Title: Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graph Generation

Hyperbolic Graph Learning

Riemannian Manifold

Graph 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.


Peer reviews:
comments: 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.

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. 

**Strengths**

- The motivation of the paper is clear, and it is to be followed by good empirical results.
- 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.
- The results in the paper seem to consistently improve multiple graph-generation and prediction benchmarks.

**Weaknesses**

- The authors could potentially add some additional benchmarks [1] to increase the validity of their results.

[1] Qin, Yiming, et al. "Defog: Discrete flow matching for graph generation." arXiv preprint arXiv:2410.04263 (2024).

QUESTIONS:
What is the computational cost of computing the gyrokernel?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-23

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\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.\n\nQUESTIONS:\nWhat 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?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed a method GeoMancer which approaches graph-structured data from a fundamentally different angle by modeling multi-level latent geometry in curved space.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\nI 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?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nGraph 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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.\n\nQUESTIONS:\n1. Could the authors provide more introductions and analyses about this kernel method?\n2. How are the dimensions and manifolds selected?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-01"}