Minimax optimal density estimation using a shallow generative model with a one-dimensional latent variable

TL;DR

This paper demonstrates that a simple shallow generative model with a one-dimensional latent variable can achieve near minimax optimal density estimation rates under certain smoothness conditions, highlighting the efficiency of minimal architectures.

Contribution

It provides the first theoretical analysis showing that a shallow VAE-type model attains near optimal convergence rates in nonparametric density estimation.

Findings

Achieves near minimax optimal convergence rates under Hellinger metric.

Shows that simple shallow models can be statistically optimal.

Provides theoretical guarantees for VAE-type density estimators.

Abstract

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.