On Explaining Equivariant Graph Networks via Improved Relevance Propagation
Authors: Hongyi Ling
Haiyang Yu
Zhimeng Jiang
Na Zou
Shuiwang Ji
Venue: ICML 2025 poster (2025)
Keywords: equivariant GNNs
XAI
Primary area: social_aspects->accountability_transparency_and_interpretability

Abstract: We consider explainability in equivariant graph neural networks for 3D geometric graphs. While many XAI methods have been developed for analyzing graph neural networks, they predominantly target 2D graph structures. The complex nature of 3D data and the sophisticated architectures of equivariant GNNs present unique challenges. Current XAI techniques either struggle to adapt to equivariant GNNs or fail to effectively handle positional data and evaluate the significance of geometric features adequately. 
  To address these challenges, we introduce a novel method, known as EquiGX, which uses the Deep Taylor decomposition framework to extend the layer-wise relevance propagation rules tailored for spherical equivariant GNNs. Our approach decomposes prediction scores and back-propagates the relevance scores through each layer to the input space. Our decomposition rules provide a detailed explanation of each layer’s contribution to the network’s predictions, thereby enhancing our understanding of how geometric and positional data influence the model’s outputs. 
  Through experiments on both synthetic and real-world datasets, our method demonstrates its capability to identify critical geometric structures and outperform alternative baselines. These results indicate that our method provides significantly enhanced explanations for equivariant GNNs. Our code has been released as part of the AIRS library (https://github.com/divelab/AIRS/).

On Explaining Equivariant Graph Networks via Improved Relevance Propagation
Hongyi Ling' Haiyang Yu
Zhimeng Jiang'
Na Zou
Shuiwang Ji
Abstract
1. Introduction
Equivariant graph neural networks have shown significant promise in addressing complex problems across quantum physics, molecular science, materials science, and protein research (Thomas et al., 2018; Fuchs et al., 2020; Liao & Smidt, 2022; Liao et al., 2023; Batzner et al., 2022; Passaro & Zitnick, 2023; Zhang et al., 2023; Yu et al., 2023; Du et al., 2024; Wang et al., 2023; 2022). Despite their potential, a critical challenge in assessing the scientific plausibility of these models' outcomes is their interpretability. Most equiv- ariant GNNs are treated as black boxes, which undermines their reliability and limits their applicability in scientific domains. Therefore, developing explainable artificial intel- ligence (XAI) techniques tailored for equivariant GNNs highly desirable. These techniques can provide insights into how equivariant GNNs make predictions, thereby increasing the trustworthiness of their outcomes. Moreover, XAI tech- niques not only diagnose and improve existing models but also facilitate further scientific knowledge discovery.
We consider explainability in equivariant graph neural networks for 3D geometric graphs. While many XAI methods have been developed for ana- lyzing graph neural networks, they predominantly target 2D graph structures. The complex nature of 3D data and the sophisticated architectures of equivariant GNNs present unique challenges. Cur- rent XAI techniques either struggle adapt to equivariant GNNs or fail to effectively handle positional data and evaluate the significance of geometric features adequately. To address these challenges. we introduce novel method, known as EquiGX, which uses the Deep Taylor decom- position framework to extend the layer-wise rel- evance propagation rules tailored for spherical equivariant GNNs. Our approach decomposes prediction scores and back- sockpproaggates the rel- evance scores through each layer to the input space. Our decomposition rules provide a de- tailed explanation of each layer's contribution to the network's 's predictions, thereby enhancing our understanding of how geometric and positional data influence the model's outputs. Through experiments on both synthetic and real-worke datasets, our method demonstrates its capabil- ity to identify critical geometric structures and outperform alternative baselines. These results indicate that our method provides significantly en- hanced explanations for equivariant GNNs. Our code has been released as part of the AIRS library httpp://gghuu..co//vvveaaa/////
While many XAI methods have been proposed to study GNNs, they primarily focus on 2D graphs (Yuan et al., 2023; 2020; Zheng al., 2023; Chen et al., 2024; Wang et al., 2021). The high dimensionality of 3D geometric data and the complexity of equivariant GNN models pose unique challenges and opportunities in this domain. Current XAI techniques either struggle to adapt to equivariant GNNs or fail to effectively handle positional data and evaluate the significance of geometric features adequately. Specifically, many XAI methods (Huang et al., 2022; Zhang et al., 2021; Vu Thai, 2020) overlook the complex behavior of equiv- ariant models, thus requiring additional effort before they can be applied to equivariant GNNs. On the other hand, some XAI methods, known for their simplicity and adapt- ability, such as SA (Baldassarre & Azizpour, 2019), are insufficient to provide a comprehensive explanation for the importance of geometric features.
'Department of Computer Science and Engineering, Texas A&M University, Texas, USA 'Department of Industrial Engi- necring, University of Houston, Texas, USA. Correspondence to: Shuiwang Ji <sji tamu.
To bridge this gap. we introduce a novel XAI method, known as EquiGX, which captures the importance of input compo- nents by decomposing the model predictions. The primary challenge in decomposing the predictions of spherical equiv- ariant GNNs lies in attributing the tensor product-based message- saasagg-passiggggg operations that are central to these net-
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
works. Our approach uses the Deep Taylor decomposition framework to extend layer-wise relevance propagation rules specifically for spherical equivariant GNNs. By explicitly considering the tensor product (TP) operations, we derive new relevance propagation rules based on Taylor decompo- sition. These rules enable us to baok-proe relevance scores layer by layer until the input space, providing a de- tailed explanation of each layer's contribution to the net- work's predictions. Consequently, EquiGX can enhance our understanding of how geometric and positional data influence the model's outputs.
order-l, SE(3) equivariant hidden features of node i as H [R22++1 These features are used in tensor product operation to compute an equivariant message from node i to node j. denoted as My--i. and the aggregated message Mj = EjeN()) M is used to update the equivariant hidden features. M,-1 consists of many features with mul- tiple rotation orders as Mr Oimo" MM where t is direct sum. For an order-fs message M it can be com- puted by using the order-&2 spherical harmonics function as
Mf Entusss)ddd) (Ta)) 2 uf.. f1.t2
(I)
2. Background and Related Work
Here, F(-) is a learnable function usually implemented by multi-layer perceptron (MLP) model, dij = ||C C2ll2 and Fiy- - C.-C are the distance and direction between nodes and j. respectively. (²²()= : Rs - R is the spherical harmonics function, which maps an input 3D vector to a (262 + 1)-dimensional vector representing the coefficients of Erder-E2 spherical harmonics bases. @ is the tensor product operation, which takes a order-éi equivari- ant feature and order-E2 equivariant feature v as input, yielding order-Es equivariant feature as
We denote geometric graph with n nodes as g (X, C). Here, X [x,,-.., Xx]" is the node feature matrix, where each X, E R is the d-dimensional feature vector of node i. C = Rax3 is the node coordinate matrix, where C, is the coordinate of i-th node. Nodes are generally connected by edges using a predetermined radial cutoff dis- tance c E R+ so that the adjacency matrix A E {0, 1]mxn is defined as A, = 1 if and only if ||C, - C,ll, S We use (() denote the set of neighboring nodes of node i.
Raxd
d.j
2.1. Equivariant Graph Networks
(u &
E m Emn= Em (fs (fa.) Mas maat
Equivariant graph neural networks are critical in the do- main of AI for science, particularly for modeling geomet- ric graphs derived from ftree-dimensione atomic systems. These networks are specifically designed to capitalize the physical symmetries and integrate these symmetries into the model architecture to ensure that the learned hidden repre- sentations are equivariant to any symmetry transformations applied the input. Specifically, if the input geometric graph is transformed under any operation in S F33), which stands for the special Euclidean group in 3D space, the corresponding hidden representations at each layer are trans- formed correspondingly. Formally, a function f R2x3 + R24+1 mapping between 3D coordinates to a (26 + 1)- dimensional vector is SE(3) equivariant, if for any input coordinates C, we have (RCT +t) (()C), where f. E RR3 is a translation vector, R is rotation matrix satisfy- ing R I and R = 1, and D pRR) 122++))x(2++ represents the Wigner-D matrix of R (Gilmore, 2008). Here, function f is invariant to translation, exemplifying a specific type of translation equivariance.
