Near-Optimal Closeness Testing of Discrete Histogram Distributions
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by
Ilias Diakonikolas, Daniel M. Kane, Vladimir Nikishkin
2017
Abstract
We investigate the problem of testing the equivalence between two discrete
histograms. A k-histogram over [n] is a probability distribution that
is piecewise constant over some set of k intervals over [n]. Histograms
have been extensively studied in computer science and statistics. Given a set
of samples from two k-histogram distributions p, q over [n], we want to
distinguish (with high probability) between the cases that p = q and
p-q_1 ≥ϵ. The main contribution of this paper is a new
algorithm for this testing problem and a nearly matching information-theoretic
lower bound. Specifically, the sample complexity of our algorithm matches our
lower bound up to a logarithmic factor, improving on previous work by
polynomial factors in the relevant parameters. Our algorithmic approach applies
in a more general setting and yields improved sample upper bounds for testing
closeness of other structured distributions as well.
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