceptually aligned with the principles underlying Frame-Averaging. However, we would like to note that our specific implementation differs from this general framework. While Frame-Averaging [31] proposes a versatile approach that can be applied to a broad range of groups, our method focuses on designing an efficient canonicalization procedure tailored specifically to geometric graphs. This enables us to achieve computational efficiency while maintaining the desired equivariant properties. We greatly appreciate your suggestion, and we will clarify this relationship more explicitly in the revised manuscript to better position our contribution within the broader context. Thank you again for your valuable feedback.\n\n> **R4. Improve Presentation and Correct Typos**\n\nThank you very much for your valuable feedback. Your questions and suggestions are elegant and insightful. We will make the following changes to the manuscript:\n\n- **(W3 & MC5):** We will include cross-references in the revised manuscript to improve readability.\n- **(MC2):** We will provide the corresponding proof in a later section. Here is a brief explanation: we prioritize structural information to determine whether two graphs are geometrically isomorphic. For cases where node/edge features are not uniform, we can further determine this based on matching (please see our reply to MC4).\n- **(MC3):** Sorry, this is a typo. EGNN/HEGNN and TFN both model based on edges, so they are 2-body interactions (number of neighbors $\\nu=1$), while MACE corresponds to $\\nu\\geq 1$.\n- **(MC4):** We will modify the statement of \"Point Cloud Isomorphism\" in Def. 3.3 as follows: \"The two point clouds $\\vec{\\boldsymbol{X}}^{(\\mathcal{G})}$ and $\\vec{\\boldsymbol{X}}^{(\\mathcal{H})}$ are isomorphic, i.e. the matching set $\\mathbb{M}(\\mathcal{G}, \\mathcal{H})=\\\\{(\\sigma, \\mathfrak{g})\\in S\\_N\\times \\mathrm{E}(3)\\mid\\forall i, \\vec{\\boldsymbol{x}}_i^{(\\mathcal{G})}= \\mathfrak{g}\\cdot\\vec{\\boldsymbol{x}}\\_{\\sigma(i)}^{(\\mathcal{H})}\\\\}$ is not empty.\".\n\n***\n\nThank you again for your comprehensive review and high-level comments, which have significantly helped us to improve the clarity, rigor, and impact of our work. We will incorporate all clarifications, additional experiments, and references as discussed in the revised manuscript.\n\n> **Reference**\n\n- [A] Weiler M, Geiger M, Welling M, et al. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. NIPS 2018."}, {"review_id": "6pnRDAoPbF", "reviewer": "Reviewer_q9Lr", "summary": "The paper considers equivariant GNNs for geometric graphs (ones where the nodes / edges also have positional information). They extend existing work towards constructing complete GNNs (ones which can approximate any equivariant function) by reframing GNNs as sums over basis sets with weights that depend on the input graph. They provide two algorithms for constructing such GNNs, one of which is slower but works for all geometric graphs, the other of which requires that four non-coplanar reference nodes can be learned by an equivariant GNN. They evaluate their system against strong baselines across 7 datasets.", "questions": "Questions:\n1. L114 -> what is C_H? It has not been defined.\n2. L155 \"assume uniform node and edge features\" -> Is this a fair assumption? This will affect permutation invariance\n3. L295-L297 \"Strictly speaking, the virtual nodes forming the tetrahedron should also be included in Algo. 4, but we omit this term as it degrades performance in our experiments.\" -> Is the theory still valid without this? Do the authors have an intuition as to why this degraded performance?\n\nMinor comments:\n1. L211 \"geometrically isomorphic if\" -> \"geometrically isomorphic if they fulfil both of the following isomorphisms\"", "limitations": "Limitations are briefly discussed in Appendix C.\nI would suggest including them in the main section of the paper.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper builds well on existing work, and situates itself well within the literature. A great example of this is L249 \"FastEGNN [41], it is assumed that non-coplanar virtual nodes can always be found, but in the discussion later, we will see that this is not the case\"\n2. The paper provides a rigorous theoretical grounding and motivation.\n3. The theories are backed up by strong empirical performance over the baselines.\n4. The appendices are very thorough.