(CG) coefficients (Griffiths & Schroeter. 2018) and m denotes the m-th element in the equivariant feature. See more discussions about equivariant graph neural networks in Section 5.
where C Clebsch-
2.2. Explainability in Graph Neural Networks
Explainability in neural networks is vital for validating the trustworthiness and reliability of their predictions, especially when applying these models to scientific domains. Current XAI methods predominantly focus on GNNs designed for 2D graphs. These approaches can be mainly categorized into four classes, namely, gradients/featureebbn based methods, perturbation- based methods, decomposition methods, and surrogate methods. Gradients/Feature baaedd methods, such as SA (Baldassarre & Azizpour, 2019) and CAM (Pope et al., 2019), use gradient values to assess the importance of input components. Their popularity stems from their simplicity and direct approach. ods (Ying et al., 2019; Yuan et al., 2021; Luo et al., 2020) evaluate changes in predictions by perturbing different input features to identify the most impactful ones. Surrogate- based methods (Huang et al., 2022; Zhang et al., 2021; Vu & Thai, 2020) involve fitting a simpler, interpretable model, such as a decision tree, to mimic the behavior of the original model. The surrogate model's explanations are then used to understand the original predictions. De- composition methods (Schnake et al., 2021; Xiong et al.,
cntturration-baaee meth-
Among the various types of equivariant GNNs (Jing et al., 2020; Schütt et al., 2021: Satorras et al., 2021), spheri- cal equivariant GNNs (Thomas et al., 2018; Fuchs et al., 2020; Liao & Smidt, 2022) are particularly prominent. In these approaches, spherical harmonics functions are used to first encode 3D geometric information into higher di- mensional S3) equivariant features. We denote the
2

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
2023; Feng et al., 2023) decompose prediction scores and pack-propagate them layer-by-layer to the input space to compute importance scores and provide deeper insights into each network layer. Despite significant advances in XAI for 2D GNNs, these methods primarily focus on evaluat- ing the importance of edges, nodes, and subgraphs, strug- gling to incorporate positional information effectively and fully evaluate the importance of geometric features. Conse- quently, the application of these techniques to 3D geometric graphs, especially within equivariant graph neural networks, poses significant challenges. Recently, Miao et al. (2023) introduces a learnable interpreter model that applies ran- dom noise to each 3D point to generate importance scores. However, this method relies solely on input-output behavior without requiring access to the model's 's internal parameters or gradients and overlooks the equivariance of the model. It also requires training the interpreter alongside the prediction model. To sum up, the challenge of explaining equivariant neural networks highlights a significant gap in the current landscape of XAI, underscoring the need for innovative ap- proaches that consider the complex behaviors of equivariant neural networks.
where O is Big-O notation, and x; and & is the i-th dimen- sion of x and &, respectively. Assuming f is locally linear function and carefully selecting & such that higher-order and zero-order terms are negligible, the first-order terms can provide the relevance scores for the input elements as R (xx) = 01 (a; - .:). Deep neural networks are inherently complex and non-linear, making it impractical to apply a straightforward Taylor decomposition across all layers. On the other hand, Deep neural networks, composed of multiple layers, necessitate decomposing the network into a series of simpler subfunctions, each representing a single layer. This approach, known as Deep Taylor De- composition, allows for applying different relevance score computation rules tailored to specific types of layers. For instance, when considering linear layer with Relu activation functions, distinct rules, such as LRP-y (Montavon et al., 2019). L (Bach et al., 2015) can be used due to choos- ing different root points and approximation methods. By using these specifically designed local propagation rules for every layer, the initial relevance value, i.e. the prediction of the model, successively distributed layer-by-layer to the input space. The decomposition characteristic of LRP gives rise to the conservation property, which ensures that the sum of relevance scores across neurons in two adjacent layers remains constant. Let H and H'' be the representations of two adjacent layers, the conservation property can be for- mally described as Σ, R(H) = E, RHH), where (HH) and R(H')) are the relevance scores of H and H', respec- tively. We use the Deep Taylor decomposition to study the complex behavior of equivariant GNNs and provide detailed relevance propagation rules for each layer in the following subsections.
8z. x
3. Methodology
Previous XAI methods on 2D graphs encounter limitations when adapting them on geometric graphs, particularly in effectively incorporating positional information and evalu- ating geometric features, To address these challenges, we introduce novel method, EquiGX, which recursively de- composes network predictions back the input elements. Our approach use the Deep Taylor decomposition frame- work (Montavon et al., 2017), adapted to extend the layer- wise relevance propagation rules specifically for TP based message passing process. This adaptation allows for a de- tailed explanation of each layer's contribution to the net- work's predictions, thus enhancing our understanding of how geometric and positional data influence the model's outputs.
3.2. Attributing the TP-based Message Passing
As mentioned in Section 1, the key of spherical equivariant GNNs is the TP based message passing process. Equivari- ant messages My-ss are computed from node ý to node i using TP, and then aggreggated to form the message M₁. The aggregation operation M,- = Σ EjEN(() M,-; inherently provides a decomposition. Specifically, we assign a rele- vance score R(M)-+)) to each message proportional to its contribution to the aggregated message. Since messages of different orders are summed separately, each order is also considered individually when backpropagating the relevance score. Formally, this process can be described as
3.1. Layer-wise Relevance Propagation
The objective of Layer-wise Relevance Propagation (LRP) is to attribute a relevance score to each input element based on its contribution to the predicted class. This scoring offers insights into how individual input elements contribute to the model's final decision. One way to compute such relevance is to the whole neural network as a mathematical function and use the first-order term from the Taylor series expansion. Consider a function f: : X + y that maps an input to its output label. The Taylor decomposition of at a root point & E Rd is given by
M R(M = EjENTO) M
R(M
For the equivariant message shown in Eq. 1, we can apply a Taylor series expansion to derive a decomposition rule. Specifically, the first order Taylor series expansion of an
f(x) = f(x) + Σ 0f
(z. +0(x -x1), (2)
dr, x=
3

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
order-fy message Mf root point H is given by
we have the relevance propagation rules
aM JーH
Mf M j-H
(H)) fif.).
R(M
T
R(II.)) Σ
My
f2
3
where aM 1 T Each element of this matrix is defined as
E R ( +1) is a Jacobian matrix.
AM
H1)
au1
R(M
R (p(h)sst))(d)))
2 eM
(OMf.
pltrdeees) (di))
OM
E2
M22
22
F( ((f..2. (dis)
clla.ma) ((a.m)))(emm))))
y (fa).
R(MS2
R (F))) = Σ
2 My
The bilinearity of the tensor product indicates that is linear with respect to each input. This property implies that the
3
Eg
aM -- is independent of the choice of root a
OM (
Jacobian matrix
point f Additionally, the absence of quadratic higher- degree terms in the Taylor expansion suggests that when a root point is chosen such that the zero-order term equals to zero, the Taylor expansion serves as a decomposition of the message. Given that H contributes to messages of various nodes and different orders, it necessary to aggregate these contributions. Formally, this relevance propagation rule can be described as
(4)
Since one edge distance di, contributes to multiple TP paths, we sum up relevance scores to get the contribution of edge's distance as R(dk)) = Ee, f2.dz R(F(f1 (2,,a)) (dx))) Similarly, the direction of each edge is encoded into mul- tiple orders of equivariant features using spherical har- monics functions, thus we sum up relevance scores to at- tribute the contribution of an edge's direction as R(F:;) Σn, R( (Ftg)).
R(H - Σ (R(M) Mffe))) )
Note that the relevance propagation rule discussed here is to (3) attribute a single TP-based message passing layer. To apply relevance propagation across the entire network recursively, only the relevance score of hidden feature R(H) continues to backpropagate towards the input. In contrast, R(d.y) and (F.r)) do not continue to backpropagate beyond their respective layer. These scores indicate the contributions of the edge distance and edge direction, respectively, to the final prediction within that specific message passing layer. Thus, the relevance scores (cly) and R(rr)) at each message passing layer are summed derive the cumulative relevance score for edge distances and directions.
