Moser Flow: Divergence-based Generative Modeling on Manifolds
This addresses the challenge of efficient generative modeling on curved surfaces for applications in fields like earth and climate sciences, representing a novel method rather than an incremental improvement.
The paper tackles the problem of learning generative models for complex geometries like manifolds, introducing Moser Flow (MF) as a new continuous normalizing flow that parameterizes density via divergence of a neural network, eliminating the need for ODE solvers during training. Empirically, it demonstrates significant improvements in density estimation, sample quality, and training complexity over existing methods on synthetic and real-world benchmarks.
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.