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arXiv · 1907.03182

Towards Testing Monotonicity of Distributions Over General Posets

Maryam Aliakbarpour·Themis Gouleakis·John Peebles·Ronitt Rubinfeld·Anak Yodpinyanee
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Abstract

In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution $p$ over a poset is monotone if, for any pair of domain elements $x$ and $y$ such that $x \preceq y$, $p(x) \leq p(y)$. To understand the sample complexity of this problem, we introduce a new property called bigness over a finite domain, where the distribution is $T$-big if the minimum probability for any domain element is at least $T$. We establish a lower bound of $Ω(n/\log n)$ for testing bigness of distributions on domains of size $n$. We then build on these lower bounds to give $Ω(n/\log{n})$ lower bounds for testing monotonicity over a matching poset of size $n$ and significantly improved lower bounds over the hypercube poset. We give sublinear sample complexity bounds for testing bigness and for testing monotonicity over the matching poset. We then give a number of tools for analyzing upper bounds on the sample complexity of the monotonicity testing problem.

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BibTeXRIS

Maryam Aliakbarpour, Themis Gouleakis, John Peebles, Ronitt Rubinfeld, Anak Yodpinyanee. 2019-07-06. Towards Testing Monotonicity of Distributions Over General Posets. https://arxiv.org/abs/1907.03182

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