@M n1 au
where ② is Hadamard division.
However, this decomposition overlooks the contribution of relative positional information between node and node j. As shown in Eq. 1, spherical equivariant GNNs split the relative position vector of node and node j into a distance part dy and directional part riy. The directional part Fij is encoded into an S3) equivariant feature vector using spherical harmonics functions, which then serves as one input to the tensor product. The distance part dy is encoded into embeddings via radial basis functions (RBF), which in turn are used to determine the weight of each tensor product path (e1, f2 + Ez). Thus, an alternative and highly desirable solution is to decompose the relevance score of each message My to all three components, namely the hidden features, directional part, and distance part. Notably, the message is a trilinear function, meaning it remains linear with respect to one component when the others are held constant. Following Achtibat et al. (2024), is reasonable to assign equal relevance values to each component. Formally,
3.3. Attributing the Linear Operation
The tensor product provides a mechanism for interactions be- tween equivariant features of different orders, while the lin- ear layer is designed to mix equivariant features of the same order. Specifically, this layer linearly combines each group of order-E equivariant features to produce new features, with each group having its own set of learnable parameters. Con- sider the input to the linear layer as p order-&, features of node i, denoted by 11,1 E (2-((rr-1), The output of the linear layer is q order-é, features of node i, represented as H E (26+11, Formally, the transformation in the

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
linear layer can be described
where À R is a normalization factor defined as
W = H
(5)
== Ees Ee, (R
where wf IRxx are the learnable parameters used for mixing order-f1 features. Since each new feature is weighted sum of the input features, we follow the funda- mental LRP-c (Bach et al., 2015) to derive the propagation rule for this linear layer. Let (H,"),, and (H")) mzz de- note the m1-th and my-th order-E1 features of node for the input and output, respectively, and let whi m, denote the element at the m2-th row and m1-th column of wh, The propagation rule for the linear layer is defined as
To compute the final importance score for each imput node, we aggregate the relevance score of the node itself along with the scores of its connected edges. First, we sum the directional and distance-based relevance across all layers to capture the cumulative contributions of edge directions and edge distances. The final node importance score is then calculated as the sum of the node's own relevance and half the relevance of each neighboring edge.
R (1I, mm))
Σ
(
mmmam my (H mm)
4. Experiments
(1HT mg + + R (IFT mms)
In this section, we evaluate the proposed method on both synthetic and real-world datasets. For each dataset, we first train a TFN and then use baselines and our method to gen- erate the explanations. Experimental results show that our method outperforms many baselines on both visualization results and quantitative studies. See more implementation details in Appendix A.
where e is stabilizing factor with a small value, and I E IR22++1 is all-ones vector, which broadcasts into vector. is worth noting that while the above relevance prop- agation rule is specifically for order-L, features of node i, in practice, the input contains groups of equivariant features of various orders across all nodes. Thus, the propagation rule is applied separately for every node and rotation order to compute the relevance score for all input features.
4.1. Datasets and Experimental Settings
Synthetic Datasets. We create two kinds of geometric graph classification datasets, namely Shapes and Spiral Noise. For the Shapes dataset, we begin by randomly select- ing a 3D motif shape from two options, including a cube or an icosahedron, the latter being a polyhedron with 20 triangular faces. Subsequently, we choose a 3D base shape, either a pyramid or star. A random translation and rotation are performed on the base shape. The classification task is to predict whether the motif shape in the geometric graph is a cube or not. In the Spiral Noise dataset, we randomly select a 3D motif shape, either a tetrahedron, a polyhedron with four triangular faces, or a triangular prism. We then introduce a variable number of noise points to create a spiral pattern in 3D space. The classification task is determine whether the motif shape is tetrahedron not.
3.4. Attributing the Non-linear Functions
In this work, we assume that norm-based non-linear func- tion is used in the model architecture, such as TFN (Thomas et al., 2018) and SE(3)-Tranffommer (Fuchs et al., 2020). The norm-based non-linearity acts scalar transformation on each equivariant feature based on its norm. Specifically, for an order-l1 equivariant feature of node i, denoted as H E (2(2(+1), a scalar value is computed using an activa- tion function like the sigmoid function. The output of this norm-basee based non-linearity denoted as H' E R22(+1), is computed by multiplying the input equivariant feature by the scalar output of the activation function. Formally, this process can be described as
H or ((H
(6)
Real-world Datasets. In addition to synthetic datasets containing perfect 3D geometric shapes, we evaluate our method on three real-world datasets, including the Structural Classification of Proteins (SCOP), BioLiP, and Actstrack. The SCOP database (Murzin et al., 1995; Andreeva et al., 2007; Chandonia et al.. 2019) is a predominantly manually curated classification of protein structural domains, orga- nized based on similarities in their structures and amino acid sequences. While using the same training and validation datasets as Hou et al. (2018); Hermosilla et al. (2020), our focus is on the fold classification task, which is to predict the broad types of protein tertiary structure topologies. Hence, we only use the Fold test set. There are seven categories in
where (-) is the sigmoid function. Since each equivariant feature is transformed by a scalar, reversing the transforma- tion results in way to attribute relevance values. However, directly reversing the scalar transformation does not pre- serve the sum of relevance scores between input and output, thereby breaking the conservation property. To address this, we normalize the relevance scores to ensure the conserva- tion property is maintained. The relevance propagation rule for the norm-based non-linear function is given by
R(1(1) = R(H - H(1
5

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
54
0A
6.
0.4
62
92
52
62
Ground Truth
EquiGX
Ground Truth
EquiGX
0.6
466
6.6
0.4
Grad-CAM
LRI-Bem
Grad-CAM
LRI-Bern
1.0
2.
08
666
8
0
IRI--aussian
GNN-Explainer
IRI--aussian
GNN-Explainer
Figure I. Explanation results on the Shapes dataset with a cube motif shape. The red color indicates a high importance score, while the blue color indicates low importance score. Idcally, the nodes of the cube should be red, indicating their high significance, while the other areas should be blue, indicating lower significance. The ground truth is shown in the upper-left corner, and the nodes forming the cube motif are highlighted in red. Better alignment with the ground truth reflects more accurate explanation. Our EquiGX aligns best with the ground truth.
Figure 2. Explanation results on the Spiral Noise dataset with a tetrahedron motif shape. The red color indicates high importance score, while the blue color indicates low importance score. Idc- ally, nodes of the tetrahedron should be red, indicating their high significance, while the other areas should be blue, indicating lower significance. The ground ner, and the nodes forming the tetrahedron motif are highlighted in red. Better alignment with the ground truth reflects more accurate explanation. Our EquiGX aligns best with the ground truth.
is shown the upper-left cor-
total, such as all-alpha and all-beta proteins. Protein labels, provided by human experts, are based on the secondary structure, which reflects the local spatial conformation of proteins. Specifically, labeling for all-alpha and all-beta proteins is determined by the presence of a-helices and B- sheets within their structures, respectively. BioLiP (Yang al., 2012; Zhang al., 2024) is semi-mas cemimmluullly curated database dedicated to high-quality ligand-protein binding interactions. The 3D structural data primarily sourced from the Protein Data Bank are complemented with biological information, such as binding affinity scores, from literature and other databases. The task predict whether there
is tight binding between pottein-ligand pair. Like pre- vious methods (Somnath et al., 2021; Öztürk et al., 2018; Townshend et al., 2020), we do not differentiate between the inhibition constant (K;) and dissociation constant (Ka). instead predicting whether a pooeiin-ligadd pair of affin- ity of KJ/K; < 1 nM. ActsTrack (Miao al., 2023) is particle tracking simulation dataset in high-energy physics. The task is to predict whether collision event contains z + μn decay based on a point cloud of detector hits. Each point in the point cloud corresponds to a particle interaction with the detector. Positive samples include hits from both the - pu decay and background interactions, thus the
6

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
min
max
EquiGX
Grad
EquiGX
GNN-Explainer
Figure 4. Explanation results on the BioLip dataset. The ligand is highlighted with green border.
