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arXiv · 2611.04207

Point interpolation and distributional universality in flow-based generative models

Antonio Álvarez-López·Jingpu Cheng·Liyan Zhang·Qianxiao Li
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Abstract

Motivated by the expressive power of neural ordinary differential equations in flow-based generative modeling, we study the approximation of probability measures by pushforwards generated by parameterized families of Lipschitz vector fields. For Wasserstein distances, we establish a quantitative relationship between finite-sample point interpolation and distributional approximation, and derive universality criteria based on $L^p$-approximation of transport maps. In particular, affine-invariant families containing a non-affine vector field are universal from compactly supported absolutely continuous base measures. For total variation, we show that coordinate-wise dynamics are insufficient in dimensions $d\geq 2$ and establish sufficient conditions based on one-dimensional Sobolev approximation and shear-type mixing. We further examine the role of point interpolation, establishing when different regimes relate to one another and when approximate interpolation can be made exact.

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Antonio Álvarez-López, Jingpu Cheng, Liyan Zhang, Qianxiao Li. 2026-11-03. Point interpolation and distributional universality in flow-based generative models. https://arxiv.org/abs/2611.04207

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