Q-malizing 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 can transport from $P$ to an arbitrary $Q$, where both $P$ and $Q$ are accessible via finite samples, would be 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 $P$ and $Q$. In this work, we propose such a ``Q-malizing flow'' by a neural-ODE model which is trained to transport invertibly from $P$ to $Q$ (and vice versa) from empirical samples and is regularized 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 flow network using classification loss, which estimates the time-partial derivative of the $\log$-density. Integrating the time-score network along time provides a telescopic DRE between $P$ and $Q$ that is more stable than a one-step DRE. The effectiveness of the proposed model is empirically demonstrated on mutual information estimation from high-dimensional data and energy-based generative models of image data.