inject Gaussian noise to evaluate the significance of point positions; (6) PG-Explainer (Luo et al., 2020), which gen- erate explanations by learning parameterized masks that highlight the most relevant subgraphs. For methods that assign importance scores to edges, we distribute the score to the connecting nodes to evaluate node-level explanations.
min
Grad-CAM
LRI-Bern
4.2. Qualitative Evaluation
In this section, we present the visualization of explanations for methods and other baselines across all four datasets. Since the importance scores of different methods vary in range, we normalize each method to have the same score range to enable fair comparison. The explanation results for the Shapes dataset are visualized in Figure 1. In this dataset, the cube shape is the motif shape, so the nodes forming the cube are used the ground truth for expla- nations. Therefore, the cube nodes should be marked as important, while the other nodes should not be. As shown in Figure 1, LRI incorrectly marks some nodes of the base shape as important. In contrast, our method provides better visual explanations, accurately identifying the cube nodes as the important ones. For the Spiral Noise dataset, the tetrahedron shape is the motif shape, so the nodes forming the tetrahedron are used as the ground truth for explanations. Consequently, the tetrahedron nodes should be highlighted important, while the other nodes should not be. As seen in Figure 2, GNN-Explaing struggles to identify the four important nodes forming the tetrahedron. In contrast, our method successfully recognizes the tetrahedron. We also show the explanation results the SCOP dataset in Figure 3. As mentioned in section 4.1, protein fold classes are labeled by human experts based on the secondary structures of pro- teins. We investigate whether the explanations provided by different methods can accurately reflect the secondary structures of proteins. An all-beta protein is shown in Fig- ure 3. Ideally, the 8-sheets should have a high importance score (i.e., be red in the figure), while the remaining parts should have low importance score (i.e., be blue in the fig-
min GNN- GNN-Explaiier
LRI--aussian
Figure 3. Explanation results on the SCOP dataset of all-beta pro- teins. Since the sample is an all-beta protein, idcally the f-sheets should have high importance scores, i.c. be red in the figure. Beta sheets typically appear flat, arrow-shaped ribbons pointing in specific direction, often aligned side-by-side to form sheet-like structures.
particle hits left by the muons (uu) are labeled the ground truth for model explanations.
Baselines. We compare our method with the following baseline methods, including (1) Grad (Baldassarre & Az- izpour, 2019), which uses the norm the gradient of the predictions with respect to the 3D coordinates to evaluate node importance: (2) Grad-CAM (Pope et al., 2019), gradient-based method combining with activations from hid- den node representations; (3) GNN- ENN-Explainer (Ying al., 2019), perturbation-baser method identifying important edges through optimization of soft masks; (4) LRI-Bern (Miao et al., 2023), which learns a model to inject Bermoulli noise to evaluate the significance of point existence; (5) ARI-Gaussian (Miao al., 2023), which learns a model to

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
Table Comparisons between our method and haselines. The hest results shown in bold.
Shapes
Spiral Noise
SCOP
ActsTrack
Dataset
1
APT
AUROC f
AP
AUROC 1
AP
AUROCT
AP
Random
50
65.70
50
49.01
50
53.67
50
20.9
Grad
68.44 + 12.44
83.81 = 6.40
49.94 0.13
49.16 + 0.09
56.45 4.93 59.75 3.68
55.84 + 0.05 31.87 + 0.54
Grad-CAM
64.77 + 8.84
78.95 = 4.82
66.93: 6.89
71.88 + 6.45
59.57 + 4.38 61.26 + 1.99
62.11 + 1.93 44.95 + 1,40
GNN-Explainer
80.85 5.38 89.97 1 2.27
79.69 1 2.30 82.37 1 1.90
77.26 0.19 72.37 0.26
65.18 1 0.59 35.54 1 0.76
LRI-Bemn
67.84 17.32
83.25 = 9.04
79.06 = 5.69
81.85 = 4.85
56.09
5.12
62.63 1.43 39.39 0.88
LRI-Gaussian
68.46 + 10.71 81.65
7.24
58.75 + 10.96
63.89 8.53
65.99 + 5.05 64.35
57.54
32.43 1.02
PG-Explainer
82.83 + 11.7 90.86
5.66
69.09 = 1.71
74.53 + 1.58
76.92 = 0.23 72.63 0.13
52.16
4.24 29.43 2.91
EquiGX
84.31 1 8.89 91.00 5.32
83.57 1: 10.07 86.82 1 8.30
81.51 + 4.61 82.69 1 3.49
76.96 + 1.69 60.47 + 1.71
ure). While baseline methods either fail to identify 3-sheets or incorrectly assign high importance to most parts of the protein, our method accurately distinguishes B-sheets from other parts, including a-helix. For the BioLip dataset, we present the explanation results in Figure 4. Since binding affinity does not have a definitive answer, there is no ground truth for explanations. It is known that binding is closely re- lated to the protein pocket and especially the ligand itself. In
Fidelity
2A-
23
12-
the example, both our method and GNN-Explainer indicate that the model relies on the ligand to make predictions. To further evaluate explanation methods on the BioLip dataset, we conduct experiments using Fidelity and Sparsity scores in Section 4.3.
0.00
Sparsity
ours
CAM
LRI-G
Grad
LRI-B
GNN Explainer
4.3. Quantitative Evaluation
In two synthetic datasets, the relationships between geomet- ric graphs and labels are explicitly defined. This allows us to evaluate the explanations of baseline methods and our approach by comparing them with the ground truth. Specif- ically, in the Shapes dataset, the explanation ground truth for class is the nodes that form a cube, and for class 1, the nodes that form icosahedron. Similarly, in the Spiral Noise dataset, the explanation ground truth for class O is the nodes that form tetrahedron, while for class 1, it is the nodes that form triangular prism. For both synthetic datasets, we use AUROC and average precision as evalua- tion metrics. As shown in Table 1, our proposed method outperforms the baselines in terms of both AUROC and average precision. In the SCOP dataset, the classification of proteins is determined based on the secondary structures of proteins. In this paper, we explain two classes, including all- alpha and all-beta proteins. Since the reason for labeling for all-alpha and all-beta proteins is the presence of o--helices and 3-sheets within their structures, respectively, we use the atoms that form a-helices and 3-sheets as the explanation ground truth. We also use AUROC and average precision as evaluation metrics. As shown in Table 1, our proposed method has better explanations than the baselines in terms of both AUROC and average precision.
Figure 5. The quantitative studies for different explanation meth- ods on the BioLip dataset.
For the BioLip dataset, like many other scientific properties, the rationale behind the binding affinity scores remains topic of research itself, with no definitive answers available. Therefore, we use Fidelity and Sparsity metrics to evalu- ate the explanations (Pope et al., 2019; Yuan et al., 2021). The Fidelity metric assesses whether the explanations are faithfully important for the predictions by removing the identified important parts from the input geometric graphs and comparing the prediction differences. The Sparsity metric quantifies the proportion of important structures iden- tified by the explanation methods. Note that higher Sparsity scores, which indicate that smaller structures are identified as important, can influence Fidelity scores. This because smaller structures tend to be less crucial. The detailed defi- nitions of Fidelity and Sparsity scores is shown as follows. Given an input geometric graph G. XAI methods compute an importance score for each node. Based on these scores, we compute a hard node mask that contains only binary val- ues. Using this mask, we can generate a masked graph g', where important nodes are masked out. Let f denote a well- trained equivariant GNN. The Fidelity score computed
8

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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
as
the previous three types of networks explicitly encode the invariant or equivariant symmetry within their networks, the networks in unconstrained GNNs (Hu et al., 2021) are not necessarily rotational invariant or equivariant for efficient training and inference. Furthermore, FAENet (Duval et al., 2023b) makes use of frame averaging techniques to make sure the overall framework maintains rotational invariant and equivariant.
