Robust Density Estimation under Besov IPM Losses

TL;DR

This paper establishes minimax convergence rates for robust nonparametric density estimation under a broad class of losses called Besov IPMs, demonstrating the effectiveness of wavelet estimators and certain GAN architectures.

Contribution

It introduces the first minimax rate results for density estimation under Besov IPM losses in the Huber contamination model, linking theory with GANs.

Findings

Wavelet thresholding estimators achieve minimax optimal rates.

Certain GAN architectures also attain these minimax rates.

Results apply to a wide range of distribution distances including Wasserstein and Kolmogorov-Smirnov.

Abstract

We study minimax convergence rates of nonparametric density estimation in the Huber contamination model, in which a proportion of the data comes from an unknown outlier distribution. We provide the first results for this problem under a large family of losses, called Besov integral probability metrics (IPMs), that includes $\mathcal{L}^p$, Wasserstein, Kolmogorov-Smirnov, and other common distances between probability distributions. Specifically, under a range of smoothness assumptions on the population and outlier distributions, we show that a re-scaled thresholding wavelet series estimator achieves minimax optimal convergence rates under a wide variety of losses. Finally, based on connections that have recently been shown between nonparametric density estimation under IPM losses and generative adversarial networks (GANs), we show that certain GAN architectures also achieve these minimax rates.