# On the best approximation by finite Gaussian mixtures

arXiv: https://arxiv.org/abs/2404.08913 · HTML: https://mlsanity.com/paper/2404.08913/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 74 references that are in the map.
- First posted on arXiv 13 Apr 2024.

## Builds on (most cited first, 9 of 9)

- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising](https://mlsanity.com/paper/1712.02009/) (2017)
- [Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models](https://mlsanity.com/paper/2008.08244/) (2020)
- [Optimal estimation of high-dimensional location Gaussian mixtures](https://mlsanity.com/paper/2002.05818/) (2020)
- [Sharp regret bounds for empirical Bayes and compound decision problems](https://mlsanity.com/paper/2109.03943/) (2021)
- [Sharper Sub-Weibull Concentrations](https://mlsanity.com/paper/2102.02450/) (2021)
- [Gaussian Mixture Reduction with Composite Transportation Divergence](https://mlsanity.com/paper/2002.08410/) (2020)
- [Empirical Bayes estimation: When does g-modeling beat f-modeling in theory (and in practice)?](https://mlsanity.com/paper/2211.12692/) (2022)
- [A New Probabilistic Distance Metric With Application In Gaussian Mixture Reduction](https://mlsanity.com/paper/2306.07309/) (2023)

## Built on by (most cited first, 3 of 3)

- [Factor Adjusted Spectral Clustering for Mixture Models](https://mlsanity.com/paper/2408.12564/) (2024)
- [Sharp Inequalities between Total Variation and Hellinger Distances for Gaussian Mixtures](https://mlsanity.com/paper/2602.03202/) (2026)
- [Sample Complexities of Estimating Gumbel--Max Watermark Proportions with and without Reduction to Pivotal Statistics](https://mlsanity.com/paper/2607.00224/) (2026)

## Abstract (arXiv)

We consider the problem of approximating a general Gaussian location mixture by finite mixtures. The minimum order of finite mixtures that achieve a prescribed accuracy (measured by various f-divergences) is determined within constant factors for the family of mixing distributions with compactly support or appropriate assumptions on the tail probability including subgaussian and subexponential. While the upper bound is achieved using the technique of local moment matching, the lower bound is established by relating the best approximation error to the low-rank approximation of certain trigonometric moment matrices, followed by a refined spectral analysis of their minimum eigenvalue. In the case of Gaussian mixing distributions, this result corrects a previous lower bound in [Allerton Conference 48 (2010) 620-628].