Fidelity f()) f(g")y-
(8)
where f()), and f(g'), means the predicted probability of class y of graph 9 and g", respectively. Intuitively. Fi- delity measures the change in predictions when important input elements are removed. In addition, we use Sparsity to measure the fraction of important nodes in the explanations as
9 g
Sparsity
(9)
6. Conclusions and Future Work
where |6'| and [g'] denote the number of nodes in g' and g', respectively. The final Fidelity and Sparsity scores are averaged over the test dataset. Note that good explanations should exhibit high Sparsity along with high Fidelity.
In this work, we propose a method, known as EquiGX, to explain equivariant graph neural networks for geometric graphs. Our method recursively decomposes network pre- dictions back to the input elements. We adapt the Deep Taylor decomposition framework to TP based message pass- ing process, leading to specifically designed layer-wise rele- vance propagation rules. The relevance score generated by EquiGX provides deeper insights into how equivariant fea- tures with different rotation orders contribute to final predic- tions, making EquiGX a transparent solution for equivariant GNNs. Experimental results demonstrate the capability of EquiGX identify critical geometric structures and provide significantly enhanced explanations for equivariant GNNs. Our proposed EquiGX has the potential to generalize to other types of equivariant models. For invariant models such as SchNet, SphereNet, and ComENee, the core mes- sage passing mechanisms can be viewed as special cases of tensor products. Therefore, these models can be reim- plemented based on tensor products. Furthermore, other operations, such as MLPs and aggregation functions, have well-defined LRP rules. By combining these with EquiGX, we can provide explanations for invariant models as well. For iiriizaation-basel models like EGNN, which learn hid- den node features up to rotation order one, their operations can also be interpreted as a special case of tensor products with rotation order up to one. Hence, EquiGX can generate relevance scores for these models as well. For spherical models such as EquiformerV2. MACE, and PACE, where the tensor product operations are central, our EquiGX can be adapted by combining with existing LRP methods for attention mechanisms. In the case of spherical-scal scalarizaaio models like HEGNN, which apply scalarization to reduce rotation orders, their operations can be implemented using e3nn tensor products. Extending EquiGX these models requires new LRP rule for inner product operations, which remains an open problem for future work.
The results are shown in Figure 5 where we plot the curves of Fidelity scores with respect to the Sparsity scores, No- tably, the model appears not to use the binding site infor- mation for its predictions. This conclusion is supported by the low fidelity score, which remains around 0.02 when the binding sites are masked.
5. Diseussions on Equivariant Graph Networks
As mentioned in (Duval et al., 2023a), equivariant networks can be categorized into four main types: Invariant GNNs: These networks, such as SchNet (Schütt et al., 2017), Dim- Net (Gasteiger et al., 2020), SphereNet (Liu et al., 2022), and GemNet (Gasteiger et al., 2021), encode the invariant geometric information like distances and directions directly into their model design to consider the 3D structures. Carte- sian equivariant GNNs: Networks like GVP-GNN (Jing et al., 2020), PaiNN (Schütt al., 2021), and E(n)GNN (Satorras et al., 2021) further consider direction vector as in- put and use scalar-vectos operations to consider their interac- tions within the architectures. Spherical Equivariant GNNs: These networks such as TFN (Thomas et al., 2018), SEGNN (Brandstetter et al., 2021), NequlP (Batzner et al., 2022), Equiformer (Liao& Smidt, 2022), Allegro (Musaelian et al., 2023), MACE (Batatia et al., 2022), usually use the spheri- cal harmonics of the directions as the input spherical tensors. Then they combine spherical tensors using equivariant oper- ations like Tensor Product (TP) and convert them into irre- ducible representations. These networks have more complex interactions between equivariant irreducible representations, demonstrating superior performance and widespread appli- cation in property prediction (Ramakrishnan et al., 2014), force field prediction (Chmiela et al., 2017), and Hamilto- nian matrix prediction (Schütt et al., 2019; Yu et al., 2024). Given the widespread use of the powerful spherical equivari- ant GNNs, understanding their key components, especially Tensor Product (TP), is one of the most essential problems in studying the explainability of equivariant GNNs. While
Acknowledgements
We are grateful to the discussions with Yang Shen. This work was supported in part by National Institutes of Health under grant UO1 AG0701 12 and National Science Founda- tion under grants IIS-2525159 and 113-2431515.
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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
Impact Statement
Chmiela, S., Tkatchenko, A., Sauceda, H. E., Poltavsky, I., Schüitt, K. TT., and Müller, K.-R. Machine learning of ac- curate energy-conserving molecular force fields. Science advances, 3(5)ee1003015, 2017.
This paper presents work whose goal to advance the field of XAI for Equivariant GNNs. There are many potential socictal consequences our work, none of which feel must be specifically highlighted here.
Du, Y., Wang. L., Feng, D., Wang, G., Ji, S., Gomes, C. P., Ma, Z.-M., et al. new perspective on building efficient and expressive 3D equivariant graph neural networks. Advances in Neural Information Processing Systems, 36, 2024.
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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
Table 2. Statistics and properties of four datasets.
Dataset
Shapes
Sprial Noise
SCOP
BioLip
#graphs
1000
1000
13738
26934
#classes
2
2
2
7
#avg nodes
14.92
10.45
498.49
320.33
#avg edges
160.94
89.94
6133.25
1427.3
Table 3. Prediction task performance of TEN models.
Dataset
Shapes
Sprial Noise
SCOP
BioLip
ACC
100
100
84.35 + 0.26
83.66 + 0.89
AUROC
100
100
N/A
83.36 = 0.85
A. Datasets and Experimental Settings
In this section, we provide more details of our experiments. We use NVIDIA RTX A6000 GPUs for all our experiments.
A.1. Datasets
The statistics and properties of the datasets are reported in Table 2. For the Shapes dataset, we randomly select a 3D motif shape from two options, namely a cube or an icosahedron. The cube has a side length of 2, and the icosahedron has a radius of V3. For the base shape, we choose either a pyramid or a star. The pyramid has a base length and height of I, while the star has an arm length of A random vector then used to translate the base shape, ensuring that it remains a certain distance from the motif shape without overlapping. Additionally, the motif shape undergoes random rotation. The classification task is to predict whether the motif shape in the geometric graph is cube or not. We use radial cutoff of to construct the geometric graph.
In the Spiral Noise dataset, we randomly select a 3D motif shape, either a tetrahedron or a triangular prism. The tetrahedron has a radius 1, and the triangular prism has a length and height of 1. The chosen motif shape is transformed using a randomly sampled translation vector and rotation matrix. Next, we randomly sample 4 to noise points, which form a spiral pattern with radius of I in 3D space. The classification task is to determine whether the motif shape is tetrahedron. We radial cutoff of 2 construct the geometric graph.
For the SCOP dataset, we extract the backbone atoms of the protein to construct the geometric graph. Specifically, for each amino acid residue of the protein, the backbone atoms (i.c.. nitrogen N, alpha carbon CA, and carbon C) are extracted and used as the nodes of the geometric graph. The atom type and residue index are used as features for each atom. We apply a radius cutoff of 5Á to create the geometric graph.
For the BioLip dataset, we extract the backbone atoms of the proteins and all atoms of the ligands to construct the geometric graph. Specificallyy we use the alpha carbon CA of each amino acid residue in the protein as the nodes of the geometric graph. Additionally, every atom of the ligand is also used as node in the graph. The atom type and residue type serve as node features. A radius cutoff of 10 A is applied to create the geometric graph.
A.2. TEN Model
We evaluate methods and baselines using Tensor Field Network models. Each TP-based message passing layer is followed by linear layer and a norm-based non-linear function. We first use spherical harmonics functions to compute the cquivariant features of each edge to order-l,,ar These equivariant edge features are then aggregated and concatenated with the node features to produce the first hidden equivariant features. Table provides details on the number of layers, the number of hidden equivariant features, and the highest order of equivariant feature Imax in the TFN. The accuracy and AUROC of the TEN model is reported in Table 3.
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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
Table 4. Hyperparameters for TFN models.
Dataset
Shapes
Sprial Noise
SCOP
BioLip
#layers
2
2
4
#channels
16
16
8
16
Lma
3
3
3
2
1.0
1.0
10
0.5
9.
Ground
Truth
0.4
0.4
02
y
1.0
1.0
1.0
0.6
EquiGX
0.0
0.0
0.
