Testing Unate Distributions

2026-07-01

AI Summary

This work presents the first systematic study of two fundamental testing problems for unate distributions over $\{±1\}^n$: uniformity testing of unate distributions and unateness testing of arbitrary distributions. Addressing the high sample complexity ($\Omega(n^2)$) of existing approaches—which rely on reductions to monotonicity testing—the authors introduce novel algorithms based on weak learning of hidden directions and new correlation bounds. Under the subcube conditional sampling model, these algorithms achieve a sample complexity of $\widetilde{O}(n^{3/2})$ for both testing problems. This result significantly improves upon naive reduction-based methods and introduces technical tools of independent interest that may advance research in both monotonicity and unateness testing.

Abstract

We initiate the study of *unate distributions* over $\{\pm1\}^n$ -- a natural analogue of unate Boolean functions -- by considering two basic testing problems that parallel well-studied questions for monotone distributions: - Uniformity Testing of Unate Distributions: We show that $\widetildeΘ(n^{3/2})$ samples are sufficient and necessary, in contrast to the $\widetildeΘ(n)$ sample complexity of the analogous problem for monotone distributions (Rubinfeld and Servedio, STOC 2005; Adamaszek, Czumaj, and Sohler, SODA 2010). - Unateness Testing of Arbitrary Distributions: We give a tester that uses $\widetilde{O}(n^{3/2})$ conditional samples in the subcube conditional model. On the other hand, every tester that draws conditional samples in a similar fashion, namely from $O(1)$-dimensional subcubes, must have an $\widetildeΩ(n^{2/3})$ complexity. In the same model, the complexity of monotonicity testing was recently shown to be $\widetildeΘ(n)$ (Chakrabarty et al., STOC 2025). Our algorithms for both problems significantly outperform the naive approach of reducing to the monotone case, which would incur $Ω(n^2)$ sample complexity. Our uniformity tester relies on a subroutine that "weakly" learns the hidden orientations of a unate distribution, together with a new correlation bound for these estimates. Both tools may be of independent interest in studying monotonicity and unateness over $\{\pm1\}^n$.

Problem

• unate distributions
• uniformity testing
• unateness testing
• sample complexity
• conditional sampling

Innovation

• unate distributions
• uniformity testing
• conditional sampling
• sample complexity
• weak learning