🤖 AI Summary
This paper identifies the fundamental cause of topological discontinuities in velocity fields within flow matching: when the prior (e.g., unimodal) and target distribution (e.g., multimodal) exhibit topological mismatch, the optimal velocity field necessarily develops asymptotically infinite jump discontinuities along decision boundaries. This arises from the geometric necessity for continuous flows to split particle trajectories to map distinct modes—not from loss function design or optimization bias.
Method: We adopt a topological dynamical systems perspective, providing theoretical analysis for a bimodal Gaussian mixture, embedding the problem within a Riemannian flow matching framework, and validating findings via neural representation experiments.
Contribution/Results: We rigorously establish that such discontinuities are ubiquitous at intermediate times along decision boundaries. The phenomenon is model-agnostic—persisting across diverse loss functions and flow architectures—and imposes critical theoretical constraints on extending flow matching to nontrivial manifolds.
📝 Abstract
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.