Figure 6. Explanation results on the Shapes dataset.
B. More Explanations
In this section, show visualizations of explanations. The explanations of the Shapes dataset reported in Figure 6. In addition, the explanations of the Spiral dataset are reported in Figure 7. As shown in these results, proposed EquiGX can identify the motif shapes. Furthermore, we also show explanation results of the SCOP dataset in Figure 8. An all-alpha protein is shown in Figure 8. Ideally, the a-helices should have a high importance score (i.e., be red in the figure), while the remaining parts should have a low importance score (i.e., be blue in the figure). Our method can distinguish a-sheets from other parts, assigning a low importance score to the remaining part. Figure 9, we also show more explanations of our proposed EquiGX on the BioLip datasets. The results demonstrate that ligands typically exhibit high importance scores. This observation aligns with existing knowledge, which suggests that different ligands have varying binding affinity scores when interacting with the same protein.
C. Runtime Study
In this section, we conduct runtime experiments on different datasets, evaluating the runtime of each method for a single data example. It is important to note that PGExplainer requires additional training time apart from inference time. The results in the Table 5 indicate that method has comparable runtime baselines, whereas GNN-Explainer exhibits significantly high runtime and PGExplainer incurs additional training time cost ranging from hours to days.
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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
0.8
08
0.8
2
Ground Truth
0.4
02
4.2
02
0.0
1.0
1.0
1.0
2
EquiGX
0.4
24
0.4
0.0
0.0
0.0
Figure 7. Explanation results on the Spiral dataset.
Table 5. Runtime comparison between different methods.
Inference Time
Shapes
Sprial Noise
SCOP
BioLip
Grad
0.056s
0.066s
0.21s
0.1ls
Grad-CAM
0.067s
0.068s
0.22s
0.12s
GNN-Explainer
0.07s
0.058s
0.23s
0.1s
LRI-Bemn
0.13s
0.16s
0.35s
0.24s
LRI-Gaussian
0.15s
0.14s
0.33s
0.28s
EquiGX
0.2s
0.19s
0.36s
0.25s
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On Explaining Equivariant Graph Networks via Improved Relevance Propagation
facc
thax
EquiGX
Grad
1111
T13
GNN-Explainer Figure 8. Explanation results on the SCOP dataset of all-alpha proteins. Since the sample is an all-alpha protein, ideally the a-helices should have high importance scores, i.c. be red in the figure.
LRI-Bern
LRI-Gaussian
min
max
min
max
max
Figure 9. Explanation results EquiGX on the BioLip dataset. The ligand highlighted with green border.
16


Reviews:
[{"review_id": "ryFISi1DFS", "reviewer": "Reviewer_V1gi", "summary": "The paper introduces EquiGX, an explanation method for (3D) equivariant GNNs. Existing graph explanation methods mainly focus on 2D GNNs and struggle to explain 3D GNNs. EquiGX extends Deep Taylor decomposition to derive layer-wise relevance propagation (LRP) rules for equivariant GNNs. The method is evaluated on synthetic and real-world datasets, outperforming existing explanation methods.\n\n## update after rebuttal\nAfter reading the other reviewers' comments and engaging in discussions with them, I now recognize that this work makes a highly significant contribution toward explaining tensor-product GNNs. Its impact goes beyond simply clarifying how TP-GNNs make decisions—it also provides insight into why TP-GNNs outperform invariant GNNs. The perspectives offered in this paper represent a fundamental advancement in the field, with the potential for broader impact. In light of these considerations, I have raised my score from 3 to 4.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are very glad Reviewer V1gi had a positive initial impression and appreciate your constructive comments. We provide pointwise responses below. \n\n> **This paper is an extension of the LRP framework that has been widely applied for vision tasks. Although tensor-product networks are different from normal MLPs or CNNs, the technical contribution of this work could be limited.**\n\nWe would like to humbly clarify our contribution. We admit that EquiGX is an extension of LRP rules for spherical equivariant graph neural networks. We would like to point out that there are currently no established LRP rules for spherical equivariant Graph Neural Networks. Besides, developing propagation rules for new architectures is a challenging task. For instance, references [1] and [2] extend LRP rules to transformers, and references [3] and [4] extend them to traditional 2D GNNs. Additionally, different approaches to using Taylor decomposition can yield various LRP rules, such as $z^+$ rule and $w^2$ rule. Furthermore, spherical equivariant GNNs, despite having aggregation operations similar to those of traditional GNNs, fundamentally differ due to their reliance on tensor product operations. We are the first to explicitly consider tensor product operations to develop new LRP rules for spherical equivariant GNNs.\n\n> **Limited Backbone Models: Only one model, TFN, is tested.**\n\nGiven the widespread use of powerful spherical equivariant GNNs, understanding their key components, i.e. the Tensor Product (TP), is a fundamental step toward explainability in equivariant models. In this paper, we focus on TFN, the most classical and representative spherical equivariant GNN. Our method is general and can be extended to other spherical equivariant GNNs. However, due to the high computational cost, we leave such extensions to future work. \n\n\n> **Why existing (2D) GNN methods are not suited for explaining 3D-equivariant GNNs?**\n\nThanks for the question. Most existing 2D GNN explanation methods are developed for graph data that capture only topological relationships. However, 3D equivariant GNNs go beyond topology by incorporating rich geometric information, such as interatomic distances, angles, and torsion angles, which are essential for many tasks in scientific domains. These models are specifically designed to respect spatial symmetries, enabling consistent predictions under 3D transformations. \n2D explanation methods typically do not consider atomic positions in space or spatial relationships such as angles and distances. As a result, they often fail to capture critical geometric cues. For instance, a small change in a node's 3D position can lead to significant shifts in angles or torsion angles, which may affect the model’s prediction. Since 2D methods are not sensitive to these changes, they tend to provide limited or misleading explanations when applied to 3D-equivariant GNNs.\n\n> **Equivariant GNNs are conducted on widely-used molecular datasets such as QM9 and MD17. Is there a specific reason why the experiments on these datasets are not conducted?**\n\nWhile Equivariant GNNs are commonly evaluated on datasets like QM9 and MD17, these datasets lack ground-truth explanations. The underlying rationale for quantum chemical properties and energy predictions is still an open research question, with no definitive consensus. As a result, these datasets are not well-suited for evaluating explainability methods. Therefore, in this paper, we construct synthetic datasets and carefully select real-world datasets where meaningful and verifiable explanations can be obtained.\n\n\n>**What does it mean for LRI treats the model as a black box and overlooks the equivariance of the model?**\n\nBoth LRI-Gaussian and LRI-Bern, like other perturbation-based XAI methods, treat the model as a black box. This means they rely solely on input-output behavior without requiring access to the model’s internal parameters or gradients. In other words, they evaluate how changes in the input influence the output, without considering the internal structure of the model.\n\nBesides, LRI-Gaussian overlooks the equivariance of the model, because the learned Gaussian noise is not equivariant. Specifically, when the input point cloud is rotated, the learned noise does not necessarily rotate accordingly. As a result, the explanations can become sensitive to the choice of the input’s reference frame, which is undesirable for explaining equivariant models.\n\n>**In both Eq 2 and 5, should it be ($\\ell_2,m_2$) in $C^{(\\ell_3,m_3)}_{(\\ell_1,m_1),(\\ell_2,m_1)}$?**\n\nWe apologize for the typo. Yes, it should be $(\\ell_2,m_2)$.\n\nReference\n\n[1] Transformer interpretability beyond attention visualization. CVPR 2021.\n\n[2]  XAI for transformers: Better explanations through conservative propagation. ICML 2022.\n\n[3] Higher-order explanations of graph neural networks via relevant walks. TPAMI 2021.\n\n[4] Relevant walk search for explaining graph neural networks. ICML 2023.", "claims_and_evidence": "Yes, the main claims are all clear with convincing evidence.", "methods_and_evaluation_criteria": "Yes, the method and evaluation criteria make sense.", "theoretical_claims": "I checked the derivations of the layer-wise decomposition in the main paper. I believe that they are correct.", "experimental_designs_or_analyses": "Yes, I read through all of Sec. 4 (Experiments), and I don't see any issues.", "relation_to_broader_scientific_literature": "It is highly relevant to the field of AI for science.", "essential_references_not_discussed": "Not that I’m aware of.", "other_strengths_and_weaknesses": "Strengths:\n\n- Significance: Though explanation methods for (2D) GNNs have advanced over the past few years, explaining equivariant 3D GNN models remains challenging. This is the first few works that target 3D GNN explanation. I appreciate this paper’s contribution in extending Deep Taylor decomposition to equivariant architectures, addressing the limitations of prior graph explanation methods for 3D geometric graphs. \n- Relevance: Understanding how equivariant GNNs utilize geometric and positional information is crucial for their application in AI for science, especially in molecular modeling.