Continuous normalizing flows are widely used in generative tasks, where a flow network transports from a data distribution $P$ to a normal distribution. A flow model that transports from $P$ to an arbitrary $Q$, where both $P$ and $Q$ are accessible via finite samples, is of various application interests, particularly in the recently developed telescoping density ratio estimation (DRE) which calls for the construction of intermediate densities to bridge between the two densities. In this work, we propose such a flow by a neural-ODE model which is trained from empirical samples to transport invertibly from $P$ to $Q$ (and vice versa) and optimally by minimizing the transport cost. The trained flow model allows us to perform infinitesimal DRE along the time-parametrized $\log$-density by training an additional continuous-time network using classification loss, whose time integration provides a telescopic DRE. The effectiveness of the proposed model is empirically demonstrated on high-dimensional mutual information estimation and energy-based generative models of image data.

翻译：连续归一化流广泛应用于生成任务中，其中流网络将数据分布 $P$ 输运至正态分布。将 $P$ 输运至任意分布 $Q$（两者均可通过有限样本获取）的流模型具有多种应用价值，尤其在近期发展的伸缩密度比估计（DRE）中，需构建中间密度以桥接两个密度。本文提出一种基于神经常微分方程的流模型，该模型通过经验样本训练，实现从 $P$ 到 $Q$ 及反向的可逆输运，并通过最小化输运成本实现最优输运。训练后的流模型允许沿时间参数化的 $\log$-密度执行无穷小DRE：通过分类损失训练附加的连续时间网络，其时间积分提供伸缩型DRE。在高维互信息估计与基于能量的图像数据生成模型上的实证结果展示了所提模型的有效性。