Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction
Authors: Yisen Gao
Xingcheng Fu
Qingyun Sun
Jianxin Li
Xianxian LI
Venue: NeurIPS 2025 poster ()
Keywords: Graph Generation
Hyperbolic Graph Learning
Riemannian Manifold
Graph Learning
Primary area: deep_learning

Abstract: Graph diffusion models have made significant progress in learning structured graph data and have demonstrated strong potential for predictive tasks. Existing approaches typically embed node, edge, and graph-level features into a unified latent space, modeling prediction tasks including classification and regression as a form of conditional generation.  However, due to the non-Euclidean nature of graph data, features of different curvatures are entangled in the same latent space without releasing their geometric potential. To address this issue, we aim to construt an ideal Riemannian diffusion model to capture distinct manifold signatures of complex graph data and learn their distribution.  This goal faces two challenges: numerical instability caused by exponential mapping during the encoding proces and manifold deviation during diffusion generation.  To address these challenges, we propose **GeoMancer**: a novel Riemannian graph diffusion framework for both generation and prediction tasks. To mitigate numerical instability, we replace exponential mapping with an isometric-invariant Riemannian gyrokernel approach and decouple multi-level features onto their respective task-specific manifolds to learn optimal representations.  To address manifold deviation, we introduce a manifold-constrained diffusion method and a self-guided strategy for unconditional generation, ensuring that the generated data remains aligned with the manifold signature. Extensive experiments validate the effectiveness of our approach, demonstrating superior performance across a variety of tasks.

Toward a Unified Geometry Understanding: Riemannian Diffusion Framework for Graph Generation and Prediction
Yisen Gao1.2.4, Xingcheng Fu'; Qingyun , Jianxin Li", Xianxian Li¹
1 Key Lab of Education Blockchain and Intelligent Technology, Guangxi Normal University Computer Science and Engineering, The Hong Kong University of Science and Technology Guangxi Key Lab of Multi-source Information Mining & Security, Guangxi Normal University *School of Computer Science and Engineering, Beihang University "gaodi@cse ust. hk, {fuxc, 1ixx) edweemuueeduee.. cn, {sunqy xliiiliiiee.e (ibaa).baaaeeeeea
Abstract
Graph diffusion models have made significant progress in learning structured graph data and have demonstrated strong potential for predictive tasks. Existing approaches typically embed node, edge, and graph-level features into a unified latent space, modeling prediction tasks including classification and regression as form of conditional generation. However, due to the non- Euclidean nature of graph data, features of different curvatures are entangled in the same latent space without releasing their geometric potential. To address this issue, we aim construt ideal Riemannian diffusion model to capture distinct manifold sig- natures of complex graph data and learn their distribution. This goal faces two challenges: numerical instability caused by exponential mapping during the encod- ing proces and manifold deviation during diffusion generation. To address these challenges, propose GeoMancer: a novel Riemannian graph diffusion frame- work for both generation and prediction tasks. To mitigate numerical instability, Riemannian gyroker-
we replace exponential mapping with an isometric-invariae nel approach and decouple multi-level features onto their respective task-specific manifolds learn optimal representations. To address manifold deviation, we introduce manifold-constrained diffusion method and a self- self-guided strategy for unconditional generation, ensuring that the generated data remains aligned with the manifold signature. Extensive experiments validate the effectiveness of our approach, demonstrating superior performance across a variety of tasks.
Introduction
Graph-structured data is widely used in real-world applications 2 such as recommendation sys- tems 3 social networks (4) and molecular modeling (5). Due the non-Euclidean structure of graphs, there are some studies [6] that have leveraged differential geometry to explore non- -unllidie geometrie spaces that better align with the intrinsic structure of graphs. In a geometric perspective, non-Euclidean manifolds can enable a deeper understanding of graph-structured data, which in turn facilitates improved performance in downstream tasks. For example, hyperbolic spaces are well-suited for modeling small-world and hierarchical graphs [ 8), whereas spherical spaces are more effective at capturing the structure of densely connected graphs (9]. Product manifold [10
is introduced to model more complex graph data by constructing a space of mixed curvatures [2]
"Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Vartons Tasks
Nade-lerel
Edge-level
Graph-leve
Nede-level
Task-ndinptation
Graph Difasion Model
Gr
G2
Panound
Gig
Semanfiehpessrviigg
repoooggnreesevving
Fasture
Tree
Grid
Graph-level
Graph
Edge-level
Overview of Graph Diffusion Model
(b) Complex Multi-level Data Manifolds
Figure 1: (a) An overview of latent graph diffusion model for prediction and generation. (b) Feature Entanglements: we visualize the multi-level latent representations learned in graph regression tasks on the ZINC12K [ dataset using t-SNE. The results reveal that representations with distinct geometric properties become entangled in shared Euclidean latent space.
Moreover, diffusion models [13 have demonstrated strong capabilities in capturing complex data distributions and understanding the inherent geometric properties of the data [14]. Recent meth- ods [13] have introduced the latent graph diffusion framework for unifying the graph generation and prediction tasks. Specifically, it embeds multi-level graph features (including node-level, edge-level, and graph-level) into unified www-dimensional latent space. Then the prediction task is reformulated as a conditional generation problem, where graph features explicitly serve as the condition guiding the generation of target graph attributes/labels.
However, although this latent diffusion framework is theoretically sound in general data, it overlooks the rich and diverse geometric structures inherently present in graph data and lacks a unified geometric perspective for graph learning tasks. As observed in Fig. [] representations across different levels are often entangled within shared latent space, despite exhibiting distinct intrinsic geometric properties. These features exhibit curvature heterogeneity across different levels and should be better modeled in spaces with varying curvature. This highlights the limitations of the existing method and calls for a new modeling paradigm capable of capturing the optimal manifold signatures [16] (such as the choice of curvature, manifold components, and dimensionality) for complex graph data.
Based on the above insights, an ideal Riemannian diffusion framework should first reconstruct the underlying data manifold using a geometry-awant autoencoder, and then model the distribution over this manifold through diffusion process. However, the inherent geometric complexity of multi-level graph features, further compounded by the demands of multi-task learning, makes modeling the underlying manifold highly challenging and well beyond the capacity of simple designs. Specifically, it faces the following two key challenges:
How assign an appropriate manifold signature during the autoencoding process? A common approach is to map features on product manifolds [10 I7 with learnable curvatures (18) using exponential mapping. However, due to curvature heterogeneity across different feature levels, this method may lead to numerical instability in the exponential map, making model optimization more range of downstream tasks.
difficult and restricting its effectiveness
How to generate an accurate manifold distribution during the diffusion process? In the generation stage, diffusion models often deviate from the original data manifold [19), leading sub-optimal performance. One approach 20 to address this issue is by incorporating manifold constraints into the condition generation process, which helps guide the model back to the desired manifold. This can be seen as a form of precise conditional control. However, in unconditional graph generation tasks, the absence of such guiding information can result in deviations from the intended manifold structure.
To address these challenges, we propose GeoMancer: a novel Riemannian Diffusion Framework for graph generation and prediction. To better choose the manifold signatures, we construct a product manifold as the latent space for each level feature. Then, we decouple the multi-level features entan- gled within it onto their corresponding task-specific manifolds. Additionally, to mitigate numerical