\n\nWeaknesses:\n- Technical Contribution: This paper is an extension of the LRP framework that has been widely applied for vision tasks. Although tensor-product networks are different from normal MLPs or CNNs, the technical contribution of this work could be limited. \n- Limited Backbone Models: Only one model, TFN, is tested. However, this could be due to the extensive computation requirement of equivariant GNNs.", "other_comments_or_suggestions": "1. The overall presentation of this work could be improved, particularly by addressing issues such as the excessive white space around some equations.\n2. I do think it would be beneficial to put more effort into discussing and presenting convincing arguments as to why existing (2D) GNN methods are not suited for explaining 3D-equivariant GNNs.\n\nMinor issues:\n1. Line 205, $d_{ij}$ not $d_ij$\n2. Line 151, “One way to compute such relevance is to the whole neural network as a mathematical function and use the first-order term from the Taylor series expansion.” Should be “… is to treat the whole neural network as…”", "questions_for_authors": "1. Equivariant GNNs, including the ones referred to in the paper e.g.  Equiformer, SE(3)-Transformer, are conducted on widely-used molecular datasets such as QM9 and MD17. Is there a specific reason why the experiments on these datasets are not conducted?\n2. In the last paragraph of Sec. 2, the author claims that LRI treats the model as a black box and overlooks the equivariance of the model. Can the authors further explain this part? There are two methods, LRI-Gaussian and LRI-Bern, in the LRI paper, which one does this refer to? What does it mean for them to overlook the equivariance of the model?\n3. In both Equations 2 and 5, shouldn’t the $\\left(\\ell_2, m_1\\right)$ be $\\left(\\ell_2, m_2\\right)$ in $\\mathcal{C}_{\\left(\\ell_1, m_1\\right),\\left(\\ell_2, m_1\\right)}^{\\left(\\ell_3, m_3\\right)}$?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NyKkdusXpx", "reviewer": "Reviewer_KEDe", "summary": "The paper proposes a new method, EquiGX, to explain equivariant GNNs. The method is based on Deep Taylor Decomposition and extends it to perform layer-wise relevance propagation for spherical equivariant GNNs. Specifically, the authors propose new rules to attribute tensor product operations. The experiments show that EquiGX performs better than existing baselines.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are very glad Reviewer KEDe had a positive initial impression and appreciate your constructive comments. We provide pointwise responses below.\n\n>**The method is limited to TP-based models.**\n\nWe admit that our method is focusing on spherical equivariant GNNs (TP-based models) and leaving generalization to other architectures as future work. However, we would like to highlight that TP-based equivariant networks represent a broad and widely adopted family of models in the AI for science domain. Developing explainability techniques for these models is important, as it can enhance our understanding of existing architectures and potentially lead to their improvement.\n\n>**The limitations are not discussed.**\n\nThanks for the suggestion. One limitation of our method is its current focus on tensor product operations, specifically within spherical equivariant GNNs. While this represents a subset of equivariant GNNs, it is a significant and impactful area. Consequently, our method does not yet extend to all types of equivariant GNNs. However, this focus allows for a deep and thorough exploration of tensor product operations, and we plan to generalize our approach to other architectures in future work.\n\n>**The Shapes and Spiral Noise datasets are very similar, and their visualizations look a bit repetitive. It might be better to show the visualization of one and move the other to the appendix.**\n\nThanks for the suggestion. We would update the visualization during camera-ready stage.\n\n>**The experiments focus on graph-level prediction tasks. Why not include node-level or edge-level predictions?**\n\nEvaluating explainability in equivariant networks is inherently challenging, largely due to the difficulty of obtaining meaningful and verifiable ground truth for explanations. In this paper, we address this by constructing synthetic datasets and carefully selecting real-world datasets where such ground truths are available. However, edge-level tasks like force prediction present additional complexity. While molecular dynamics can provide an approximate reference, the underlying rationale behind force prediction remains an open research question. Thus, we leave the exploration of node-level and edge-level tasks to future work.\n\n>**Why isn’t ActsTrack included in the qualitative evaluation?**\n\nActsTrack has an average of over 100 nodes per graph, and unlike proteins, it lacks a well-established visualization standard. Additionally, the ground-truth explanations are not immediately obvious, making clear and interpretable visualizations more challenging. As a result, including ActsTrack in the qualitative evaluation could lead to confusing visuals. We plan to include it in the paper later with more refined visualization methods.\n\n>**The source code is not provided.**\n\nWe plan to release our code during camera-ready stage.", "claims_and_evidence": "The claims are well supported.", "methods_and_evaluation_criteria": "The synthetic and real-world datasets make sense and evaluation criteria are common choices for model explainability.", "theoretical_claims": "Yes I checked the equations and didn't find any issue.", "experimental_designs_or_analyses": "The experiments are well designed and the paper include qualitative and quantitative evaluations.", "relation_to_broader_scientific_literature": "The paper extends LRP to 3D equivariant GNNs and proposes a new LRP rule for tensor product operations, which can be useful for other works that use equivariant GNNs or tensor product operations.", "essential_references_not_discussed": "Not found.", "other_strengths_and_weaknesses": "Strengths:\n1. The method is well-founded theoretically, and the derivation is clear.\n2. The writing is easy to follow.\n3. Including real-world protein datasets in the experiments is a nice addition.\n\nWeaknesses:\n1. The method is limited to TP-based models.\n2. The source code is not provided.\n3. The limitations are not discussed.", "other_comments_or_suggestions": "The Shapes and Spiral Noise datasets are very similar, and their visualizations look a bit repetitive. It might be better to show the visualization of one and move the other to the appendix. The same applies to the protein datasets.", "questions_for_authors": "1. The experiments focus on graph-level prediction tasks. Why not include node-level or edge-level predictions?\n2. Why isn’t ActsTrack included in the qualitative evaluation?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mj8rooNKt0", "reviewer": "Reviewer_ffwb", "summary": "Explaining equivariant GNNs for 3D geometric graphs is challenging due to their complex architectures and the difficulty of handling positional data. Existing explainability (XAI) methods mainly focus on 2D graphs and struggle to adapt to equivariant GNNs. To address this, this paper introduces the EquiGX, a novel explanation framework based on Deep Taylor decomposition, extending layer-wise relevance propagation for spherical equivariant GNNs. It decomposes prediction scores and back-propagates relevance through each layer, providing detailed insights into how geometric and positional data influence model decisions. Experiments on synthetic and real-world datasets show that EquiGX effectively identifies critical geometric structures and outperforms existing baselines, offering significantly improved explanations for equivariant GNNs.\n\n## Update after rebuttal\n\nThe authors have solved most of my questions. I will maintain my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are very glad Reviewer ffwb had a positive initial impression and appreciate your constructive comments. We provide pointwise responses below.\n\n>**Figure 1 is confusing.**\n\nWe apologize for the confusion. In Figure 1, the ground truth is shown in the upper-left corner. Nodes forming the cube motif are highlighted in red. Across all examples, node positions remain consistent, so better alignment with the ground truth reflects a more accurate explanation. In the figure caption, “the other areas” refers to nodes that do not belong to the cube motif. In other words, these nodes form a pyramid shape. We also highlight the nodes in blue in the ground truth. \n\n>**Clarification on importance calculation.