Gaussian Mean Testing Made Simple

10/25/2022

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by   Ilias Diakonikolas, et al.

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We study the following fundamental hypothesis testing problem, which we term Gaussian mean testing. Given i.i.d. samples from a distribution p on ℝ^d, the task is to distinguish, with high probability, between the following cases: (i) p is the standard Gaussian distribution, 𝒩(0,I_d), and (ii) p is a Gaussian 𝒩(μ,Σ) for some unknown covariance Σ and mean μ∈ℝ^d satisfying μ_2 ≥ϵ. Recent work gave an algorithm for this testing problem with the optimal sample complexity of Θ(√(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.