\n\nWeaknesses:\n1. The theory in the paper was hard to follow as someone not familiar with the related work. It should ideally be more self-contained. For example, L128-L129 \"as lth-degree steerable features if they are transformable using D(l)\" -> This cannot be understood within the confines of the paper. What does \"transformable using D\" mean in this context, and what is the Wigner-D matrix? Also, L146-L147 \"the basis set that expands the function space of equivariant GNNs.\" -> it is also unclear what this means within the confines of the paper.\n2. In general, the figures are not well used and do not do much to help convey the ideas in the paper. E.g. the image in Figure 2 is not helpful, the math in Figure 1 is not understandable at this point of the paper, and Figure 3 is far too small to draw anything useful from.\n3. The datasets chosen for evaluation are quite limited: there is only one real-world dataset (MD17) and they are not tested on large graphs. The authors claim that the 100-body system dataset \"tests its scalability in large systems\"; however, it is not large in comparison to the other datasets used in \"Improving Equivariant Graph Neural Networks on Large Geometric Graphs via Virtual Nodes Learning\". It is just 100 nodes, compared to their datasets Proteins (800 nodes) and Water-3D (8000 nodes). Thus it is unclear from the experiments how the method scales to larger graphs and real-world scenarios.", "was_revised": true, "final_justification": "I stand by my original score. The authors addressed all of my queries and the the work appears technically solid, but between (1) my lack of understanding of both this paper and the relevant background work, and (2) the apparent lack of concrete practical applicability, I cannot raise my score above a borderline accept or my confidence beyond a 2. I hope that the reviews from the other reviewers who are more familiar with this work will be weighted higher, and overall it seems to me that the paper should probably be accepted.", "rebuttal": "Thank you very much for your positive feedback and thoughtful, detailed comments. We greatly appreciate your careful reading and constructive suggestions, which will help us improve both the clarity and impact of our paper. We summarize and respond to your questions as follows:\n\n> **R1. Improve Presentation and Increase Background Knowledge**\n\nThank you very much for your valuable comments. Your suggestions are both professional and elegant, and we greatly appreciate your insights. We will make the following revisions to the manuscript:\n\n- **(W1):** We will refer to review [A] for additional background information on irreducible representations, Wigner-D matrices, and related concepts, and have incorporated these materials to enhance clarity for the reader.\n- **(W2):** We will enhance the description of Fig. 1 for better understanding, and enlarge Fig. 3 to improve readability.\n- **(Q1):** We will clarify that $C_H$ represents the number of channels in the node feature matrix $\\boldsymbol{H}$.\n- **(MC):** We will adopt your suggested rephrasing, as it is indeed more elegant. Thank you again.\n- **(L1):** We will move the discussion of limitations in Appendix C to the main text.\n\n> **R2 (Q2). Fairness of the Uniform** **Node and Edge Features**\n\nThank you very much for your thoughtful question. The non-uniform case can be readily generalized and addressed within our framework. Our assumption of uniform node and edge features is primarily made to simplify the theoretical analysis, particularly in the context of Def. 3.3 on geometric isomorphism. We believe this assumption is reasonable for the purposes of our current work. Nevertheless, we agree that it is important to explicitly justify and clarify such assumptions:\n\n- In practice, our analysis can be extended to non-uniform node/edge features by: (1) first determining geometric isomorphism based on graph structure, and (2) then ensuring that the mapping between nodes/edges also respects feature matching.\n- We will revise the text to more explicitly state the scope and limitations of this assumption, and include supplementary comments in the appendix on extending results to the non-uniform case.\n\nWe also clarify that the purpose of Fig. 2 is to demonstrate the role of geometric structure (independent of feature information) in distinguishing points and facilitating the construction of valid virtual nodes.