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Testing Unate Distributions

Authors Daeho Lee , Shivam Nadimpalli , Mingda Qiao , Ronitt Rubinfeld

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ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Mathematics of computing → Probability and statistics
Keywords
  • Distribution testing
  • unate distributions
  • monotone distributions
  • uniformity testing
  • subcube conditioning

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Abstract

We initiate the study of unate distributions over {±1}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 Θ̃(n^{3/2}) samples are sufficient and necessary, in contrast to the Θ̃(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 Õ(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 Ω̃(n^{2/3}) complexity. In the same model, the complexity of monotonicity testing was recently shown to be Θ̃(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(n²) 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 {±}ⁿ.

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Daeho Lee, Shivam Nadimpalli, Mingda Qiao, and Ronitt Rubinfeld. Testing Unate Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 57:1-57:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.57

Author Details

Daeho Lee
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Shivam Nadimpalli
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Mingda Qiao
  • University of Massachusetts Amherst, MA, USA
Ronitt Rubinfeld
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Lee, Daeho: Supported by the MIT Undergraduate Research Opportunities Program (UROP).
  • Nadimpalli, Shivam: Partially supported by Elchanan Mossel’s Vannevar Bush Faculty Fellowship ONR-N00014-20-1-2826.
  • Rubinfeld, Ronitt: Supported by the NSF TRIPODS program (award DMS-2022448) and CCF-2310818.

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