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instability caused by exponential and logarithmic mappings, we employ a Riemannian kernel method based on generalized Fourier transforms. This approach preserves the isometric geometric properties of Riemannian spaces while remaining compatible with well-established Euclidean models. To guide the diffusion model to generate features on a more desirable manifold, we leverage the rich geometric information in the latent space to produce pseudo-labels for unconditional graph generation. This allows us to reformulate all graph-related tasks as conditional generation problems. During the generation process, we further enhance this guidance by introducing manifold-constrained sampling strategy. Our contributions are summarized as follows:
We propose a unified Riemannian diffusion framework that provides geometric understanding of graph generation and prediction tasks by assigning geometric spaces aligned with the intrinsic property of multi-level graph data.
We introduce key improvements to both the encoding and generation stages of the Riemannian latent diffusion framework, including multi-level product manifold modeling, a numerically stable Riemannian kernel method, and a manifold-constrained conditional generation strategy, enabling more accurate and robust learning of the underlying geometry in graph data.
Extensive experiments demonstrate that our model achieves excellent performance across multiple levels of tasks in generation, classification, and regression.
2 Related Work
Graph diffusiin model for generation. Graph diffusion models can be broadly categorized into two approaches: discrete diffusion and latent diffusion. In discrete diffusion, GDSS [1 employs stochastic differential equation (SDE)-based diffusion techniques to model both node features and adjacency matrices. GSDM [22] extends this framework by incorporating diffusion in the spectral domain, further enhancing its capability to capture graph structures. DiGress (23] adapts the diffusion process specifically for discrete data, while GruM 24 introduces a Schrödinger bridge to preserve the effective structural properties of graphs during generation. Defog [25] proposes discrete flow matching method for generating discrete graph-structured data. In latent diffusion, Graphusion [26] utilizes variational autoencoders map graph structures into latent representation space, assigning soft labels through structural clustering to capture spatial relationships. HypDiff [27 projects graph structures into hyperbolic space and applies geometrically constrained diffusion to maintain the anisotropic properties of graphs, ensuring that the generated structures align with their intrinsic geometric characteristics.
Graph diffusson model for representation. Compared to their widespread use in generative tasks, graph diffusion models have been relatively under-explored in the context of representation learning. Among the few existing approaches, DDM 28 focuses on denoising node features and leverages a Unet-based architecture to extract effective representations. Meanwhile, LGD [15] represents a significant advancement as the first model to unify generation and diffusion within single framework. It achieves representation learning by reformulating downstream tasks as conditional diffusion processes, thereby bridging the gap between generative and discriminative objectives.
Riemannian representation model. Representation learning in Riemannian space primarily relies on exponential and logarithmic mappings. For example, HGCN (7] uses exponential mapping project representations generated by GCN into hyperbolic space, performing operations like aggregation and activation in Euclidean space after logarithmic mapping. [22 extends this to Riemannian spaces with multiple curvatures, enabling more flexible representation learning. HyLA 29 introduces hyperbolic framework that replaces exponential mappings with isometric invariant kernel mappings, preserving hyperbolic geometric properties. Building this, MotifRGC [18) generalizes the ap- proach to arbitrary Riemannian spaces and uses contrastive learning to assign node-specific curvatures, significantly improving representation expressiveness. The Riemannian MOE architecture 30 31
has also been introduced to capture different graph structure features recently.
3 Method
In this section, we propose our model GeoMancer, a Riemannian diffusion model for graph generation and prediction. In Section 3.I we first introduce how to use graph diffusion model as a unified
framework for both generation and prediction tasks. In Section 3.2 propose the Riemannian graph
3

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Gruph Encoding
Graph Dillusion Process
Geo. Decoupled Decoding
Task
Forward Process
Hrperrblllc
Gr
Gr-s
Ge
Sphertoal
Edge level
Goomente C
Reverse Process
Za
Fimeer
Unconditional Denoising P(X
Cond Coniiicution Hearaaios
T.A.0.
Tacideas
Graph-lerd
Demoising P(XI))
ZE
Prediction Denoising (
Figure An Illustration of GeoMancer Architecture.
autoencoder. We first introduce Riemannian kernel based on the generalized Fourier transform to replace the exponential map. Then, describe how the encoder captures the complex geometry of the latent space, and how the decoder projects data onto the task-specific manifold. In Section 3.3 we introduce self-gut self-guided strategy for unconditional graph generation, allowing all graph tasks to be unified under a conditional generation framework, We further incorporate manifold constraint guidance method to ensure the generation aligns with clean data manifolds. The preliminaries for the methods are presented in the Appendix B.
3.1 Notation and Problem Definition
graph with N nodes is defined as G - (X,E,Xx, YG). where [21, T2. IN| E Nxx,, denotes the node features and z; denotes the feature of node i. NNNNxd. denotes the feature of edges and eij is the edge feature between node and node j. Yx E RR denotes labels or properties of the nodes, YG E TK denotes the label or property of the graph
Our model is a generative framework that unifies general generation and multi-level prediction tasks under diffusion paradigm. Specifically, can be conceptualized as follows: (1) For the unconditional graph generation task, the objective is to learn graph generative model that can capture the distribution p(G) of the target graph set (G1, Gz.... GN J. (2) For the conditional graph generation task, it seeks to generate synthetic graphs conditioned on specific label or property Yg- The goal model the conditional distribution p(G)YG). (3) For the downstream prediction task. can be viewed special form of the conditional generation task where the generation target is label Y and the condition is (X, F). Thus, the generation model is to learn the conditional distribution of p(Y|X, F).
3.2 Riemannian GyroKernel Autoencoder
The goal of Riemannian autoencoder is to encode complex multi-level graph features into a unified low-dimensional latent space that preserves the geometric heterogeneity and decode them to the task-specific manifold. However, mapping data onto product manifold often relies on exponential and logarithmic maps, which prone to severe numerical instability, making the optimization difficult.
To address this, we use Riemannian kernel 18 as a substitute, which maps Riemannian prior representations to Euclidean space while preserving isometry for encoding Euclidean features. Guided by Bochner's Theorem (See the Appendix A), isometry-invarian kernels can be constructed through Fourier mappings based on eigenfunctions of Laplace operators. this work, we utilize generalized Fourier mapping Øgr (z) defined in the gyrovector ball G. of Riemannian manifold with the learnable curvature x Specifically, the eigenfunction gF A.A. (x) in the gyrovector ball G can be formulated
gF (.) = Aw cOS (N(w, x)nx + b), G
(I)

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where Aw.== exp [
(w,))))) (w, at), log |+*||=: [[==w|
is the signed distance in the gyrovector
ball. w denotes the phase vector, uniformly sampling from a a n-dimensional unit ball. denotes the bias, uniformly sampling from [0, 2m).
Then the generalized Fourier mapping (gF()) can be denoted
Oer (1)
gg Vm
Widdn (x),
F
Sm (1r] Rd
We initialize product manifold Riemannian representation vectors Vx (Ve. Vayy.... Ve,.. with different curvatures Hi, Using equation (2), we compute the isometric-invarian Euclidean features V. after generalized Fourier mapping:
V = (VF(VV) = (ogg )
OgF (Vrm))
(3)
Then, we can aggregate each dimension of any graph representation Z by taking ad advantage of the geometric properties of the product manifold: = ZV= Unlike simple feature mapping, this approach enables each dimension of the features capture distinct geometric information, thereby more effectively revealing the intrinsic geometric properties of graph features. We have provided more detailed introduction this method in Appendix B.
Encoder. Here, we need to build a powerful graph encoder that can embed the node features X and edge features E of the graph G into unified www-dimession latent space Z [Zx, Zæ} R[N+N*N) The graph-level feature Zg can be obtained by aggregating Zx and Zg. To better represent edge features E, we adopt a flexible graph transformer [32 as the backbone network for edge enhancement. It constructs relevant attention mechanisms for both nodes Zu, and edges Zer, to efficiently underlying the relationships between them. Specifically, the 1-th graph transformer layer be represented as:
241 a ( ((QE2) Kz @ Erozer)) + Eozles)
E.y
0ij = Softmax (W (+1),
Deij
(4)
J+1 2g.
Σ Oiis Cvis(V (V 2,, + Enzett) 141),
JEV
where Q, K, V, W, E., Es. E, are learnable weight matrices; O denotes the elementwise multi- plication; is nonlinear activation and p(x:) (ReI.Uxx))1/2 (ReI.U-xx)) is function used for training stability.
Then, the geometrie latent representation Z is obtained by endowing the embeddings with product manifold geometric properties with given V.. To simplify notation, denote the geometric latent representation Z as Z in the remainder.
Decoder.
The decoder aims to project the latent representations Z onto task-specific manifolds, enabling effective adaptation to downstream tasks. To achieve this, we design a dedicated decoupling modle for each level feature. The core idea of this method is split a complex product manifold into multiple simple manifolds: M + Mi ... X Mm and selects the most appropriate geometric representation based on the specific requirements of each task.
21, L} be diieeeiifirenttiable functions such that AM >r where AM; Laplace operators and X; the eigenvalue. Take n; :M + M; to be the projection of M onto M,, We can then define the natural extension of Ji to M via g = St It follows that
Proposition 3.1. Let fi - iR,
L 99
L
AM
I
>
II
9h-
(5)
According to the proposition 3.1 a Euclidean representation endowed with the geometric prior of complex product manifold can be decoupled into simpler representations over its constituent manifolds.
Then, we decompose Z [2x,, Zxg...., 2r into the representation of each component Zx,- Since they are still in Fuclidean space after the generalized Fourier mapping, we directly use the attention or linear layers at the end to capture the most effective manifold representation for the task.
5

