Point interpolation and distributional universality in flow-based generative models
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 Lp-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≥ 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.
Related
- Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression
- Mark distance correlation functions: from moment-based to distributional mark summary characteristics in spatial point processes
- Generative Refinement Learning for Temporal Interpolation in Scientific Machine Learning
- A dynamic point of view on universality for random matrices over finite local rings
- Universality and Convergence of Generative Flows
BibTeX
@misc{alvarezlopez2026point,
title = {{Point interpolation and distributional universality in flow-based generative models}},
author = {Antonio Álvarez-López and Jingpu Cheng and Liyan Zhang and Qianxiao Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04207},
eprint = {2611.04207},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04207},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Point interpolation and distributional universality in flow-based generative models
AU - Antonio Álvarez-López
AU - Jingpu Cheng
AU - Liyan Zhang
AU - Qianxiao Li
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04207
UR - https://arxiv.org/abs/2611.04207
AB - 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≥ 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.
ER -