# Sharp Inequalities between Total Variation and Hellinger Distances for Gaussian Mixtures

arXiv: https://arxiv.org/abs/2602.03202 · HTML: https://mlsanity.com/paper/2602.03202/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 40 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising](https://mlsanity.com/paper/1712.02009/) (2017)
- [Two Modeling Strategies for Empirical Bayes Estimation](https://mlsanity.com/paper/1409.2677/) (2014)
- [Robust Kernel Density Estimation with Median-of-Means principle](https://mlsanity.com/paper/2006.16590/) (2020)
- [Sharp regret bounds for empirical Bayes and compound decision problems](https://mlsanity.com/paper/2109.03943/) (2021)
- [On the best approximation by finite Gaussian mixtures](https://mlsanity.com/paper/2404.08913/) (2024)

## Built on by (most cited first, 1 of 1)

- [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 study the relation between the total variation (TV) and Hellinger distances between two Gaussian location mixtures. Our first result establishes a general upper bound: for any two mixing distributions supported on a compact set, the Hellinger distance between the two mixtures is controlled by the TV distance raised to a power 1-o(1), where the o(1) term is of order 1/loglog(1/TV). We also construct two sequences of mixing distributions that demonstrate the sharpness of this bound. Taken together, our results resolve an open problem raised in Jia et al. (2023) and thus lead to an entropic characterization of learning Gaussian mixtures in total variation. Our inequality also yields optimal robust estimation of Gaussian mixtures in Hellinger distance, which has a direct implication for bounding the minimax regret of empirical Bayes under Huber contamination.