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Training Objective. The training objective encompasses a target loss Ligt and regularization constraints Cregi
(6) In the generation task, the target loss Ligt is composed of the reconstruction cross-entropy loss of nodes and edges. For regression or classification tasks, the target loss Ligt is the MSE a regression task or the cross-entropy of classification task. The regularization loss Greg repre- sents the regularization constraints, which pushes the representation Z towards standard normal distribution, thus preventing high variance in the latent space. Specifically, it can be written as: Lreg = DRL (g(Z E)) || I)).
L Ligt + Freg.
3.3
Manifold-Constrainet Diffusion
After obtaining geometric latent representation Z [Zx, Ze. Za), conduct diffusion training and generation within this unified latent space. In addition to incorporating geometric priors, a notable advantage of the Riemannian kernel method is that the representations remain in Euclidean space. This allows for the direct application of classical diffusion techniques without the design for more complex Riemannian diffusion. The forward process [13] of the diffusion model gradually turns the data toward noise. Specifically, this process can be defined as:
N N(X, v 0z-17)
where (.) is the Gaussian distribution and 6;-1 is calculated by noise schedule.
To better capture the complex data manifolds, we hope to adopt a manifold-constrial
conditional generation approach [20]. While tasks like conditional generation and prediction can be naturally redefined as conditional generation problems, unconditional graph generation lacks explicit label guidance. To bridge this gap, we introduce self-guided mechanism that leverages the rich geometric information embedded in the latent space to generate pseudo-labels thereby providing effective guidance to the model during generation.
Self-Guidance. Even in unconditional generation, structural variations inherently reflect differences on the underlying geometric manifold. Therefore, natural approach is to leverage the rich geometric features encoded in the latent space to guide the graph generation process. By applying k-means clustering the latent graph-level representation Zg, we assign pseudo-label C' each graph. Consequently, the unconditional generation of graphs can be reformulated as a new conditional generation task to learn P(GIC). the sampling stage, is selected randomly for each graph.
Further, we unify graph generation and prediction task by modeling them within a general conditional generation framework to learn the distribution P(x|y) v) For unconditional generation, generate pseudo labels that capture complex geometric property through self- gufdaneee.ee.. In conditional graph generation, explicit graph properties serve conditions. Predictive tasks, such as classification and regression, are reformulated as conditional generation processes based on known representations.
We then adopt CFG++ [20). a manifold-conssrrineed classifier-free guidance method that formulates conditional generation as an inverse problem, enabling the model to better capture the clean manifold of the data. Specifically, the reverse generation process can be formulated as:
2. = (Z: - v1 C Z., Z,,77(770 (1))))
Z,-1 Var-120 v1 -;-160
where y is the condition and r(y) is the latent embedding of y. éo is derived from the outputs of the model co under both conditional and unconditional settings. Specifically, it can be written as:
Eo (τ( r()) = (1- (2, ((y) x
where À is a hyperparameter that controls the strength of condition guidance.
The model eg is optimized via noise prediction strategy, learning to estimate the added noise in the forward process:
Casfs E2.. [lleo t,,, ( -cel
(10)

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Table 1: Unconditional generation results QM9. (Bold: best; Underline: runner-up.)
Model
Validity (%)1
Uniqueness (%)}
FCD
NSPDK +
Novelty (%)
MoFlow
91.36
98.65
4.47
0.0170
94.72
GraphAF
74.43
88.64
5.27
0.0200
86.59
GraphDF
93.88
98.58
10.93
0.0640
98.54
GDSS
95.72
98.46
2.90
0.0003
86.27
DiGress
99.01
96.34
0.25
0.0003
35.46
HGGT
99.22
95.65
0.40
0.0003
24.01
GruM
99.69
96.90
0.11
0.0002
24.15
LGD
98.46
97.53
0.32
0.0004
56.35
GeoMancer
100.00
95.74
0.09
0.0002
90.43
Table 2: Conditioal generation results
QM9 (MAE 4). (Bold: best; Underline: runner-up.)
Method
μ
<HOMO
ELUMO
A
Cv
W
0.043
0.10
39
36
64
0.040
@ (LGD)
0.058
0.06
18
24
28
0.038
(GeoMancer)
0.054
0.06
16
22
27
0.037
Random
1.616
9.01
645
1457
1470
6.857
Natom
1.053
3.86
426
813
866
1.971
EDM
1.111
2.76
655
1.101
GeoLDM
1.108
2.37
340
522
587
1.025
LGD
0.879
2.43
313
641
586
1.002
GeoMancer
0.832
2.38
304
628
S81
1.002
Model Architechture. Since there no prior information about edges during noise sampling, do not use message passing or positional encoding. Instead, we directly employ graph transformer as the denoising network. For incorporating conditional information, follow existing methods [T5 and introduce the condition through the cross-attention:
(Qn 2
(K (T))
[+1 softmax
VNr(y),
Vd
(11)
1+1 sofumax Ze;
Ver(y).
dd
Experiment
4.1 Experimental Setup
We evaluate the effectiveness of our mode conducting experiments across multiple tasks. For graph structure generation, we assess both unconditional and conditional molecular generation. For prediction tasks, we evaluate the model's 's performance on node classification and graph regression. These tasks collectively cover node-level, edge-level and graph-level tasks, providing a thorough assessment of our model's capabilities.
For all baselines, report their results based the results presented in their papers the optimal parameters provided. All experiments were conducted using PyG, and the reported results are averaged over five runs. All models were trained and evaluated an Nvidia A800 80GB GPU. More experimental details are reported in Appendix C.
4.2 Generation Task
We conduct molecular graph generation experiments the QM9 dataset (33. widely used bench- mark machine learning for molecular data. QM9 contains 133,885 molecular graphs with 12
Our code is available http://ghhabcccmm/iiggggSsssssss

