Gaussian Mean Testing Made Simple

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 $\mathcal{N}(μ,Σ)$ for some unknown covariance $Σ$ and mean $μ\in \mathbb{R}^d$ satisfying $\|μ\|_2 \geq ε$. Recent work gave an algorithm for this testing problem with the optimal sample complexity of $Θ(\sqrt{d}/ε^2)$. 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.