当分布拓扑不匹配时，流模型的最优速度场会出现跳跃，影响生成质量。
On the continuity of flows
- 分析发现拓扑不匹配导致速度场在决策边界出现跳跃
- 理论证明跳跃幅度随时间逼近目标分布趋于无穷
- 对流模型在流形上建模及神经网络学习不连续表示有启示
流匹配作为连续归一化流的生成建模框架已崭露头角。本文研究了一个潜在的拓扑约束：当先验分布与目标分布拓扑不匹配（如单峰到多峰）时，标准流匹配目标下的最优速度场可能表现出空间不连续性。我们指出，这种不连续性源于连续流必须分叉以将单一模式映射到多个模式，迫使粒子在中间时间点做出离散路由决策。通过对双峰高斯混合模型的理论分析，我们证明最优速度场在决策边界处存在跳跃间断，其幅度随时间趋近目标分布而趋于无穷。分析表明该现象并非仅限于 $L^2$ 损失，而是分布间拓扑不匹配的必然结果。我们通过实证验证了该理论，并讨论了其对流匹配在流形上的应用、与近期黎曼流匹配工作的关联，以及神经网络学习不连续表示所面临的挑战。
原文摘要 · Abstract (English)
Flow matching has emerged as a powerful framework for generative modeling through continuous normalizing flows. We investigate a potential topological constraint: when the prior distribution and target distribution have mismatched topology (e.g., unimodal to multimodal), the optimal velocity field under standard flow matching objectives may exhibit spatial discontinuities. We suggest that this discontinuity arises from the requirement that continuous flows must bifurcate to map a single mode to multiple modes, forcing particles to make discrete routing decisions at intermediate times. Through theoretical analysis on bimodal Gaussian mixtures, we demonstrate that the optimal velocity field exhibits jump discontinuities along decision boundaries, with magnitude approaching infinity as time approaches the target distribution. Our analysis suggests that this phenomenon is not specific to $L^2$ loss, but rather may be a consequence of topological mismatch between distributions. We validate our theory empirically and discuss potential implications for flow matching on manifolds, connecting our findings to recent work on Riemannian flow matching and the challenge of learning discontinuous representations in neural networks.
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