Chenxi Tao and Seung-Kyum Choi's paper analyzes the impact of velocity field properties on integration error within flow matching, a method for generating data by integrating a learned velocity field; the number of integration steps (NFE) determines inference cost [1].
The study decomposes the velocity Jacobian into symmetric (strain rate, S) and antisymmetric (vorticity, Omega) components to understand their respective roles in integration error [1].
The authors prove that strain controls exponential error amplification through the logarithmic norm, while vorticity contributes linearly to local truncation error [1].
The research indicates that the optimal transport velocity field is irrotational and has a zero material derivative, which implies second-order Euler accuracy; for exact displacement interpolation, the associated Lagrangian particle dynamics are integrated exactly by Euler [1].
The paper investigates weighted Jacobian regularization with strain weight alpha and vorticity weight beta, motivated by the analysis [1].
Experiments on 2D synthetic data confirm the theoretical predictions, showing up to 2.7x lower integration error at NFE=5 [1].
Preliminary CIFAR-10 experiments show consistent trends, with a lightweight fine-tuning procedure improving FID by 14 percent at NFE=10 while preserving high-NFE quality [1].
The paper includes 16 pages and 7 figures, and it is a preliminary version that includes qualitative CIFAR-10 comparisons and supporting synthetic experiments [1].
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