The deep generative model yields an implicit estimator for the unknown distribution or density function of the observation. This paper investigates some statistical properties of the implicit density estimator pursued by VAE-type methods from a nonparametric density estimation framework. More specifically, we obtain convergence rates of the VAE-type density estimator under the assumption that the underlying true density function belongs to a locally H\"{o}lder class. Remarkably, a near minimax optimal rate with respect to the Hellinger metric can be achieved by the simplest network architecture, a shallow generative model with a one-dimensional latent variable. The proof of the main theorem relies on the well-known result from the nonparametric Bayesian literature that a smooth density with a suitably decaying tail can efficiently be approximated by a finite mixture of normal distributions. We also discuss an alternative proof, which offers important insights and suggests a potential extension to structured density estimation.

翻译：深度生成模型为观测数据的未知分布或密度函数提供了隐式估计器。本文从非参数密度估计框架出发，研究基于VAE类方法获得的隐式密度估计量的统计性质。具体而言，在底层真实密度函数属于局部Hölder类的假设下，我们推导了VAE类密度估计量的收敛速率。值得关注的是，通过最简单的网络结构——具有一维隐变量的浅层生成模型，即可实现关于Hellinger距离的近似极小极大最优速率。主要定理的证明依赖于非参数贝叶斯文献中的一个经典结论：具有适当尾部衰减的光滑密度函数可被有限正态混合分布高效逼近。此外，我们还讨论了一种替代证明方法，该方法提供了重要见解并可能拓展至结构化密度估计。