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Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry
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Proposes DiffeoCFM, which performs conditional flow matching on SPD and correlation matrix manifolds by pulling back Euclidean geometry through global diffeomorphisms (matrix log and normalized Cholesky), so training and sampling run in Euclidean space.
Problem
Generating realistic brain connectivity matrices (SPD covariance or correlation matrices) with manifold constraints is computationally expensive with Riemannian flow matching, which requires geodesics, norms and manifold integration.
Approach
Use pullback metrics induced by global diffeomorphisms (matrix logarithm for SPD, normalized Cholesky for correlation) so that Riemannian CFM is exactly equivalent to standard Euclidean CFM applied to transformed data; sampling then maps back by the inverse diffeomorphism, guaranteeing manifold-valid outputs without post-processing.
Key results
- On fMRI datasets DiffeoCFM obtains the highest alpha,beta-F1 (e.g., 0.68 on ADNI vs 0.04 for DiffeoGauss) and best CAS F1 across ABIDE, ADNI, OASIS-3.
- Compared to RiemCFM on EEG, DiffeoCFM delivers similar performance while training about 8x faster and sampling about 10x faster (e.g., 319.83 s vs 2753.93 s training on BNCI2015-001).
- TriangDDPM and TriangCFM achieve near-zero alpha,beta-F1 after projection onto the manifold (e.g., 0.00 on ABIDE), showing post-hoc projection degrades quality.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DiffeoCFM instead of DiffeoGauss | Brain connectivity generation · ABIDE | α,β-F1: 0.38 ± 0.06 -> 0.59 ± 0.08 | Helped |
| Use DiffeoCFM instead of DiffeoGauss | Brain connectivity generation · ADNI | α,β-F1: 0.04 ± 0.02 -> 0.68 ± 0.06 | Helped |
| Use DiffeoCFM instead of DiffeoGauss | Brain connectivity generation · OASIS-3 | α,β-F1: 0.38 ± 0.04 -> 0.44 ± 0.04 | Helped |
| Use DiffeoCFM instead of RiemCFM | Brain connectivity generation · BNCI2014-002 | α,β-F1: 0.63 ± 0.03 -> 0.62 ± 0.02 | No clear effect |
| Use DiffeoCFM instead of RiemCFM | Brain connectivity generation · BNCI2015-001 | α,β-F1: 0.88 ± 0.01 -> 0.89 ± 0.01 | Helped |
| Use DiffeoCFM instead of RiemCFM | Brain connectivity generation · EEG datasets | training time: 8× faster | Helped |
| Use DiffeoCFM instead of RiemCFM | Brain connectivity generation · EEG datasets | sampling time: 10× faster | Helped |
| Add manifold projection to TriangCFM outputs (ablation) | TriangCFM · ABIDE | α,β-F1: -0.50 | Hurt |
| Add manifold projection to TriangCFM outputs (ablation) | TriangCFM · ADNI | α,β-F1: -0.69 | Hurt |
| Add manifold projection to TriangCFM outputs (ablation) | TriangCFM · OASIS-3 | α,β-F1: -0.64 | Hurt |
| Add manifold projection to TriangCFM outputs (ablation) | TriangCFM · BNCI2014-002 | α,β-F1: -0.19 | Hurt |
| Add manifold projection to TriangCFM outputs (ablation) | TriangCFM · BNCI2015-001 | α,β-F1: -0.09 | Hurt |
Surprising findings
- Post-hoc projection of triangular-part generative models onto the manifold severely degrades sample quality, e.g. alpha,beta-F1 drops of up to -0.69 on ADNI.
- The simple DiffeoGauss baseline is competitive in ROC-AUC on several datasets despite poor quality metrics.
Limitations
- Relies on a global diffeomorphism to Euclidean space, which does not exist for all manifolds (e.g., Stiefel).
- Higher parcellation granularity makes the problem harder; manifold dimension and sample complexity grow with region count.
- Evaluation metrics like alpha-precision and beta-recall are geometry-agnostic and may miss neurophysiological structure.
Experimental setup
Datasets: ABIDE ADNI OASIS-3 BNCI2014-002 BNCI2015-001
Benchmarks: ABIDE ADNI OASIS-3 BNCI2014-002 BNCI2015-001
Metrics: alpha-precision beta-recall alpha,beta-F1 ROC-AUC F1 training time sampling time
Baselines: Real Data DiffeoGauss TriangDDPM TriangCFM RiemCFM
Builds on: Riemannian flow matching conditional flow matching log-Euclidean metric torchcfm
Closest papers by meaning
- MAGNET: Manifold-Aware Graph Diffusion Network for Connectome Generation (NeurIPS 2026 · similarity 0.88)
- Exploring the Enigma of Neural Dynamics Through A Scattering-Transform Mixer Landscape for Riemannian Manifold (ICML 2024 · similarity 0.88)
- Implicit Gaussian process representation of vector fields over arbitrary latent manifolds (ICLR 2024 · similarity 0.86)
- Geodesic Optimization for Predictive Shift Adaptation on EEG data (NeurIPS 2024 · similarity 0.86)
- Geometry-Aware Similarity Metrics for Neural Representations on Riemannian and Statistical Manifolds (NeurIPS 2026 · similarity 0.85)
- B[FM]$^2$: Brain Foundation Model via Flow Matching with SplitUNet (NeurIPS 2026 · similarity 0.85)
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