Matching Normalizing Flows and Probability Paths on Manifolds

TL;DR

This paper introduces a novel training method for Continuous Normalizing Flows on manifolds using probability path divergence, enabling scalable, efficient, and high-quality generative modeling for complex data structures.

Contribution

It proposes probability path divergence (PPD) for training CNFs on manifolds, avoiding ODE solutions per iteration and improving scalability and applicability.

Findings

Achieves state-of-the-art likelihoods and sample quality on manifold benchmarks.

First to scale generative models to moderately high-dimensional manifolds.

Demonstrates theoretical bounds relating PPD to classical divergences.

Abstract

Continuous Normalizing Flows (CNFs) are a class of generative models that transform a prior distribution to a model distribution by solving an ordinary differential equation (ODE). We propose to train CNFs on manifolds by minimizing probability path divergence (PPD), a novel family of divergences between the probability density path generated by the CNF and a target probability density path. PPD is formulated using a logarithmic mass conservation formula which is a linear first order partial differential equation relating the log target probabilities and the CNF's defining vector field. PPD has several key benefits over existing methods: it sidesteps the need to solve an ODE per iteration, readily applies to manifold data, scales to high dimensions, and is compatible with a large family of target paths interpolating pure noise and data in finite time. Theoretically, PPD is shown to bound classical probability divergences. Empirically, we show that CNFs learned by minimizing PPD achieve state-of-the-art results in likelihoods and sample quality on existing low-dimensional manifold benchmarks, and is the first example of a generative model to scale to moderately high dimensional manifolds.