Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry

2025-05-20

AI Summary

Modeling brain functional connectivity matrices—symmetric positive-definite (SPD) or correlation matrices—on Riemannian manifolds suffers from low computational efficiency and difficulty preserving intrinsic geometric constraints. Method: We propose DiffeoCFM, the first framework establishing equivalence between Riemannian conditional flow matching (under a pullback metric) and Euclidean flow matching, enabling efficient, structure-preserving generation. Leveraging global diffeomorphisms—specifically matrix logarithm and normalized Cholesky decomposition—we construct the pullback metric and tightly couple conditional flow matching with ODE solvers, supporting multimodal fMRI and EEG data. Results: Evaluated on five large-scale neuroimaging datasets (>34,000 samples), DiffeoCFM achieves state-of-the-art performance with significantly accelerated training and sampling speeds, while rigorously respecting manifold geometry and SPD constraints.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Generating realistic brain connectivity matrices efficiently
• Modeling constrained connectivity spaces as Riemannian manifolds
• Enabling fast sampling while preserving manifold constraints

Innovation

• Uses pullback metrics for Riemannian flow matching
• Applies standard CFM after data transformation
• Leverages matrix logarithm and Cholesky decomposition