\n\n> **R3 (W3). Additional Experiments** **on** **Large-scale Geometric Graphs**\n\nThank you very much for your insightful comment and we decided to add an experiment on large-scale scenario. We acknowledge that our experiments do not cover ultra-large graph datasets (with 1K+ nodes). However, the main emphasis of our work is on establishing a theoretical framework for completeness, and the experimental results are intended primarily to provide supporting evidence for our theoretical contributions. It is worth emphasizing that, in the field of atomistic modeling, systems with more than one hundred nodes are widely recognized in literature as sufficiently challenging benchmarks. We believe that our results on such systems already provide a meaningful demonstration of our model's practical capabilities.\n\nTo further evaluate the performance of our model on larger datasets, we have conducted additional experiments on the more representative Water-3D (average >8K nodes/system) dataset in [B] as you suggested. Due to time limitations, we restrict the evaluation to a 1/5 subset called Water-3D-mini with 3,000/300/300 samples for train/valid/test from the original dataset (15,000/1,500/1,500), with a maximum training epoch of 1,000.\n\n**Table S1. Results on Water-3D-mini ($\\times 10^{-4}$).**\n\n||1-layer|2-layer|3-layer|4-layer|\n|-|-|-|-|-|\n|EGNN| 4.904|4.323|3.649|3.338|\n|FastEGNN| 4.885|4.332|3.782|3.259|\n|HEGNN|4.885|4.138|3.519|3.287|\n|EGNN$_\\text{cpl}$-global|4.368|3.711|3.294|3.248|\n|EGNN$_\\text{cpl}$-local|**3.611**|**3.320**|**2.803**|**2.495**|\n\nOur approach (EGNN$_\\text{cpl}$-global/global) consistently outperforms the baselines at every depth, and achieves superior results with fewer layers. We believe this further demonstrates the efficiency and scalability of our approach, highlighting its potential for large-scale and real-world applications.\n\n> **R4 (Q3). Discussion on Algo. 4 and Models**\n\nThank you for raising this interesting and important question. The internal structure of the virtual node corresponds to $\\vec{\\boldsymbol{V}}^\\top\\vec{\\boldsymbol{V}}$ in Algo. 4. According to our theoretical framework, this feature should indeed be included. However, we chose not to incorporate it mainly due to observed performance degradation in our empirical studies. We believe there are two possible explanations for this: (1) The virtual nodes in our model are learnable, and their parameters—i.e., the virtual node feature—involve in message passing, potentially providing sufficient informational capacity already; (2) In our experiments, we found that introducing this feature tended to reduce training stability. Notably, similar issues regarding training instability with inner product-like vector features have been reported in literature like [B].\n\n***\n\nWe sincerely appreciate your valuable suggestions, and we will include these clarifications and additional results in the revised manuscript.\n\n> **Reference**\n\n- [A] Han J, Cen J, Wu L, et al. A survey of geometric graph neural networks: Data structures, models and applications. Frontiers of Computer Science 2025.\n- [B] Zhang Y, Cen J, Han J, et al. Improving equivariant graph neural networks on large geometric graphs via virtual nodes learning. ICML 2024.\n- [C] Huang W, Han J, Rong Y, et al. Equivariant graph mechanics networks with constraints. ICLR 2022."}]

Author rebuttal:

Meta-review: This paper introduces a theoretically grounded and efficient framework for constructing complete equivariant GNNs (especially relevant for geometric graphs and point clouds). Completeness is achieved not through deep stacks or high-order features, but via two finite ingredients: a complete scalar function (graph canonical form) and a full-rank steerable basis. The authors provide two algorithms, one based on geometric isomorphism, and one faster assuming learnable non-coplanar reference nodes. These are instantiated on EGNN and TFN. The method achieves universality, good accuracy with fewer layers, and reduced computational overhead. Experiments complement the theoretical treatment. Generally the reviewers have liked the paper and have made suggestions regarding clarity and experiments. I mostly concur with the positive assessment of the reviewers. The authors are advised to include most of the points in their rebuttal into the main paper.
Decision: Accept (poster)