用几何变换让脑连接矩阵生成又快又准。
Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry
- 通过坐标变换将矩阵流形转为欧氏空间，用标准方法生成
- 在3个fMRI和2个EEG数据集上达到顶尖性能
- 适合需要快速生成真实脑网络的研究者
生成真实的脑连接矩阵对于分析脑组织的群体异质性、理解疾病机制以及在困难分类问题中扩充数据至关重要。功能连接矩阵位于对称正定或相关矩阵等受限空间中，可建模为黎曼流形。然而，使用黎曼工具通常需要重新定义核心操作（测地线、范数、积分），导致生成建模计算效率低下。本文提出 DiffeoCFM，通过利用欧氏空间上全局微分同胚诱导的拉回度量，在矩阵流形上实现条件流匹配（CFM）。我们证明，采用此类度量的黎曼型CFM等价于对数据进行变换后应用标准CFM。该等价性使得向量场学习高效，采样可借助标准常微分方程求解器快速完成。我们以矩阵对数处理协方差矩阵，以归一化乔列斯基分解处理相关矩阵来实例化DiffeoCFM。在包含超过4600次扫描的三个大规模fMRI数据集（ADNI、ABIDE、OASIS-3）及两个EEG运动想象数据集（BNCI2014-002、BNCI2015-001，共超30000次试验）上评估，DiffeoCFM实现了快速训练与最先进性能，同时严格保持流形约束。代码开源：https://github.com/antoinecollas/DiffeoCFM
原文摘要 · Abstract (English)
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
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