让生成模型理解数据背后的拓扑结构，提升复杂网络数据建模效果
Topological Flow Matching
- 将流匹配框架重构为退化薛定谔桥问题，引入拉普拉斯漂移注入拓扑信息
- 在脑图、洋流等结构化数据上实现更优生成质量，保持确定性采样路径
- 适用于需保留空间拓扑的生成任务，如神经影像、气候模拟与交通预测
流匹配是一种强大的生成建模框架，因其简洁性和出色的实证性能而备受青睐。然而，其标准形式将结构化空间（如脑图上的fMRI数据）中的信号视为欧氏空间中的点，忽略了其定义域丰富的拓扑特征。为此，我们提出拓扑流匹配，一种对流匹配的拓扑感知推广。我们将流匹配解释为求解一个退化的薛定谔桥问题，并通过引入基于拉普拉斯算子的漂移项来增强参考过程，从而注入拓扑信息。这一原则性修改在保留流匹配优良性质的同时——稳定且无需模拟的损失函数、确定性的样本路径——有效捕捉了底层域的结构特征。结果，我们的框架可作为标准流匹配的即插即用替代方案。我们在多种结构化数据集上验证了其有效性，包括脑fMRI、海洋洋流、地震事件和交通流量。
原文摘要 · Abstract (English)
Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces, such as fMRI data on brain graphs, as points in Euclidean space, overlooking the rich topological features of their domains. To address this, we introduce topological flow matching, a topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topological information by augmenting the reference process with a Laplacian-derived drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: a stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a drop-in replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.
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