# Gaussian Mean Testing Made Simple

arXiv: https://arxiv.org/abs/2210.13706 · HTML: https://mlsanity.com/paper/2210.13706/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 26 references that are in the map.
- First posted on arXiv 25 Oct 2022.

## Builds on (most cited first, 16 of 16)

- [Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures](https://mlsanity.com/paper/1611.03473/) (2016)
- [Optimal Algorithms for Testing Closeness of Discrete Distributions](https://mlsanity.com/paper/1308.3946/) (2013)
- [Optimal Testing for Properties of Distributions](https://mlsanity.com/paper/1507.05952/) (2015)
- [Learning and Testing Causal Models with Interventions](https://mlsanity.com/paper/1805.09697/) (2018)
- [Testing Ising Models](https://mlsanity.com/paper/1612.03147/) (2016)
- [Testing Bayesian Networks](https://mlsanity.com/paper/1612.03156/) (2016)
- [Testing Conditional Independence of Discrete Distributions](https://mlsanity.com/paper/1711.11560/) (2017)
- [Testing Identity of Structured Distributions](https://mlsanity.com/paper/1410.2266/) (2014)
- [Optimal Identity Testing with High Probability](https://mlsanity.com/paper/1708.02728/) (2017)
- [Optimal Testing of Discrete Distributions with High Probability](https://mlsanity.com/paper/2009.06540/) (2020)
- [Random Restrictions of High-Dimensional Distributions and Uniformity Testing with Subcube Conditioning](https://mlsanity.com/paper/1911.07357/) (2019)
- [Learning and Testing Junta Distributions with Subcube Conditioning](https://mlsanity.com/paper/2004.12496/) (2020)
- [Optimal Algorithms and Lower Bounds for Testing Closeness of Structured Distributions](https://mlsanity.com/paper/1508.05538/) (2015)
- [Testing Shape Restrictions of Discrete Distributions](https://mlsanity.com/paper/1507.03558/) (2015)
- [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)

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

- [Learning multivariate Gaussians with imperfect advice](https://mlsanity.com/paper/2411.12700/) (2024)
- [Monotonicity Testing of High-Dimensional Distributions with Subcube Conditioning](https://mlsanity.com/paper/2502.16355/) (2025)
- [Sample Complexity of Nonparametric Closeness Testing for Continuous Distributions and Its Application to Causal Discovery with Hidden Confounding](https://mlsanity.com/paper/2503.07475/) (2025)

## Abstract (arXiv)

We study the following fundamental hypothesis testing problem, which we term Gaussian mean testing. Given i.i.d. samples from a distribution p on ℝd, the task is to distinguish, with high probability, between the following cases: (i) p is the standard Gaussian distribution, N(0,Id), and (ii) p is a Gaussian N(μ,Sigma) for some unknown covariance Sigma and mean μ ∈ ℝd satisfying |μ|2 ≥ ε. Recent work gave an algorithm for this testing problem with the optimal sample complexity of Theta(√d/ε2). Both the previous algorithm and its analysis are quite complicated. Here we give an extremely simple algorithm for Gaussian mean testing with a one-page analysis. Our algorithm is sample optimal and runs in sample linear time.