aiqbench analysis
This paper examines how different velocity fields can produce the same continuous marginal evolution in generative flows yet diverge after numerical discretization. It offers analytical tools, including projected modified equations, reachability conditions, and Wasserstein error bounds, and applies them to Gaussian, mixture, and diffusion probability-flow ODE settings. For practitioners, the key point is that sampler quality depends on more than the learned marginal path, so solver choice and discretization error can change results even when the training objective is identical.
Finite-Step Geometry of Marginal-Preserving Velocity Fields in Generative Flows
arXiv:2610.37936v1 Announce Type: new Abstract: Generative flows can admit different velocity fields that induce the same continuous marginal evolution but behave differently after numerical discretization. We study this finite-step non-equivalence while keeping the marginal path and numerical method fixed. Projected modified equations characterize the component of local integration error that changes the marginal law and give sufficient conditions for higher marginal convergence order. For affine fixed-shape distributions, we characterize exact finite-step reachability and derive spectral obstructions and sharp nearest-reachable Wasserstein errors. For diffusion probability-flow ODEs, we construct skew-score perturbations that preserve the exact marginal path under exact scores and quantify their leakage under score error. Controlled Gaussian and mixture experiments recover the predicted order, reachability, and obstruction phenomena.
Abstract:Generative flows can admit different velocity fields that induce the same continuous marginal evolution but behave differently after numerical discretization. We study this finite-step non-equivalence while keeping the marginal path and numerical method fixed. Projected modified equations characterize the component of local integration error that changes the marginal law and give sufficient conditions for higher marginal convergence order. For affine fixed-shape distributions, we characterize exact finite-step reachability and derive spectral obstructions and sharp nearest-reachable Wasserstein errors. For diffusion probability-flow ODEs, we construct skew-score perturbations that preserve the exact marginal path under exact scores and quantify their leakage under score error. Controlled Gaussian and mixture experiments recover the predicted order, reachability, and obstruction phenomena.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2610.37936 [cs.LG]
(or arXiv:2610.37936v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.37936 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 210 words