Density estimation on smooth manifolds with normalizing flows

TL;DR

This paper introduces a novel normalizing flow framework for density estimation on complex, non-trivial manifolds by combining local models to effectively learn distributions on data manifolds of unknown topology.

Contribution

It proposes a new approach that 'glues' local models to learn distributions on manifolds, overcoming limitations of existing methods that require Euclidean topology or strong priors.

Findings

Better sample efficiency on synthetic data

Competitive performance on high-dimensional manifolds

Effective handling of unknown manifold topology

Abstract

We present a framework for learning probability distributions on topologically non-trivial manifolds, utilizing normalizing flows. Current methods focus on manifolds that are homeomorphic to Euclidean space, enforce strong structural priors on the learned models or use operations that do not easily scale to high dimensions. In contrast, our method learns distributions on a data manifold by "gluing" together multiple local models, thus defining an open cover of the data manifold. We demonstrate the efficiency of our approach on synthetic data of known manifolds, as well as higher dimensional manifolds of unknown topology, where our method exhibits better sample efficiency and competitive or superior performance against baselines in a number of tasks.