ManiFlow: Implicitly Representing Manifolds with Normalizing Flows

TL;DR

ManiFlow introduces a method to generate samples on lower-dimensional manifolds using normalizing flows trained on perturbed data, enabling better modeling of complex data distributions like 3D point clouds.

Contribution

This work shows that normalizing flows implicitly represent manifolds and proposes an optimization to recover the most likely manifold point from noisy samples.

Findings

Normalizing flows trained on perturbed data implicitly model data manifolds.

The proposed optimization effectively recovers points on the original data manifold.

Application to 3D point clouds improves surface reconstruction quality.

Abstract

Normalizing Flows (NFs) are flexible explicit generative models that have been shown to accurately model complex real-world data distributions. However, their invertibility constraint imposes limitations on data distributions that reside on lower dimensional manifolds embedded in higher dimensional space. Practically, this shortcoming is often bypassed by adding noise to the data which impacts the quality of the generated samples. In contrast to prior work, we approach this problem by generating samples from the original data distribution given full knowledge about the perturbed distribution and the noise model. To this end, we establish that NFs trained on perturbed data implicitly represent the manifold in regions of maximum likelihood. Then, we propose an optimization objective that recovers the most likely point on the manifold given a sample from the perturbed distribution. Finally, we focus on 3D point clouds for which we utilize the explicit nature of NFs, i.e. surface normals extracted from the gradient of the log-likelihood and the log-likelihood itself, to apply Poisson surface reconstruction to refine generated point sets.