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Table 3: Node-level classification tasks (aceuracy 1) out of memory.)
Bold: best; Underline: runner-up. OOM: cuda
Model
Photo
Physics
Pubmed
Cora
Citeseer
GCN
92.70 0.20
96.18 0.07
88.9 0.32
81.60 0.40
71.60 = 0.40
GAT
93.87 + 0.11
96.17
83.28 = 0.12
$
72.10 = 1.10
GraphSAINT
01.72
96.43 0.05
85.64 0.26
81.82
72.30 0.17
Graphormer
0.13
OOM
92.64 + 0.96
82.62
71.60 + 0.32
GraphGPS
95.06
00M
90.28 0.62
8.18
72.73 + 0.23 0.23
Exphormer
05.35
96.89 0.099
91.44
8
71.63 = 0.29
NAGphormer
95.49 4 0.11
97.34
91.76
8
71.40 0.30
LGD
96.94 0.14
98.55 #.11
92.88 + 0.29
82.81 1.18
72.40 + 0.30
GeoMancer
97.05 # 0.13
98.78 0.12
93.10 = 0.29
83.50 0.23
72.60 + 0.20
Table 4: Ablation study on unconditional generation. (Bold: best; Underline: runner-up.)
Model
Validity (%)*
Uniqueness (%)f
FCD
NSPDK
Novelty (%)f
GeoMeneer(w/o self-guid -ufdguuddncce)
98.99
97.61
0.12
0.0003
54.06
GooMancer( w/o cfg++)
100.00
91.74
0.09
0,0010
76.43
GeoMancer(W/o Riemannian)
100.00
92.53
0.25
0.0004
90.26
GooMancer
100.00
95.74
0.09
0.0002
90.43
quantum chemical properties limited to 9 heavy atoms. Unconditional Generation. In the un- conditional molecular generation task, we evaluate the model's ability capture the distribution of molecular data and generate chemically valid and structurally diverse molecules. Specifically, validity is the fraction of valid molecules without valency correction or edge resampling. Uniqueness quantifies the proportion of unique valid molecules among the generated set. Novelty assesses the fraction of valid molecules that do appear in the training set. To further evaluate the quality of the generated molecules, we employ two additional metrics: the Neighborhood Subgraph Pair- wise Distance Kernel (NSPDK) MMD [34, which computes the Maximum Mcan Discrepancy (MMD) between the generated and test molecules by considering both node and edge features, and the Fréchet ChemNet Distance (FCD) 35) which evaluates the distance between the training and generated graph sets using the activations of the penultimate layer of ChemNet, providing measure of similarity in the feature space. For the baseline models, we selected the classical and recent state-of-the-art approaches, including MoFlow [36], GraphAF (37, GraphDF 38 GDSS (2]]
DiGress (23], HGGT (39] GruM 24 and LGD
15
The results have been reported in Table I It can be observed that our model achieves significant improvements in the task of unconditional molecular generation. Specifically, we achieve state-of- the-art performance in terms of validity and the distributional similarity of molecular structures. In addition, our model shows competitive results uniqueness and novelty. Notably, compared to LGD [15] which also employs a latent graph diffusion framework, our approach achieves substantial improvement in novelty. We will further investigate the underlying reasons for this phenomenon through detailed ablation studies.
Conditional Generation. For the conditional generation task, its objective is generate target molecules with specific chemical properties. Following the experimental setup outlined in (5, we split the training set into two halves, each containing 50,000 molecules. We train a latent graph diffusion model and a separate property prediction network on each subset. During evaluation, we generate molecule using the latent graph diffusion model conditioned on given property and then use the property prediction network to predict the target property for the generated molecule. We calculate the mean absolute error (MAE) between the predicted property and the true value, conducting experiments across six properties: Dipole moment 14, polarizability a, orbital energies éHOMO, ÉL.O their gap Ae and heat capacity Cv. Following [15), we establish EDM [40) GeoL.MM (5) and LGD [15] as baseline models. Additionally, we include the following reference points for comparison: (a) the MAE of the regression model w of ours which serve as a lower bound of the generative models; (b) Random, which shuffle the labels and evaluate representing an upper
8

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Edgelleed
Grapplnd
Hyperbolic
Spherical
Exclidean
Figure 3: Visualization of the embeddings with different downstream task in the sub-spaces of the geometric decoupled decoders.
bound for the MAE metric; (c) Natoms, which predicts the properties based only on the number of atoms in the molecule. The results in Table 2
First, w our model achieves lower MAE compared to other models, demonstrating that the Rie- mannian autoencoder excels in regression capability. Additionally, our approach delivers outstanding performance across multiple properties in the generation task. Notably, even without incorporating 3D information, we successfully achieve high-quality conditional generation.
4.3 Prediction Task
For predictive tasks, evaluated the graph re- gression task and node classification task.
GCN
COOD
GNN-AK+
Graph Regression. For graph regression task, we select ZINC [T], which is a subset of ZINC250k containing 12k molecules. The task focuses on predicting molccular properties, particularly constrained solubility, with perfor- mance evaluated by MAE. Here, we use the official split of the dataset. As baselines, consider a range of representative graph regres- sion models. GIN [41] employs simple aggregator with learnable bias followed by MLP updates, achieving expressive power equivalent to the 1-WL test. PNA [42] combines multiple
0.20
PNA
0.15
DeePLRP
LGD
GeoMancer
MAE
KP-GIN
0.10
0.05
0.00
Figure 4: Comparison of graph regression task on Zinc dataset.
neighborhood aggregators with degree-scalers, allowing the model to capture diverse structural statis- tics. DeepLRP (43 encodes local structural patterns by pooling over permutations of nodes within
small subgraphs. OSAN introduces ordered subgraph sampling and aggregation to enhance message passing with subgraph-level information. KP-GIN+ [4 extends GIN with edge-sensitive updates while aggregating peripheral K-hop subgraphs for richer context. GNN-AK+ [46] improves expressiveness by applying subgraph GNN to each node's local induced neighborhood. CIN [47 generalizes message passing beyond edges by propagating information across nodes, edges, and higher-order cells within a cell complex. GPS [48] integrates local MPNN aggregation with global Transformer attention a hybrid design. Finally, LGD [15] applics latent-space diffusion to jointly model graph structure and multi-level features.
As shown in Fig. our model demonstrates superior performance compared traditional regression models. This fully demonstrates that GeoMancercan achieve better data understanding capabilities by effectively capturing the underlying geometric manifolds of complex multi-level data. The improvement over baselines highlights the model's 's ability to integrate both local and global geometric information in representations. Furthermore, GeoMManceronssistently produces accurate predictions across different datasets and experimental settings, reflecting the stability of its learned embeddings. In addition, Geeoannceexhhiiis an excellent convergence rate, as described in Appendix D, making it efficient to train while maintaining high performance.

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Node Classification. For node classification tasks, we evaluate our model on several widely used benchmark datasets 49: Amazon Photo, co-purchase network where nodes represent products and edges indicate frequent co-purchases; PubMed, biomedical citation network; and Physics, citation network from the arXiv Physics section;Cora, citation network of scientific publications; Citeseer, citation network of six research fields related to computer science. These datasets provide diverse scenarios for evaluating our model's ability to capture node-level semantics. The common 60%, 20%, 20% random split is adopted for the first three dataset, with the remaining adopting the standard split. We use Accuracy as the classification metric. Baselines include classic GNNs (GCN (50).
GAT [5T], GraphSAINT [52)) and graph transformers (Graphormer [53] SAN [54], GraphGPS 48 Exphormer (55), NAGphormer [5
The results are reported in Table Our model achieves state-of-the performance on the node classification task, surpassing not only GNNs but also Graph Transformers. This demonstrates the model's ability to effectively capture the conditional probability distribution for regression tasks. Furthermore, our approach outperforms LGDD, highlighting that the Riemannian diffusion mechanism successfully identifies the Riemannian manifold better suited for node classification tasks. However, since node classification only involves modeling node-level manifolds and is relatively simple, the performance improvement is less significant compared to graph regression tasks.
4.4 Ablation Study
As shown in Table we further investigate the contribution of each component in GeoMancer. Notably, the substantial improvement in molecular validity primarily results from the eeff-guidacce mechanism, which effectively exploits the complex geometry of the latent space to guide the gen- eration process. In contrast, the increase in Novelty arises from the joint effect of self-guidance and manifold-constraiie
conditional generation. Additionally, the Riemannian model significantly enhances the model's 'ss ability to capture the underlying data distribution, showing better performance FCD and NSPDK.
4.5
Visualization
To demonstrate the effectiveness of manifold selection, we visualize the decoupled manifolds and their weights on the ZINC12k dataset. As shown in Fig. our approach captures geometric priors with diverse curvature representations, dynamically leveraging them across task levels. For example, at the graph level, molecular solubility mainly influenced by hyperbolic and spherical features, with Euclidean features playing a smaller role. At the node and edge levels, each manifold contributes more evenly, with the visualization highlighting how different spaces adapt to the data's structural characteristics. These findings show that manifold selection enhances representational diversity and reveals how geometry impacts multi-level features.
Conclusion
In this work, we present GeoMancer, Riemannian graph diffusion framework that unifies generation and prediction by explicitly modeling manifold signatures in graph data. By replacing unstable exponential mappings with a Riemannian gyrokernel and decoupling multi-level features across task-specific manifolds, our method mitigates numerical instability while preserving non-Euclidean geometric priors. Moreover, manifold-constrainee diffusion and self-guided generation ensure that samples remain consistent with their underlying manifold distributions. Experiments on several datasets demonstrate the advantages of GeoMancer classification, regression, and generation.
Acknowledgement
The corresponding authors are Xingcheng Fu. This paper is supported by Beijing Natural Science Foundation (QY24129), and the National Natural Science Foundation of China 2No.62462007 and No. No.62302023), Research Fund of Guangxi Key Lab of Multi-source Information Mining & Security (MIMS24- 12),, and The Basic Ability Enhancement Program for Young and Middle-aged Teachers of Guangxi (No. 20240033)) We owe sincere thanks to all co-authors for their valuable efforts and contributions.
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A Proof and Derivation
In this section ,we proof the Proposition 3.1.
Definition Given a Riemannian manifold (M* g) and a function mapping E differential map is linear for all M. The gradient Vgy of a function is denoted
(M), the
X = (XX.))
(12)
where d :T, + R is linear for all Xz. E M. The divergence is the operator div, I (TM) + C (M) making
d((xw,) = d wy for allX E Te (TM). where T' (TM) represents the space smooth sections the tangent bundle.
(13)
The Laplacian on (M, g) is the operator defined as Ag - -div o gg Lemma A.1. For any smooth functions p, the following product rule:
v E C (M), the Laplace-Beltrami Beltrami operator Ag satisfies
A v/)
=b) &b &dgb 2
2(V gi, Vgth..
(14)
This identity generalizes the classical product rule of the Laplacian to Riemannian manifolds, wherel,.)) , represents the Riemannian inner product with the metric g.
Proof. To derive the formula for the divergence of vector field on Riemannian manifold, we begin by considering a smooth vector field
a
x=
Σ bj dx,
E EIc (TM(
We express the contraction of X with the volume form wg:
a
0
1x Wg
Dx:
Dzn
Wg
X,
Dr.
drn
0
= (-11)-- -wi
Ozi
Ozn
= (-1)²-1 / det A ... A don
X,
Om1
Dr.
b,( 1 gl
(15)
Next, compute the exterior derivative of the contraction:
12
d(lxwg) 1x(-1) det g dæy A... di, dxn
(1)" θ
(b, / det g/) A dx A A dæ; A dxn
dz,
(16)
(biV det gl)dxr ^ ... dxn
Dz,
i (biv det gl) wg.
det
Dx.
1=1
21

