Neural Implicit Manifold Learning for Topology-Aware Density Estimation

TL;DR

This paper introduces a neural implicit manifold approach for density estimation on complex, topology-aware manifolds, overcoming limitations of traditional pushforward models by representing the manifold as neural zeros and employing energy-based models for density learning.

Contribution

The paper proposes modeling data manifolds as neural implicit sets and learning densities with energy-based models, enabling accurate topology-aware density estimation.

Findings

Outperforms pushforward models on complex topologies

Accurately learns distributions supported on manifolds

Handles complex manifold geometries effectively

Abstract

Natural data observed in $\mathbb{R}^n$ is often constrained to an $m$-dimensional manifold $\mathcal{M}$, where $m < n$. This work focuses on the task of building theoretically principled generative models for such data. Current generative models learn $\mathcal{M}$ by mapping an $m$-dimensional latent variable through a neural network $f_\theta: \mathbb{R}^m \to \mathbb{R}^n$. These procedures, which we call pushforward models, incur a straightforward limitation: manifolds cannot in general be represented with a single parameterization, meaning that attempts to do so will incur either computational instability or the inability to learn probability densities within the manifold. To remedy this problem, we propose to model $\mathcal{M}$ as a neural implicit manifold: the set of zeros of a neural network. We then learn the probability density within $\mathcal{M}$ with a constrained energy-based model, which employs a constrained variant of Langevin dynamics to train and sample from the learned manifold. In experiments on synthetic and natural data, we show that our model can learn manifold-supported distributions with complex topologies more accurately than pushforward models.