Gaussian Mean Testing Made Simple

TL;DR

This paper introduces a simple, sample-optimal algorithm for Gaussian mean testing that is easier to understand and implement than previous complex methods, achieving optimal sample complexity and linear runtime.

Contribution

It presents an extremely simple, one-page algorithm for Gaussian mean testing with proven optimal sample complexity and linear runtime, simplifying prior complex approaches.

Findings

Algorithm is sample optimal with $ ilde{O}(rac{ ext{sqrt}(d)}{ ext{epsilon}^2})$ samples.

Algorithm runs in linear time relative to sample size.

Analysis is concise, only one page long.

Abstract

We study the following fundamental hypothesis testing problem, which we term Gaussian mean testing. Given i.i.d. samples from a distribution $p$ on $\mathbb{R}^d$, the task is to distinguish, with high probability, between the following cases: (i) $p$ is the standard Gaussian distribution, $\mathcal{N}(0,I_d)$, and (ii) $p$ is a Gaussian $\mathcal{N}(\mu,\Sigma)$ for some unknown covariance $\Sigma$ and mean $\mu \in \mathbb{R}^d$ satisfying $\|\mu\|_2 \geq \epsilon$. Recent work gave an algorithm for this testing problem with the optimal sample complexity of $\Theta(\sqrt{d}/\epsilon^2)$. 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.