---

Since the divergence of a vector field is defined by div, X - d(exwy)/wg we obtain:
R a dr,
divgA X
(b, / gl).
(17)
V
Now, we generalize this formula to the case where the vector field is multiplied by a smooth function Ap:
div,(pX)) :dii, X (V)g,
(18)
This formula plays a crucial role in deriving the Laplace- Laplace-Beltrami operator's product rule. Using the definition Ag = div, Vg. we proceed to compute (p(r))
(g(())) -div, (V)(pv))
div, (4 nb Vg
(v vb) div sP
div, , ( 9 V b div (V,,, Vopio]
(V,b,
(19)
( / 2 gf V y
&49e + 2 2(V g+f gb)
Thus, we have established the product rule for the Laplace- Belltram operator Riemannian manifold.
Here we begin to proof the proposition 3.1:
Proof. According mathematical induction,
Base Case: I==1
For I = 1, the theorem reduces to the action of the Laplace- Beltrami operator on single function
Ø1- By definition, if 91 is eigenfunction of AM with eigenvalue A₁ have:
AMgi = A191.
(20)
which trivially satisfies the theorem. This establishes the base case.
Inductive Hypothesis: L==
Assume the theorem holds for L N, f,rr any product of N eigenfunctions I1s ............. 9N with corresponding eigenvalues '₁ A2, AN. the Laplace-Beltrami operator acts as:
N
A M
g
A
I
gt-
(21)
Inductive Step: L=N+1
We now prove the theorem for N + 1, we can get:
AM II gi) AM
9:)gN+1
(22)
Considering the Eq. 21 and Lemma A.I
can derive it as:
22

---

N+1
AM
II
9b) AM
Oi
9N+1
i=1
N
= 1N++1M M
9: +
II
M
V. II Dis
MMN++1
9N+1
9:
9t 19
(23)
N+11
N+1
X4)
94 + AN+1
II 9
N+1
N-1
II
9:
1=1
It can be observed that the theorem holds for L = N + Thus, we have successfully proven the proposition 3.1.
B
Preliminary
B.1 Preliminary of Riemannian Geometry
In this section, we provide a detailed introduction to Riemannian geometry.
A smooth manifold M is termed a Riemannian manifold when equipped with a Riemannian metric crucial measure that quantifies the extent of geodesic bending. For each point E M, there exists a tangent space T.M C IRd that surrounds x, where the metric is applied to determine the manifold's ' shape. The relationship between the tangent space and the manifold is established through exponential and logarithmic maps. Specifically, the exponential map at point represented as e (+) T.M + M, transforms points from the tangent space into the manifold, while the logarithmic map log (-) (-)pf(-)) serves as its inverse..
g. Curvature
In this paper, manifold space: Euclidean space (c = 0), hyperbolic space (c < 0), and spherical space (c > 0).
three geometric spaces of different curvature form a product Riemannian
Hyperbolic space. A hyperbolic space is defined as [H {xp represents the dimension and the inner product is defined as (x, y) c = 11!1 + プド RzHji) In
Rd+11 (Xp, xp) 1/c], where d
hyperbolic space, The geodesic distance between the two points is:
1 arccosh (c (x,))c).
d(x, wy)
(24)
V-c
The exponential map hyperbolic space is defined as:
x exps, (x) = cosh (V-cl|x()) xp + sinh (V-c|x[) V-cl|x|]
(25)
Sphere space. Sphere space is defined as sd. = (x, E RR++1 (Xp, xp)s = 1/c), where the inner product is the standard Euclidean inner product (x,y)s Ei Zz The geodesic distance between the two points is:
The
d(x, v) arecos (c(x, v)
(26)
Tc
The exponential map in spherical space is defined as:
expt, (x) cosh (vdlx(|) x, + sinh (ve[x()) v-cl|x|
(27)
23

---

A product manifold is the Cartesian product P = x uD x Ma where e; and d; are the curvature and dimensionality of the manifold M respectively. If we restrict be composed of the Euclidean
plane Ede hyperboloids H and hyperspheres $ of constant curvature, represent
arbitrary product manifold of model spaces as such:
Ed ci x
X
(28)
In Riemannian machine learning, each layer requires the conversion between exponential and logarithmic mappings. Although this process conceptually natural, it computationally complex and prone to instability. Notably, the equations associated with exponential mappings often lead to instability issues, such as values becoming excessively large or small, resulting in NaN (Not Number) problems. Consequently, it is frequently necessary to meticulously identify appropriate hyperparameters to mitigate these issues.
B.2 Kernel Method
Theorem B.1. Bochner's theorem (57]: For any shift-invariant continuous kernel k(x, y) k(x y defined on IR"" p(w) is its Fourier transform and (w(x) = exp(i(w, x)). then is positive definite if and only i O. In this case if we sample w according to the distribution proportional to plw), the kernel & can be expressed as:
k(x y) - p(w) expliww x- y) = k(()) Ewwp (ar)
(29)
Since both the probability distribution p(w) and and the kernel h are real, the integral unchanged when we replace the exponential with a cosine. Leveraging this property. (29) developed hyperbolic Laplacian feature function within hyperbolic space H yielding hyperbolic Laplacian feature that approximates an invariance kernel in H
n 11-1-
HyLa (z) exp
(w, 2) cos (N(w, () b).
(30)
18 generalized it to the more general Riemannian manifold. The Laplacian features of the Riemannian space are extracted by deriving the eigenfunction in the gyrovector ball G
gF (x)) = Aw, cos (N(w, x)xx + b). E G
(31)
where Aw.. exp (w, w),, (w) log 1 -
1+0|
Using the Eq. (31). generalized Fourier map dgF (a:) can be constructed estimate equidistant- invariant kernel on a Riemannian space. Moreover, this kernel can be seen as generalization of Poisson's kernel in hyperbolic space.
This kernel method can be applied to two types features: node embeddings and feature embeddings. For node embeddings, Riemannian feature representation must first be constructed for each indi- vidual node 2,. Then, the inner product (dgF (z1), ogF (2,)) between nodes v; and Uy approximates kernel function k( 2z). The optimization of is driven by the goal of learning kernel over the product space to support downstream tasks. For feature embeddings, we represent normalized node feature X E R and inital Riemannian embedding 2;- Then it can be
effective
calculated by Σd & (z+). Its inner product between two features is:
d Σx x (z)))((2)) Σ
X X.ndeF
(xX)
(32)
[=1
k.1=1
24

