Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifolds

TL;DR

This paper introduces Generalised Flow Maps (GFM), a novel few-step generative modeling framework on Riemannian manifolds that improves efficiency and quality over existing geometric models, with theoretical unification and state-of-the-art results.

Contribution

It extends Euclidean flow models to Riemannian manifolds, providing a unified framework with three training methods and theoretical insights.

Findings

Achieves state-of-the-art sample quality on geometric datasets.

Unifies and elevates existing Euclidean models to Riemannian settings.

Demonstrates superior or competitive log-likelihoods.

Abstract

Geometric data and purpose-built generative models on them have become ubiquitous in high-impact deep learning application domains, ranging from protein backbone generation and computational chemistry to geospatial data. Current geometric generative models remain computationally expensive at inference -- requiring many steps of complex numerical simulation -- as they are derived from dynamical measure transport frameworks such as diffusion and flow-matching on Riemannian manifolds. In this paper, we propose Generalised Flow Maps (GFM), a new class of few-step generative models that generalises the Flow Map framework in Euclidean spaces to arbitrary Riemannian manifolds. We instantiate GFMs with three self-distillation-based training methods: Generalised Lagrangian Flow Maps, Generalised Eulerian Flow Maps, and Generalised Progressive Flow Maps. We theoretically show that GFMs, under specific design decisions, unify and elevate existing Euclidean few-step generative models, such as consistency models, shortcut models, and meanflows, to the Riemannian setting. We benchmark GFMs against other geometric generative models on a suite of geometric datasets, including geospatial data, RNA torsion angles, and hyperbolic manifolds, and achieve state-of-the-art sample quality for single- and few-step evaluations, and superior or competitive log-likelihoods using the implicit probability flow.