On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising

TL;DR

This paper analyzes the nonparametric maximum likelihood estimator (NPMLE) for Gaussian mixture densities, demonstrating its near-optimal risk performance and effectiveness in Gaussian denoising without prior knowledge of mixture components.

Contribution

It provides finite sample guarantees for NPMLE accuracy and shows its application to empirical Bayes estimation in Gaussian denoising tasks.

Findings

NPMLE achieves near-parametric risk for Gaussian mixtures.

Empirical Bayes estimates based on NPMLE are nearly optimal for denoising.

Theoretical bounds are established for NPMLE accuracy in finite samples.

Abstract

We study the Nonparametric Maximum Likelihood Estimator (NPMLE) for estimating Gaussian location mixture densities in $d$-dimensions from independent observations. Unlike usual likelihood-based methods for fitting mixtures, NPMLEs are based on convex optimization. We prove finite sample results on the Hellinger accuracy of every NPMLE. Our results imply, in particular, that every NPMLE achieves near parametric risk (up to logarithmic multiplicative factors) when the true density is a discrete Gaussian mixture without any prior information on the number of mixture components. NPMLEs can naturally be used to yield empirical Bayes estimates of the Oracle Bayes estimator in the Gaussian denoising problem. We prove bounds for the accuracy of the empirical Bayes estimate as an approximation to the Oracle Bayes estimator. Here our results imply that the empirical Bayes estimator performs at nearly the optimal level (up to logarithmic multiplicative factors) for denoising in clustering situations without any prior knowledge of the number of clusters.