On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising
This work addresses density estimation and denoising challenges in statistics and machine learning, offering a method with strong theoretical guarantees but is incremental as it builds on existing NPMLE frameworks.
The paper tackles the problem of estimating Gaussian location mixture densities in high dimensions using the Nonparametric Maximum Likelihood Estimator (NPMLE), proving that it achieves near-parametric risk up to logarithmic factors without prior knowledge of mixture components, and applies this to Gaussian denoising where the empirical Bayes estimator performs nearly optimally in clustering scenarios.
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.