---

B.3 Preliminary of Diffusion Model
Denoising Diffusion Probabilistic Models (DDPMs) are class of generative models based on diffusion processes that generate high-quality samples by simulating a step-by step-by-step denoising process, transforming noise into realistic data. The core idea of DDPMs to construct generative model through two complementary processes: forward process that gradually adds noise to data, and reverse process that learns to iteratively denoise and reconstruct the data distribution.
Forward Process: the forward process ((2[z) is the parierne Markov process:
(2xx/x) = (a I),, wherea = 1 1 a3
(33)
For intermediate steps, the transition between noise levels is given by:
((xx|2x) (N((ax/xxxx I whered X (1 cベー~aOR
(34)
Reverse Process: The reverse process is generative model that starts from prior distribution Pa(Z (Z = (0, I) and reconstructs the real data distribution by iteratively denoising the data. The reverse transition modeled as:
pa (2)) Ni (xx,) (1)1))
(1 " (1)
(35)
where (2A, e (ax/xxxxx)) + (1- ex and o xIja (1 (1-c~-X'os
The model is trained predict the noise each step by minimizing the objective function, which measures the discrepancy between the predicted noise and the actual noise added during the forward process. Specifically, the training objective is formulated as: E,,x [ileo(z)) E
To enhance the controllability and quality of generated samples, DDPMs often employ guidance mechanisms:
Classifier Guidance: Classifier guidance (58 introduces conditional diffusion process by leverag- ing a pre-trained classifier po (c/z>). The guided noise prediction is given by:
(Z(2), c) = co(2). c) wox 22, loglo(()zz) = - ZA, [lo p(zxlc) + w log po(c/z))) (36)
where w controls the strength of the guidance. This can be interpreted as:
Zy, (z,, c) 0o V., logp(zz)c) + w log (c
(37)
Classifie--free Guidance: Classifier-free guidance (59] eliminates the need for separate classifier by jointly training conditional and unconditional models. The guided noise prediction is computed as:
Ez c) (1 + w)eo(z,, c) wee(zx)
(38)
where w is a hyperparameter that controls the strength of conditional guidance. This approach simplifies the training process while maintaining high controllability.
Manifold-Constranned Classifier-free Guidance: inverse problem:
20 model condition generation as solving an
min (ode (ads(x) VooooVo vI vI- āze, =el
(39)
This implies that goal to identify solutions on the clean manifold M that optimally aligns with the condition The resulting sampling process from reverse diffusion is then given by
Rt-1
vā--1 (3z 7o V (ade + vI- a.-1Eg.
(40)
25

---

Table 5: Overview of the datasets and metrics used in the paper.
Dataset
#Graphs
Avg. #nodes
Avg. #edges
Prediction Level
Task
Metric
QM9
130,000
18.0
37.3
graph
regression
Mean Absolute Error
ZINC
12,000
23.2
24.9
graph
regression
Mean Absolute Error
Cora
2708
10,556
node
7-way classification
Accuracy
PubMed
19717
88648
node
3 w-w classification
Accuracy
Photo
7650
238,162
node
8- way classification
Accuracy
Physics
34,493
495,924
node
5-way classification
Accuracy
we can equivalently write the loss as (x))( =
xell? so can be written as:
I-a;
#t-1 - (6g M(Be f2)) + v1 āc-160
(41)
C Experiment details
Diffusion Process. In all experiments, we employ diffusion process with T = 1000 diffusion steps, diffusion steps, parameterized by linear schedule for a, and corresponding decay for â₁. For inference, we adopt the DDPM framework.
Model Architechture. For the node-level task, including all node classification datasets, we use an MPNN-centric model as the encoder, which consists of GCN, GIN and GAT. Typically, we use a 5-layer architecture with residual connections and normalization layers to facilitate optimization. The optimizer Adam W, with learning rate of Ic-3 and a weight decay of le-5. The dimension of final hidden layer is set to 4. Since the node classification task only considers the node features, only study the reconstruction loss and Riemann decoupling of the node features. Here, we initialize the curvature of product manifold consisting of three gyroscopic space vectors with curvature -1,0,1, respectively, with dimension 4. The random Laplacian map is then calculated separately and the underlying spatial features transformed into Riemannian Spaces. Finally, they weighted by linear layer and then decoded by a linear layer.
For graph-level tasks, such as graph generation and regression, we employ edge-enhanced graph transformers as the backbone network, incorporating position embeddings to capture structural information. For the decoder, we design Riemannian decoupling layer for each subtask, including node-level reconstruction, edge-level reconstruction, and graph-level property reconstruction. After the Riemannian decoupling layers, linear layer aggregates representations from each Riemannian space, and the final results are produced through an additional linear layer. The optimizer is Adam W, with learning rate of le-4 and a weight decay of 1e-6. The dimension of final hidden layer is set to 16 or 32.
D Experiment Analysis
D.1 Analysis of Riemannian Autoencoder
To better analyze the impact of our Riemannian autoencoder, we conduct a detailed evaluation on the ZINC12 graph regression task. Figure D.I the convergence speed of MAE during model training. We compare our approach with the model without Riemannian block. The learning rate setting of the optimizer remains consistent, all being le-5. The results show that the convergence speed of this model is significantly accelerated. Furthermore, our final convergence result is superior to removing the Riemannian block. We attribute this improvement to the Riemannian decoupler, which effectively decouples each feature onto the appropriate product manifold, thereby promoting more efficient learning.
E Limitation
Due to the limitation of computing resources, our model is not large enough and lacks the verification of the scaling law of the diffusion method graph tasks.
26

---

MAE of Zinc Regression
GrsMancemeh Nemaniinn Cocllancer
1071
200 400
300
D00
1003
Figure 5: MAE during the training in graph regression.
F Broader Impact
This paper aims to advance the field of graph generation technologies. Our work contributes to the field Machine Learning and has many potential societal consequences. It may play an important role in understanding fields such drug generation and recommendation systems based graph structures from a geometric perspective. However, we believe that there are no negative impacts that need clarification.
27


