Title: Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry
Venue: NeurIPS 2025
Accepted: yes
Keywords: brain connectivity

Riemannian geometry

covariance matrix

fmri

diffeomorphism

generative model
Abstract: Generating realistic brain connectivity matrices is key to analyzing population heterogeneity in brain organization, understanding disease, and augmenting data in challenging classification problems. Functional connectivity matrices lie in constrained spaces—such as the set of symmetric positive definite or correlation matrices—that can be modeled as Riemannian manifolds. However, using Riemannian tools typically requires redefining core operations (geodesics, norms, integration), making generative modeling computationally inefficient. In this work, we propose DiffeoCFM, an approach that enables conditional flow matching (CFM) on matrix manifolds by exploiting pullback metrics induced by global diffeomorphisms on Euclidean spaces. We show that Riemannian CFM with such metrics is equivalent to applying standard CFM after data transformation. This equivalence allows efficient vector field learning, and fast sampling with standard ODE solvers. We instantiate DiffeoCFM with two different settings: the matrix logarithm for covariance matrices and the normalized Cholesky decomposition for correlation matrices. We evaluate DiffeoCFM on three large-scale fMRI datasets with more than $4600$ scans from $2800$ subjects (ADNI, ABIDE, OASIS‑3) and two EEG motor imagery datasets with over $30000$ trials from $26$ subjects (BNCI2014‑002 and BNCI2015‑001). It enables fast training and achieves state-of-the-art performance, all while preserving manifold constraints.
Code: https://github.com/antoinecollas/DiffeoCFM


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new method for generating realistic brain connectivity matrices, i.e. symmetric positive definite (SPD) and correlation matrices. Generative modeling with such data requires one to take into account their Riemannian geometries, which is a computationally expensive task. To reduce the computational complexity of the task, the authors focus on Pullback metrics which turns Riemannian computations into Euclidean computations. Generative modeling can then be easily and efficiently performed on Euclidean space. Experimental results on fMRI and EEG motor imagery datasets show the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The proposed method relies on Pullback metrics which are well-established and have proven efficient in extending traditional machine learning models to the Riemannian settting. \n- Theoretical justifications are provided in the supplemental material. \n- The considered applications and datasets are relevant. Experimental results are promising.\n\nWeaknesses:\n- Important baselines are missing, making it hard to evaluate real benefits of the proposed method w.r.t. existing works.\n\nQUESTIONS:\n- State-of-the-art works for generative modeling on Riemannian manifolds such as [9,14] use NLL (Negative log-likelihood) scores \nand NFE (number of function evaluations) as evaluation metrics. I think those metrics are relevant in the context of the present work\nand are worth investigating (e.g. by using synthetic data as in [9,14]). This would further demonstrate the advantage of the proposed \nmethod w.r.t. [9,14]. Currently, those metrics are not considered in the experiments. Could the authors elaborate on this choice of \nevaluation metrics ?\n\n- The authors do not use the methods in [9,14] for comparison with the proposed method. I carefully checked those references [9,14] and noticed that both works are applicable (and can be easily adapted) to SPD manifolds, making them strong baselines. The authors of [9] even included SPD manifolds as part of their code. However, the authors  state in Appendix G that \"its current implementations are limited to a few specific manifolds such as spheres and tori, which are not directly applicable to our task.\". Could the authors elaborate on their choice of baselines ?\n\n- The work in [A] is closely related to the present work but is missing in the references. The proposed generative method \nfor SPD manifolds can be directly derived from [A], since SPD manifolds with Log-Euclidean metrics have Lie group structure. \nI think it should be cited in the related works and eventually be used in comparison with the proposed method.\n\n**References**\n\nLuca Falorsi, Pim de Haan, Tim R. Davidson, Patrick Forré: Reparameterizing Distributions on Lie Groups. AISTATS 2019: 3244-3253.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

comments: SUMMARY:
The paper studies the problem of generating brain connectivity with Riemannian flow matching. Authors provide comprehensive descriptions on the methodology. Experiments on fMRI and EEG data demonstrate the effectiveness of the proposed work.

STRENGTHS AND WEAKNESSES:
Strengths:
- The problem being resolved is interesting and of great value. 
- The idea of optimizing over objective in the Euclidean space from perspective of using pullback metric is interesting. It simplified the problem and better utilized the power of neural networks.
- The paper is well-written with comprehensive details.
- Experiments show the proposed work can generate decent results on real data.

Weakness: 
Since the generation is on connectivity where the connectivity matrices are obtained from brain signal data (such as fMRI), the way of defining connectivity could possibly influence the "ground-truth" thus introduce bias. It would be great if authors can discuss more on it.

