Moser Flow: Divergence-based Generative Modeling on Manifolds

TL;DR

Moser Flow introduces a divergence-based continuous normalizing flow model that efficiently learns generative models on complex manifolds, improving density estimation and sample quality without costly ODE solvers.

Contribution

The paper presents Moser Flow, a novel CNF that uses divergence of neural networks for density modeling on manifolds, enabling efficient training and universal approximation.

Findings

Effective sampling from curved surfaces demonstrated.

Significant improvements in density estimation and sample quality.

Reduced computational complexity compared to existing CNFs.

Abstract

We are interested in learning generative models for complex geometries described via manifolds, such as spheres, tori, and other implicit surfaces. Current extensions of existing (Euclidean) generative models are restricted to specific geometries and typically suffer from high computational costs. We introduce Moser Flow (MF), a new class of generative models within the family of continuous normalizing flows (CNF). MF also produces a CNF via a solution to the change-of-variable formula, however differently from other CNF methods, its model (learned) density is parameterized as the source (prior) density minus the divergence of a neural network (NN). The divergence is a local, linear differential operator, easy to approximate and calculate on manifolds. Therefore, unlike other CNFs, MF does not require invoking or backpropagating through an ODE solver during training. Furthermore, representing the model density explicitly as the divergence of a NN rather than as a solution of an ODE facilitates learning high fidelity densities. Theoretically, we prove that MF constitutes a universal density approximator under suitable assumptions. Empirically, we demonstrate for the first time the use of flow models for sampling from general curved surfaces and achieve significant improvements in density estimation, sample quality, and training complexity over existing CNFs on challenging synthetic geometries and real-world benchmarks from the earth and climate sciences.