On the continuity of flows
This work addresses a fundamental limitation in generative modeling for researchers, highlighting that standard flow matching may fail for certain distribution pairs, which is incremental as it builds on existing flow matching frameworks.
The paper investigates how topological mismatches between prior and target distributions in flow matching can cause optimal velocity fields to become discontinuous, particularly when mapping unimodal to multimodal distributions, with theoretical and empirical evidence showing infinite magnitude jumps at decision boundaries.
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.