Minimax bounds for estimating multivariate Gaussian location mixtures

TL;DR

This paper establishes the optimal rates for estimating multivariate Gaussian mixture models under different loss functions, revealing how tail behavior influences the minimax bounds in high-dimensional settings.

Contribution

It provides the first sharp minimax bounds for Gaussian location mixture estimation under squared $L^2$ and Hellinger losses, considering tail conditions of the mixing measure.

Findings

Minimax rate under squared $L^2$ loss is proportional to $n^{-1}( ext{log } n)^{d/2}$.

For sub-Gaussian tails, the Hellinger loss minimax rate is at least $( ext{log } n)^d / n$.

With bounded $p$-th moments, the rate is at least $n^{-p/(p+d)}( ext{log } n)^{-3d/2}$.

Abstract

We prove minimax bounds for estimating Gaussian location mixtures on $\mathbb{R}^d$ under the squared $L^2$ and the squared Hellinger loss functions. Under the squared $L^2$ loss, we prove that the minimax rate is upper and lower bounded by a constant multiple of $n^{-1}(\log n)^{d/2}$. Under the squared Hellinger loss, we consider two subclasses based on the behavior of the tails of the mixing measure. When the mixing measure has a sub-Gaussian tail, the minimax rate under the squared Hellinger loss is bounded from below by $(\log n)^{d}/n$. On the other hand, when the mixing measure is only assumed to have a bounded $p^{\text{th}}$ moment for a fixed $p > 0$, the minimax rate under the squared Hellinger loss is bounded from below by $n^{-p/(p+d)}(\log n)^{-3d/2}$. These rates are minimax optimal up to logarithmic factors.