TR17-132 Authors: Ilias Diakonikolas, Daniel Kane, Alistair Stewart

Publication: 8th September 2017 14:00

Downloads: 405

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We study the problem of {\em generalized uniformity testing}~\cite{BC17} of a discrete probability distribution: Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbf{\Omega}$, we want to distinguish, with probability at least $2/3$, between the case that $p$ is uniform on some {\em subset} of $\mathbf{\Omega}$ versus $\epsilon$-far, in total variation distance, from any such uniform distribution.

We establish tight bounds on the sample complexity of generalized uniformity testing. In more detail, we present a computationally efficient tester whose sample complexity is optimal, up to constant factors,

and a matching information-theoretic lower bound. Specifically, we show that the sample complexity of generalized uniformity testing is $\Theta\left(1/(\epsilon^{4/3}\|p\|_3) + 1/(\epsilon^{2} \|p\|_2) \right)$.