# Optimal estimation of high-dimensional location Gaussian mixtures

arXiv: https://arxiv.org/abs/2002.05818 · HTML: https://mlsanity.com/paper/2002.05818/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2020.

## Built on by (most cited first, 16 of 16)

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- [Sample Complexities of Estimating Gumbel--Max Watermark Proportions with and without Reduction to Pivotal Statistics](https://mlsanity.com/paper/2607.00224/) (2026)
- [Characterizing Heterogeneous Rates in Finite Mixture Estimation via Partial Optimal Transport](https://mlsanity.com/paper/2609.16622/) (2026)
- [Model Selection and Parameter Estimation of One-Dimensional Gaussian Mixture Models](https://mlsanity.com/paper/2404.12613/) (2024)
- [Low-degree Lower bounds for clustering in moderate dimension](https://mlsanity.com/paper/2602.23023/) (2026)
- [Model Selection and Parameter Estimation for Multidimensional Gaussian Mixture Models with a Common Covariance Matrix](https://mlsanity.com/paper/2603.19657/) (2026)

## Abstract (arXiv)

This paper studies the optimal rate of estimation in a finite Gaussian location mixture model in high dimensions without separation conditions. We assume that the number of components k is bounded and that the centers lie in a ball of bounded radius, while allowing the dimension d to be as large as the sample size n. Extending the one-dimensional result of Heinrich and Kahn citeHK2015, we show that the minimax rate of estimating the mixing distribution in Wasserstein distance is Theta((d/n)1/4 + n-1/(4k-2)), achieved by an estimator computable in time O(nd2+n5/4). Furthermore, we show that the mixture density can be estimated at the optimal parametric rate Theta(√d/n) in Hellinger distance and provide a computationally efficient algorithm to achieve this rate in the special case of k=2. Both the theoretical and methodological development rely on a careful application of the method of moments. Central to our results is the observation that the information geometry of finite Gaussian mixtures is characterized by the moment tensors of the mixing distribution, whose low-rank structure can be exploited to obtain a sharp local entropy bound.