# Density estimation for Hellinger via minimum-distance estimators: mixtures of Gaussians, log-concave, and more

arXiv: https://arxiv.org/abs/2606.11469 · HTML: https://mlsanity.com/paper/2606.11469/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 60 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## Builds on (most cited first, 26 of 26)

- [Testing Identity of Structured Distributions](https://mlsanity.com/paper/1410.2266/) (2014)
- [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)
- [Sample-Optimal Density Estimation in Nearly-Linear Time](https://mlsanity.com/paper/1506.00671/) (2015)
- [Recent progress in log-concave density estimation](https://mlsanity.com/paper/1709.03154/) (2017)
- [Optimal Algorithms and Lower Bounds for Testing Closeness of Structured Distributions](https://mlsanity.com/paper/1508.05538/) (2015)
- [Communication-constrained hypothesis testing: Optimality, robustness, and reverse data processing inequalities](https://mlsanity.com/paper/2206.02765/) (2022)
- [Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression](https://mlsanity.com/paper/1903.05315/) (2019)
- [Testing Shape Restrictions of Discrete Distributions](https://mlsanity.com/paper/1507.03558/) (2015)
- [Efficiently Learning Structured Distributions from Untrusted Batches](https://mlsanity.com/paper/1911.02035/) (2019)
- [Testing Closeness of Multivariate Distributions via Ramsey Theory](https://mlsanity.com/paper/2311.13154/) (2023)
- [Near-Optimal Closeness Testing of Discrete Histogram Distributions](https://mlsanity.com/paper/1703.01913/) (2017)
- [Testing Identity of Multidimensional Histograms](https://mlsanity.com/paper/1804.03636/) (2018)
- [Optimal empirical Bayes estimation for the Poisson model via minimum-distance methods](https://mlsanity.com/paper/2209.01328/) (2022)
- [Learning Multivariate Log-concave Distributions](https://mlsanity.com/paper/1605.08188/) (2016)
- [Efficient Robust Proper Learning of Log-concave Distributions](https://mlsanity.com/paper/1606.03077/) (2016)
- [Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities](https://mlsanity.com/paper/1802.10575/) (2018)
- [Robust hypothesis testing and distribution estimation in Hellinger distance](https://mlsanity.com/paper/2011.01848/) (2020)
- [Robust Model Selection and Nearly-Proper Learning for GMMs](https://mlsanity.com/paper/2106.02774/) (2021)
- [Nonparametric MLE for Gaussian Location Mixtures: Certified Computation and Generic Behavior](https://mlsanity.com/paper/2503.20193/) (2025)
- [The Sample Complexity of Distributed Simple Binary Hypothesis Testing under Information Constraints](https://mlsanity.com/paper/2506.13686/) (2025)
- [Nonparametric inference under shape constraints: past, present and future](https://mlsanity.com/paper/2509.26040/) (2025)
- [Ratio Covers of Convex Sets and Optimal Mixture Density Estimation](https://mlsanity.com/paper/2602.16142/) (2026)
- [A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential Families](https://mlsanity.com/paper/1907.08306/) (2019)
- [Attainability of Two-Point Testing Rates for Finite-Sample Location Estimation](https://mlsanity.com/paper/2502.05730/) (2025)
- [Instance-Optimal Uniformity Testing and Tracking](https://mlsanity.com/paper/2508.02637/) (2025)

## Abstract (arXiv)

We study the task of density estimation, where we hope to accurately estimate a probability density from n samples. A textbook method for density estimation in total variation distance is the minimum-distance estimator approach, where we conclude both the algorithm and the analysis merely from bounding the VC dimension of a particular concept class (the so-called Yatracos class). While this technique has originally yielded sharp guarantees primarily for total variation distance, in this work we extend the minimum-distance estimator approach for learning within Hellinger distance. Our main observation is that we may produce an analogous recipe for Hellinger (where we only require bounding the VC dimension of a related concept class) by drawing connections to recent results yielding reverse data processing inequalities. This recipe is flexible enough to accommodate fast algorithms originally designed for total variation distance; by modifying the approach of Acharya et al. (2017) we conclude the first near-linear time algorithm for learning classes including univariate mixtures of log-concave densities and mixtures of Gaussians (with arbitrary variances), with near-optimal sample complexity.