Reviews:
[{"review_id": "72OgZtHMn9", "reviewer": "Reviewer_bYtP", "summary": "Graph diffusion models offer a unified framework for tackling both graph generation and prediction tasks. To account for the non-Euclidean nature of graph-structured data, this paper proposes GeoMancer, a Riemannian diffusion approach that captures the intrinsic geometric nature of multi-level features. GeoMancer integrates Riemannian kernel autoencoders with manifold-constrained diffusion processes.  Experiments across multiple tasks demonstrate that GeoMancer achieves strong performance in both generation and prediction tasks.", "questions": "1. Could the authors provide more introductions and analyses about this kernel method?\n2. How are the dimensions and manifolds selected?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "## Strength:\n\n1. The motivation of this work is well-grounded. While prior researches[1] on non-Euclidean graph data have primarily focused on modeling the geometric structure of individual features, this paper takes a significant step forward by exploring the complex geometry spaces of multi-level features. \n2. The Riemann gyroscope kernel method is interesting. As the authors point out, the traditional exponential map often suffers from instability during optimization and requires meticulous parameter tuning. The gyroscope-based alternative offers a promising and potentially more robust replacement.\n3. The paper provides solid theoretical grounding, including detailed derivations that justify the use of the Riemannian gyroscope kernel to model geometric structures over product manifolds. \n\n## Weakness:\n1. There is a lack of a more detailed introduction to the Riemannian kernel method, such as how much its distortion is and the influence of the number of basis functions on the mapping.\n2. The author of the manifold signature only considered setting the learnable curvature to obtain it, but did not make a more detailed design for the selection of dimensions and manifolds.\n3. In order for readers to better understand the background of geometric deep learning[1], more preliminary about Riemannian geometric methods should be added in the article and citations of relevant literature should be supplemented like [2-4] .\n\n---\n\n[1] Geometric deep learning: going beyond Euclidean data.\n\n[2] Neural embeddings of graphs in hyperbolic space\n\n[3] Computationally tractable riemannian manifolds for graph embeddings.\n\n[4] Mixed-curvature Variational Autoencoders.", "was_revised": true, "final_justification": "I have read the rebuttal as well as the comments from other reviewers and found no major flaws. Concerns regarding the background introduction and learnable curvature are well addressed.  Meanwhile, I suggest the author incorporate the several revisions for their manuscript.", "rebuttal": "Dear Reviewer bYtP,\n\nWe sincerely thank the reviewer for the detailed comments and insightful questions. Our response to your comments one by one as follows.\n\n> W1: There is a lack of a more detailed introduction to the Riemannian kernel method, such as how much its distortion is and the influence of the number of basis functions on the mapping.\nQ1. Could the authors provide more introductions and analyses about this kernel method?\n> \n\nReply to W1&Q1: There are many studies that use kernel methods or Laplace operators to analyze data manifold learning [1][2][3][4]. **This is an interesting mapping method that it essentially preserves the isometric invariance between two points while the feature still remain in Euclidean space.** [1] showed that the Laplace eigenfunctions can approximate the estimation of the kernel and replace the exponential mapping in hyperbolic space. Although it is not necessarily mathematically elegant enough, it is a very important property for model training, which means that we can simplify many additional Riemannian operators.\n\n[1] Random Laplacian Features for Learning with hyperbolic space. ICLR 2023\n\n[2] Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning. AAAI 2024\n\n[3] Analysis on Manifolds via the Laplacian.\n\n[4] Analysis of the Laplacian on the complete Riemannian manifold.\n\n---\n\n> W2: The author of the manifold signature only considered setting the learnable curvature to obtain it, but did not make a more detailed design for the selection of dimensions and manifolds.\nQ2: How are the dimensions and manifolds selected?\n> \n\nReply to W2: As we explained in Section 3.2, for the learnable curvature, we adopt the gyrovector ball as the underlying manifold. Regarding the dimensionality, we currently lack an effective way to optimize it during training and can only set it as a fixed hyperparameter in advance. It is such a valuable comment worth exploring in the future.\n\n---\n\n> W3: In order for readers to better understand the background of geometric deep learning[1], more preliminary about Riemannian geometric methods should be added in the article and citations of relevant literature should be supplemented like [2-4] .\n> \n\nReply to W3: We sincerely appreciate the reviewer’s valuable suggestions.  These recommendations are instrumental in strengthening the background of our work. We will carefully incorporate this feedback by expanding the background discussion and adding relevant citations in the revised version."}, {"review_id": "4S7WuMw4J8", "reviewer": "Reviewer_c5kY", "summary": "This paper proposed a method GeoMancer which approaches graph-structured data from a fundamentally different angle by modeling multi-level latent geometry in curved space.", "questions": "I am not clear about the ability of the diffusion model to handle heterogeneous manifolds. Therefore, I hope the author can discuss the following issues with me in detail. I will consider improving my score.\n\nQ1: In Fig1(b),in what aspects is the complexity of multi-level data manifolds reflected? Are their curvatures the same? For example, for node classification or link prediction tasks, are node features and edge features completely different?\n\nQ2: Can the diffusion model effectively capture data manifolds? Are there any previous studies or theoretical derivations?\n\nQ3: What is the computational complexity of the gyroscope kernel method mentioned in the paper?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strength**\n1. The paper introduces a novel approach by leveraging Riemannian geometry to model the feature space of multi-level graph data, offering a novel perspective on geometric representation in graph learning.\n2. The integration of a self-guided mechanism with manifold-constrained diffusion is particularly innovative. This design effectively utilizes both explicit and implicit geometric cues to enhance representation quality.\n3. The visualization experiments are both informative and aesthetically presented, clearly demonstrating the model’s ability to capture and interpret geometric structures.\n\n**Weakness**\n1. The explanation of Figure 1(b) could be expanded. It remains unclear whether different feature levels correspond to distinct curvatures, dimensions, or structural complexities, and how these aspects interact.\n2. I'm not very familiar with the ability of diffusion to capture data manifolds. The paper lacks a thorough discussion on how well the diffusion process captures data manifolds, particularly in the context of modeling multiple, potentially heterogeneous, geometric levels.\n3. The computational cost of the proposed method is not addressed.  A discussion of its time complexity would help assess the scalability and practicality of the approach.", "was_revised": true, "final_justification": "Embedding graph geometry through Riemannian geometry and curvature computation is highly challenging and interesting. The authors have done an excellent job. After reading other reviews, in my opinion, all major issues have been well addressed. Therefore, I keep my score at 5.", "rebuttal": "Dear Reviewer c5kY,\n\nWe sincerely thank the reviewer for the detailed comments and insightful questions. Our response to your comments one by one as follows.\n\n---\n\n> W1: The explanation of Figure 1(b) could be expanded. It remains unclear whether different feature levels correspond to distinct curvatures, dimensions, or structural complexities, and how these aspects interact. Q1: In Fig1(b),in what aspects is the complexity of multi-level data manifolds reflected? Are their curvatures the same? For example, for node classification or link prediction tasks, are node features and edge features completely different?\n> \n\nReply to W1: On the one hand, features at different levels exhibit varying curvatures. We calculated the ricci curvature of different features as reported in Table 1. This indicates that the spatial curvatures of features across levels are distinct, highlighting the complexity of the data manifold. On the other hand, from the task perspective, different tasks encourage  different manifolds. Node classification encourages nodes with the same label to learn similar features, whereas link prediction primarily emphasizes structural features. Thus, the differing task objectives further contribute to the complexity of the data manifold.\n\nTable 1 curvature of multi-level embeddings\n\n| Node-level | Edge-level | Graph-level |\n| --- | --- | --- |\n| 0.3 | -0.6 | 0.1 |\n\n---\n\n> W2:  I'm not very familiar with the ability of diffusion to capture data manifolds. The paper lacks a thorough discussion on how well the diffusion process captures data manifolds, particularly in the context of modeling multiple, potentially heterogeneous, geometric levels. Q2: Can the diffusion model effectively capture data manifolds? Are there any previous studies or theoretical derivations?\n> \n\nReply to W2: To our knowledge, many studies[1][2][3] have demonstrated that diffusion models can effectively capture data manifolds. **The key insight is that the score function learned by a diffusion model tends to point in the normal direction towards the data manifold.** This property has been further leveraged to investigate intrinsic dimensions and other geometric characteristics of data manifolds.\n\n[1] Diffusion Models Encode the Intrinsic Dimension of Data Manifolds.\n\n[2] Adaptivity of Diffusion Models to Manifold Structures.\n\n[3] Convergence of Diffusion Models Under the Manifold Hypothesis in High‑Dimensions.\n\n---\n\n> W3: The computational cost of the proposed method is not addressed. A discussion of its time complexity would help assess the scalability and practicality of the approach. Q3: What is the computational complexity of the gyroscope kernel method mentioned in the paper?\n> \n\nReply to W3: Here, we analyze the complexity of our Riemannian mappings from theory and experiment.\n\nTheory: Suppose that a general Fourier mapping $\\phi_{\\mathrm{gF}}(x)$ consists of m eigenfunctions and the number of representation vectors $V$ in different Riemannian spaces is $n$. First, the computation of each eigenfunction $\\mathrm{gF}_{\\boldsymbol{\\omega},b,\\lambda}^{\\kappa}(\\boldsymbol{x})$ involves only simple operations with the time complexity of $O(1)$. Thus ,the whole process of calculating $\\overline{V}$ is O(m*n).  