Natural data observed in $\mathbb{R}^n$ is often constrained to an
$m$-dimensional manifold $\mathcal{M}$, where $m < n$. Current generative
models represent this manifold by mapping an $m$-dimensional latent variable
through a neural network $f_\theta: \mathbb{R}^m \to \mathbb{R}^n$. Such
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. To learn
the data distribution within $\mathcal{M}$, we introduce constrained
energy-based models, which use a constrained variant of Langevin dynamics to
train and sample within the learned manifold. The resulting model can be
manipulated with an arithmetic of manifolds which allows practitioners to take
unions and intersections of model manifolds. In experiments on synthetic and
natural data, we show that constrained EBMs can learn manifold-supported
distributions with complex topologies more accurately than pushforward models.

本论文提出一种通过神经隐式流来表示数据流形分布的方法，称为神经隐式流分布。为了解决基于前向模型的限制，引入了约束能量模型，该方法使用约束的 Langevin 动力学来训练和采样，能比前向模型更精确地学习复杂拓扑结构的流形支持分布，并允许将模型流形的并集和交集进行运算。