# Lower Bounds on the Total Variation Distance Between Mixtures of Two Gaussians

arXiv: https://arxiv.org/abs/2109.01064 · HTML: https://mlsanity.com/paper/2109.01064/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 35 references that are in the map.
- First posted on arXiv 2 Sep 2021.

## Builds on (most cited first, 11 of 11)

- [Mixture Models, Robustness, and Sum of Squares Proofs](https://mlsanity.com/paper/1711.07454/) (2017)
- [Private Hypothesis Selection](https://mlsanity.com/paper/1905.13229/) (2019)
- [Ten Steps of EM Suffice for Mixtures of Two Gaussians](https://mlsanity.com/paper/1609.00368/) (2016)
- [Tight bounds for learning a mixture of two gaussians](https://mlsanity.com/paper/1404.4997/) (2014)
- [Private Identity Testing for High-Dimensional Distributions](https://mlsanity.com/paper/1905.11947/) (2019)
- [Robust Learning of Mixtures of Gaussians](https://mlsanity.com/paper/2007.05912/) (2020)
- [Optimal Average-Case Reductions to Sparse PCA: From Weak Assumptions to Strong Hardness](https://mlsanity.com/paper/1902.07380/) (2019)
- [A Two-round Variant of EM for Gaussian Mixtures](https://mlsanity.com/paper/1301.3850/) (2013)
- [Learning Distributions Generated by One-Layer ReLU Networks](https://mlsanity.com/paper/1909.01812/) (2019)
- [Uniform Convergence Rates for Maximum Likelihood Estimation under Two-Component Gaussian Mixture Models](https://mlsanity.com/paper/2006.00704/) (2020)
- [Algebraic and Analytic Approaches for Parameter Learning in Mixture Models](https://mlsanity.com/paper/2001.06776/) (2020)

## Built on by (most cited first, 4 of 4)

- [Discriminative Estimation of Total Variation Distance: A Fidelity Auditor for Generative Data](https://mlsanity.com/paper/2405.15337/) (2024)
- [Optimal Estimation and Computational Limit of Low-rank Gaussian Mixtures](https://mlsanity.com/paper/2201.09040/) (2022)
- [Lower Bounds for the Total Variation Distance Given Means and Variances of Distributions](https://mlsanity.com/paper/2212.05820/) (2022)
- [Approximating the Total Variation Distance between Gaussians](https://mlsanity.com/paper/2503.11099/) (2025)

## Abstract (arXiv)

Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically difficult to obtain tight lower bounds for mixtures. Exploiting a connection between total variation distance and the characteristic function of the mixture, we provide fairly tight functional approximations. This enables us to derive new lower bounds on the total variation distance between pairs of two-component Gaussian mixtures that have a shared covariance matrix.