Which Geometry Governs Few-Step Degradation in Latent Flow Matching?

Abstract

Few-step sampling can substantially reduce the cost of flow-matching generation, but coarse numerical integration introduces degradation whose geometric origin is not well understood. This question is especially ambiguous for latent models, where the dynamics are integrated in a latent space and the resulting trajectory is subsequently transformed through nonlinear mappings before evaluation. We study which representation-space geometry predicts few-step degradation across latent, decoded motion, and Cartesian motion spaces. Using matched latent and direct conditional flow-matching models, we find that trajectory geometry measured in the solver's integration space is consistently more predictive of few-step degradation than geometry measured after downstream mappings. Numerical analysis explains this separation: discretization error is generated along the integrated trajectory, while downstream maps affect the resulting error through their local sensitivity. Empirically, output-map gain strongly predicts degradation, whereas downstream trajectory geometry loses predictive power. Geometry-based arc-length timestep schedules do not outperform a cosine schedule, indicating that representation choice and scheduling statistic are distinct issues. Finally, reflow substantially straightens the predictive integration-space trajectory and yields large improvements in the few-step regime, while the guided teacher remains better at the largest step budgets. These results identify integration-space geometry as the primary object for analyzing and controlling few-step flow-matching sampling. 𝗣𝗿𝗼𝗷𝗲𝗰𝘁 𝗽𝗮𝗴𝗲: https://azizkhadraoui.github.io/few-step-geometry/ The project page includes the PDF paper, OpenReview page, open-source code, interactive motion visualizations, and additional reproducibility and clarification notes.