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.

翻译：在 $\mathbb{R}^n$ 中观测到的自然数据通常被约束在一个$m$维流形$\mathcal{M}$上，其中$m < n$。本文聚焦于为此类数据构建具有理论基础的生成模型。当前的生成模型通过神经网络 $f_\theta: \mathbb{R}^m \to \mathbb{R}^n$ 将$m$维隐变量映射来学习$\mathcal{M}$。这些我们称为前推模型的方法存在明显局限：流形通常无法用单一参数化表示，试图这样做将导致计算不稳定或无法学习流形内的概率密度。为解决此问题，我们提出将$\mathcal{M}$建模为神经隐式流形——神经网络的零点集。随后，我们通过约束能量模型学习$\mathcal{M}$内的概率密度，该模型采用约束朗之万动力学的变体来训练并从所学流形中采样。在合成数据和自然数据的实验中，我们证明该模型能够比前推模型更准确地学习具有复杂拓扑结构的流形支撑分布。