Neural Implicit Manifold Learning for Topology-Aware Density Estimation

Published: 05 Jan 2024, Last Modified: 05 Jan 2024Accepted by TMLREveryoneRevisionsBibTeX
Authors that are also TMLR Expert Reviewers: ~Gabriel_Loaiza-Ganem1
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.
Certifications: Expert Certification
Submission Length: Regular submission (no more than 12 pages of main content)
Assigned Action Editor: ~marco_cuturi2
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 1391