Neural Implicit Manifold Learning for Topology-Aware Density Estimation

Understand

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.