Optimal transport flow and infinitesimal density ratio estimation

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.