流模型理论研究：Lipschitz 向量场逼近概率分布的普适性条件
Point interpolation and distributional universality in flow-based generative models
搞流模型或连续归一化流的理论同学可以看看，讲清楚了什么样的向量场族能逼近任意分布，Wasserstein 和总变差两种距离下的结论都给了。
这篇 arXiv 论文研究 flow-based 生成模型中，参数化 Lipschitz 向量场的 pushforward 对概率测度的逼近能力。作者在 Wasserstein 距离下建立了有限样本点插值与分布逼近之间的定量关系，并给出基于传输映射 L^p 逼近的普适性判据。结果表明，包含非仿射向量场的仿射不变族从紧支撑绝对连续基测度出发即具有普适性。在总变差距离下，论文证明坐标式动力学在维度 d≥2 时不充分，并给出基于一维 Sobolev 逼近与剪切型混合的充分条件。
Point interpolation and distributional universality in flow-based generative models
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.