It is worth noting that since the settings of m and n are very small, typically m*n<10. the process is therefore very fast. \n\nExperiment: We report the time consumption in Table 1. Experimental results show that our mapping remains almost identical to the original method without significant time loss.\n\nTable 1 time consumption on photo dataset in one epoch\n\n| Method | Time(s) |\n| --- | --- | \n| GeoMancer(w/o riemannian) | 1.5 |\n|GeoMancer | 1.6 |"}, {"review_id": "vvxc49najm", "reviewer": "Reviewer_aMiG", "summary": "This work introduces a geometry-based diffusion mechanism that adapts to the intrinsic curvature of multi-level graph features, from node, edge and graph feature. It first embeds multi-level graph data into a product manifold using a specially designed gyroscopic kernel.", "questions": "What is the denoising process like under the self-guided mechanism? Can the author provide visualization results? Why not use other Riemannian diffusion models? Is there any relationship between the pseudo-labels generated by molecules and their chemical properties? Can the author calculate the jaccard coefficients of the two categories after clustering based on their chemical properties?", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strength:\nS1: The motivation behind this work is clearly articulated, with well-justified explanations for the design of each component and the specific challenges they address.\nS2: The article is grounded in solid theoretical foundations, offering rigorous proofs on the construction and decoupling of product manifolds through the use of gyroscopic kernels.\nS3: The self-guidance mechanism based on geometric features is particularly well-conceived. Experimental results suggest it plays a significant role in enhancing generative performance.\nS4: The experimental evaluation is comprehensive, with the model’s effectiveness validated across multiple tasks. The ablation studies convincingly demonstrate the contribution of each individual component.\nWeakness:\nW1: The lack of visual representation of the generation process makes it impossible to vividly demonstrate how this self-guiding mechanism is generated.\nW2: The paper lacks a comparative discussion of existing Riemannian diffusion approaches[1][2]. It is important to explain why alternative Riemannian diffusion models were not adopted or compared.\nW3: The self-guiding mechanism warrants a deeper analysis.  For instance, it remains unclear whether the generated labels reflect underlying chemical properties of the molecules and, if so, how such relationships are captured or influenced by the guidance strategy.\n[1] Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes.\n[2] Riemannian Diffusion Models.", "was_revised": true, "final_justification": "I have carefully reviewed the authors’ rebuttal, and all of my previous questions have been clearly addressed. Thus, I will maintain my score.", "rebuttal": "Dear Reviewer aMiG,\n\nWe sincerely thank the reviewer for the detailed comments and insightful questions. Our response to your comments one by one as follows.\n\n> W1: The lack of visual representation of the generation process makes it impossible to vividly demonstrate how this self-guiding mechanism is generated. Q1: What is the denoising process like under the self-guided mechanism?\n> \n\nReply to W1: We are very sorry that due to the latest rebottal notification, we are unable to provide the relevant figures. The self-guiding mechanism enables the model to generate features that are more similar to the original representation. In the later stage of denoising, the features of the same label will be closer. However, if the self-guiding mechanism is removed, the generated features will be scattered at various locations. We will add relevant visualization content in the revised version.\n\n---\n\n> W2: The paper lacks a comparative discussion of existing Riemannian diffusion approaches[1][2]. It is important to explain why alternative Riemannian diffusion models were not adopted or compared. Q2: Why not use other Riemannian diffusion models?\n> \n\nReply to W2: We clarify that one of the benefits for this design is to consider the efficiency issue of implementing diffusion directly on Riemannian geometry. **Benefiting from the fact that the feature remains in the Euclidean space after the generalized Fourier mapping, this design does not require a complex execution of the diffusion process on the Riemannian space no matter whether to use Riemannian autoencoder.** In addition, the Riemannian kernel method can indirectly enhance the representation of the latent space and thus improve the performance of diffusion model. This flexibility allows our architecture to be compatible with pre-trained graph latent diffusion models in the future without training a specialized Riemannian diffusion model from scratch.\n\n---\n\n> W3:  The self-guiding mechanism warrants a deeper analysis. For instance, it remains unclear whether the generated labels reflect underlying chemical properties of the molecules and, if so, how such relationships are captured or influenced by the guidance strategy. Q3: Can the author calculate the jaccard coefficients of the two categories after clustering based on their chemical properties?\n> \n\nReply to W3: This is indeed a very valuable question. We conducted an analysis on the QM9 dataset to examine the relationship between clustering labels and the chemical attribute $\\mu$. Since the chemical attribute values are continuous numerical data, we first sorted them based on their magnitudes and divided them into corresponding sets. We then computed the Jaccard index between these sets and those derived from the clustering labels. **The average Jaccard score obtained was 0.1286, indicating that the self-guiding mechanism is only weakly related to the underlying chemical properties.** \n\nAlthough there exists a certain correlation between molecular structure and chemical properties, the representations used for generation may exhibit weaker correlations with chemical attributes, as they focus on different levels of information. We believe that the pseudo-labels produced by the self-guiding mechanism are primarily influenced by structural topology, which include the reconstructed representation information of nodes and edges."}, {"review_id": "0aJa645Ylu", "reviewer": "Reviewer_8zMr", "summary": "The paper introduces GeoMancer, a framework for graph generation and prediction using Riemannian diffusion. This framework explicitly models various geometric curvatures within distinct task-specific manifolds. To address numerical instability during the encoding process, the authors use an isometry-invariant Riemannian \"gyrokernel\" instead of the traditional exponential map. To avoid manifold drift during generation, they implement a manifold-constrained diffusion process along with a self-guided strategy for unconditional sampling. The experiments conducted on multiple benchmark datasets demonstrate advancements over both Euclidean and previous hyperbolic graph generation methods.", "questions": "What is the computational cost of computing the gyrokernel?", "limitations": "Yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "I am not very familiar with ideas of diffusion, but I have done my best to identify the strengths and weaknesses of the paper. I hope this might help the authors improve their paper slightly. \n\n**Strengths**\n\n- The motivation of the paper is clear, and it is to be followed by good empirical results.\n- I found the use of the Riemannian gyrokernel in place of the exponential map an interesting approach to address numerical instability, which deviates from the traditional approach in the literature.\n- The results in the paper seem to consistently improve multiple graph-generation and prediction benchmarks.\n\n**Weaknesses**\n\n- The authors could potentially add some additional benchmarks [1] to increase the validity of their results.\n\n[1] Qin, Yiming, et al. \"Defog: Discrete flow matching for graph generation.\" arXiv preprint arXiv:2410.04263 (2024).", "was_revised": true, "final_justification": "I am not an expert in this topic, but to the best of my knowledge, I think the authors did a good job in the paper.", "rebuttal": "Dear Reviewer 8zMr,\n\nWe sincerely thank the reviewer for the detailed comments and insightful questions. Our response to your comments one by one as follows.\n\n---\n\n> W1: The authors could potentially add some additional benchmarks [1] to increase the validity of their results.\n> \n\nReply to W1: Thank you for pointing out the model and benchmarks we had previously overlooked. We will cite this work in the revised version and try to add the relevant metrics to the MOSES dataset, and incorporate the baseline model.\n\n---\n\n> Q1: What is the computational cost of computing the gyrokernel?\n> \n\nReply to Q1: Here, we analyze the complexity of our Riemannian gyrokernel mappings from theory and experiment.\n\nTheory: Suppose that a general Fourier mapping $\\phi_{\\mathrm{gF}}(x)$ consists of m eigenfunctions and the number of representation vectors $V$ in different Riemannian spaces is $n$. First, the computation of each eigenfunction $\\mathrm{gF}_{\\boldsymbol{\\omega},b,\\lambda}^{\\kappa}(\\boldsymbol{x})$ involves only simple operations with the time complexity of $O(1)$. Thus ,the whole process of calculating $\\overline{V}$ is O(m*n).  It is worth noting that since the settings of m and n are very small, typically m*n<10. the process is therefore very fast. \n\nExperiment: We report the time consumption in Table 1. Experimental results show that our mapping remains almost identical to the original method without significant time loss.\n\nTable 1 time consumption on photo dataset in one epoch\n\n| Method | Time(s) |\n| --- | --- |\n| GeoMancer(w/o riemannian） | 1.5 |\n| GeoMancer | 1.6 |"}]

Author rebuttal:

Meta-review: This paper introduces GeoMancer, a Riemannian diffusion framework for graph generation and prediction. The main contributions are: (1) a gyroscope-based Riemannian kernel that mitigates numerical instability compared to the exponential map, (2) a manifold-constrained diffusion process with a self-guided mechanism that ensures generated graphs remain aligned with manifold structures, and (3) comprehensive empirical results showing improvements over Euclidean and hyperbolic baselines across multiple tasks.  Reviewers highlighted the novelty of modeling multi-level geometric structures, the sound theoretical foundation, and the strong empirical validation. Notably, the unified framework proposed by this work closes the gap between two important problems via an interesting perspective.

For the camera-ready version, I encourage the authors to (1) expand the introduction and related work on Riemannian kernel methods and geometric deep learning (see the suggested literature by Reviewer bYtP), (2) provide more discussion and visualization of the self-guided mechanism and its implications for interpretability, and (3) include complexity analysis to further strengthen the practical impact of the framework.
Decision: Accept (poster)