QUESTIONS:
Please see above

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces DIFFEOCFM, a novel conditional flow matching framework designed for efficient generative modelling on Riemannian manifolds, specifically targeting brain connectivity matrices (SPD and correlation matrices). DIFFEOCFM utilises global diffeomorphisms to map manifold data to Euclidean spaces, thus enabling simpler computations and avoiding expensive manifold-specific operations. Experiments were conducted on three large-scale fMRI datasets (ADNI, ABIDE, OASIS-3) and two EEG motor imagery datasets (BNCI2014-002, BNCI2015-001), demonstrating superior performance in both realism and downstream classification tasks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nQuality: The manuscript is technically sound, and the mathematical framework is well-constructed. The pullback approach significantly reduces computational overhead without compromising geometric fidelity, which is convincingly supported by theoretical propositions (Propositions 1-3).\nClarity: The manuscript is clearly structured, thoroughly described, and easy to follow, with well-defined notation and coherent transitions between sections.\nSignificance: DIFFEOCFM significantly advances generative modelling on Riemannian manifolds by simplifying complex geometric computations. This methodological innovation is impactful, particularly in fields like neuroimaging, where preserving manifold constraints is crucial.\nOriginality: The paper's primary contribution—leveraging pullback geometry via global diffeomorphisms—is novel and well-motivated, clearly delineating itself from prior works such as SPD-DDPM and traditional Riemannian CFM.\n\nWeaknesses:\n1. Key implementation and experimental details—e.g., hyperparameters and projection analyses—are buried in the appendices (J, L); a concise summary in the main text would greatly improve self-containment.\n2. Algorithms 1 and 2 omit pseudocode for computing crucial terms such as Dϕ(x) and Dϕ⁻¹(z), which are necessary for understanding Eq. (7) and ensuring reproducibility.\n3. No ablation is reported on ODE step size h or network depth, so the robustness–efficiency trade-off of DIFFEOCFM remains unclear.\n\nQUESTIONS:\n1. The vector field transformation in Eq. (7) relies on computing the differential of the diffeomorphism and its inverse. What are the computational costs and numerical stability issues, particularly for high-dimensional Corr matrices? Have the authors observed any instability in Dϕ(Σ) when Σ is near singular?\n2. How would the method extend to manifolds without known diffeomorphic embeddings (e.g., the Stiefel manifold)? Could approximate embeddings or learned bijections (e.g., neural ODEs) be considered?\n3. During back-propagation through Dϕ, did you encounter exploding/vanishing gradients?\n4. The CSP maps in Figure 2(b) are presented with colour bars and topographic projections. It would be helpful to also report quantitative similarity metrics (e.g., structural similarity index) between real and synthetic filters for more rigorous comparison.\n5. α-precision and β-recall are geometry-agnostic. Could authors provide a discussion of a manifold-aware version (e.g., geodesic-based recall) in future work?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces DIFFEOCFM, a method designed to address the challenge of generating realistic brain connectivity matrices, such as symmetric positive definite or correlation matrices, reside on Riemannian manifolds. Traditional manifold-aware generative models incur heavy computational overhead. DIFFEOCFM uses a global diffeomorphism to map these manifolds to Euclidean space, enabling efficient application of standard generative techniques while ensuring geometric constraints are maintained. Evaluated on 3 fMRI datasetsand 2  EEG datasets, DIFFEOCFM demonstrates superior performance in quality and classification metrics than baseline methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is well motivated and well written. The integration of pullback geometry with conditional flow matching is a novel approach that simplifies generative modeling on Riemannian manifolds while preserving geometric constraints.\n- DIFFEOCFM achieves state-of-the-art results on multiple fMRI and EEG datasets, with significant improvements in quality and classification metrics over baselines.\n- The biological plausibility analysis adds credibility, and generated samples are neurophysiologically meaningful, supporting applications in brain disease research and classification tasks.\n\nWeaknesses:\n- The impact of diffeomorphism choice (log-Euclidean vs. affine-invariant, other Cholesky normalizations) or vector-field capacity is not discussed.\n- Comparisons with other true manifold baselines such as SPD-DDPM in terms of quality and runtime would be helpful.\n- Additional literature could be discussed, such as Jo et al. and Kapusniak et al., which also generate data on manifolds through learned or mixture-based interpolants.\n\nJo et al. Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes. ICML 2024\n\nKapusniak et al. Metric Flow Matching for Smooth Interpolations on the Data Manifold. NeurIPS 2024\n\nQUESTIONS:\n1) Discussion on scalability to high-dimensional manifolds would be helpful.\n2) The choice of a two-layer 512-unit MLP is not fully justified. Could the authors try transformer-style vector fields or neural ODEs, and also explore alternative pullbacks?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}