A 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.

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一种隐式估计方法。本文从非参数密度估计框架出发，探究VAE类型方法所追求的隐式密度估计量的若干统计性质。具体而言，在真实密度函数属于局部Hölder类的假设下，我们获得了VAE类型密度估计量的收敛速率。值得注意的是，采用最简单的网络架构——即带有一维潜变量的浅层生成模型——即可实现关于Hellinger度量的近极小极大最优速率。