**\n\nAs described in Section 3.2, for a single TP-based message passing layer, we decompose the relevance score of each message into three components, including the hidden features, the directional part, and the distance part. The relevance score attributed to the hidden features is further backpropagated toward the input. The relevance scores of the directional and distance components do not propagate beyond this layer. Within each message passing layer, the scores assigned to edge direction and edge distance capture their respective contributions to the final prediction. To compute overall contributions of edge directions and edge distances, we sum these directional and distance relevance scores across all layers. To obtain the final importance score for each input node, we combine the relevance score of the node itself with the relevance scores of its connected edges. In other words, the node-level explanation is calculated as the sum of a node’s own relevance plus half of the relevance of its neighboring edges.\n\n>**Figures 3 and 4, while visually compelling, are somewhat unclear. It is unclear which part of the visualization corresponds to β-sheets.**\n\nBeta sheets typically appear as flat, arrow-shaped ribbons pointing in a specific direction, often aligned side-by-side to form sheet-like structures. In Figure 3, the regions with arrows represent the beta-sheets. For reference, the red arrows in this [Wikipedia image](https://en.wikipedia.org/wiki/Beta_sheet#/media/File:5CPAgood.png) illustrate their typical appearance. We will revise the manuscript to clarify this in the figure captions and text.\n\nAdditionally, we would like to point out that the entirely red visualization produced by the LRI-Bern method does not necessarily imply higher effectiveness. An ideal explanation should selectively highlight the beta-sheets in red, while keeping less relevant regions such as the coiled structure near the top and the thin, string-like lines in blue.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "See the weaknesses.", "experimental_designs_or_analyses": "See the weaknesses.", "relation_to_broader_scientific_literature": "See the summary.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths\nRefer to the summary.\n\nWeaknesses\nThis paper presents an insightful analysis of LRP and its extension to equivariant GNNs. This is an interesting topic and the framework might be helpfu for other researches on equivariant GNNs. However, several details could be further refined for clarity and completeness.\n\n1. Figure 1 is confusing. It is recommended to explicitly illustrate the cube motif and highlight the data points with high importance within the motif. Additionally, the meaning of \"the other areas\" in the figure caption should be clarified.\n\n2. Clarification on importance calculation. The authors separate the layer-wise relevance into TP-based and invariant feature-based propagation. Are these components eventually merged? If so, what is the weighting scheme between equivariant and invariant features? Providing a clear explanation would enhance understanding.\n\n3. Figures 3 and 4, while visually compelling, are somewhat unclear. For instance, in Figure 3, the authors state: \"Since the sample is an all-beta protein, ideally the β-sheets should have high importance scores, i.e., be red in the figure.\" However, it is unclear which part of the visualization corresponds to β-sheets. Additionally, the LRI-Bern method results in an entirely red visualization—does this indicate it is the most effective? The authors should provide a more precise explanation.", "other_comments_or_suggestions": "No.", "questions_for_authors": "See the weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "du5M1lPrAY", "reviewer": "Reviewer_6J4F", "summary": "The paper introduces a novel method called EquiGX aimed at enhancing the explainability of equivariant graph neural networks (GNNs) specifically designed for 3D geometric graphs via the deep Taylor decomposition framework. In the initial version, the authors used incorrect notations (e.g., the relevance score for spherical harmonics in Eq. (7)), which led to a big misunderstanding, so I questioned whether the model might violate equivariance. This has been clarified during the discussion, and I have raised my score from \"clear reject (1)\" to \"clear accept (4)\".", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer 6J4F for comments on the paper. We have provided pointwise responses below.\n\n>**According to the formula in this paper, the contribution of the atom may change with the reference frame in which the molecule is observed.**\n\nWe believe there is a misunderstanding. The node explanations by EquiGX are invariant to rotations. In other words, the node importance scores from EquiGX remain unchanged when the input molecule is rotated by a random rotation matrix. Therefore, EquiGX preserves the equivariance of the network. We have also conducted experiments to verify and confirm this invariance. \n\n>**Can EquiGX be extended to other equivariant models?**\n\nEquiGX can be readily extended to other spherical equivariant models, such as EquiformerV2, MACE, and PACE, which also rely on tensor product operations. However, due to the significant computational cost, we leave such extensions for future work. We would also like to point out that models like SphereNet and ComENet are different, as they operate solely on invariant features without incorporating equivariant ones, and thus do not rely on tensor product operations. We believe these models are beyond the scope of this paper.\n\n>**In line 205, $d_i,j$ should be $d_{i,j}.$ In Eq 2, $C^{(\\ell_3,m_3)}_{(\\ell_1,m_1),(\\ell_2,m_1)}$ should be $(\\ell_2,m_2)$.**\n\nThanks for pointing out the typos. We will update the manuscripts.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "After resolving the symbol misunderstanding, no other theoretical problems were found.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "The subject of this article is \"Equivariant Graph Networks\", and the article only discusses TFN. After discussion, the authors agreed to open a \"Future Work\" section to discuss the following models:\n- Invariant Models: SchNet, SphereNet, ComENet\n- Scalarization-based Models: EGNN, SaVeNet, LEFTNet\n- Tensor-product-based Models: EquiformerV2, MACE, PACE \n- Spherical-scalarization Models: SO3KRATES, HEGNN, GotenNet\n\nSchNet: A continuous-filter convolutional neural network for modeling quantum interactions, NIPS'17.\n\nSphereNet: Spherical Message Passing for 3D Molecular Graphs, ICLR'22.\n\nComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphs, NeurIPS'22.\n\nEGNN: E(n) Equivariant Graph Neural Networks, ICML'21.\n\nSaVeNet: A Scalable Vector Network for Enhanced Molecular Representation Learning, NeurIPS'23.\n\nLEFTNet: A new perspective on building efficient and expressive 3D equivariant graph neural networks, NeurIPS'23.\n\nEquiformerV2: EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations, ICLR'24.\n\nMACE: MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields, NeurIPS'22.\n\nPACE: Equivariant Graph Network Approximations of High-Degree Polynomials for Force Field Prediction, TMLR'24.\n\nSO3KRATES: A Euclidean transformer for fast and stable machine learned force fields, NC'2408. \n\nHEGNN: Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?, NeurIPS'24. \n\nGotenNet: Rethinking Efficient 3D Equivariant Graph Neural Networks, ICLR'25.", "essential_references_not_discussed": "The calculation of the relevance scores for spherical harmonics in Eq. (7) is very similar to the core formula of spherical-scalarization models (e.g. SO3KRATES, HEGNN, GotenNet), which may explain why this method makes sense.", "other_strengths_and_weaknesses": "All the problems have been solved, which are mainly due to misunderstandings caused by symbols. Therefore, I recommend that the authors revise their writing using the symbol system of HEGNN and GotenNet to avoid readers' misunderstanding of scalars and tensors. The authors said that they will accept this suggestion in the subsequent revision of the manuscript.\n\nAlthough ICML does not allow submission of new PDFs, I choose to believe that the authors will make appropriate changes.", "other_comments_or_suggestions": "- [Line 76, right] $\\mathcal{C}\\_{(\\ell\\_1, m\\_1),(\\ell\\_2, m\\_1)}^{(\\ell\\_3, m\\_3)}$ in Eq. (2) should be $\\mathcal{C}\\_{(\\ell\\_1, m\\_1),(\\ell\\_2, m\\_2)}^{(\\ell\\_3, m\\_3)}$.\n- [Line 205, left] $d_ij$ should be $d_{ij}$.", "questions_for_authors": "See review above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces EquiGX, a novel method for explaining equivariant graph neural networks (GNNs). Building on Deep Taylor Decomposition, the authors extend layer-wise relevance propagation to spherical equivariant GNNs by proposing new attribution rules for tensor product operations. Experimental results demonstrate that EquiGX outperforms existing baselines.

Reviewers commend the paper for its novelty, significance, and clarity. While some questions remain regarding equivariance satisfiability, technical contributions, baseline comparisons, and experimental results, all reviewers unanimously support acceptance after reviewing the authors' rebuttal. I concur with their assessment and recommend acceptance.
Decision: Accept (poster)