Monotonicity Testing of High-Dimensional Distributions with Subcube Conditioning

2025-02-22

AI Summary

This paper studies monotonicity testing of high-dimensional Boolean distributions and uniformity testing under a monotonicity promise, in the subcube conditioning query model. It introduces directed isoperimetric inequalities to distribution testing for the first time, extends the KMS inequality to real-valued functions, and constructs information-theoretic lower bounds to show that a monotonicity promise does not substantially accelerate uniformity testing. The main results are tight complexity bounds: monotonicity testing requires $widetilde{Theta}(n/varepsilon^2)$ subcube queries, while uniformity testing under monotonicity promise requires $widetilde{Theta}(sqrt{n}/varepsilon^2)$. Technically, the work integrates directed isoperimetric analysis, probabilistic distance estimation, and adversarial lower-bound construction. These advances significantly extend the theoretical frontiers of property testing for high-dimensional distributions.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study monotonicity testing of high-dimensional distributions on ${-1,1}^n$ in the model of subcube conditioning, suggested and studied by Canonne, Ron, and Servedio~cite{CRS15} and Bhattacharyya and Chakraborty~cite{BC18}. Previous work shows that the emph{sample complexity} of monotonicity testing must be exponential in $n$ (Rubinfeld, Vasilian~cite{RV20}, and Aliakbarpour, Gouleakis, Peebles, Rubinfeld, Yodpinyanee~cite{AGPRY19}). We show that the subcube emph{query complexity} is $ ilde{Theta}(n/varepsilon^2)$, by proving nearly matching upper and lower bounds. Our work is the first to use directed isoperimetric inequalities (developed for function monotonicity testing) for analyzing a distribution testing algorithm. Along the way, we generalize an inequality of Khot, Minzer, and Safra~cite{KMS18} to real-valued functions on ${-1,1}^n$. We also study uniformity testing of distributions that are promised to be monotone, a problem introduced by Rubinfeld, Servedio~cite{RS09} , using subcube conditioning. We show that the query complexity is $ ilde{Theta}(sqrt{n}/varepsilon^2)$. Our work proves the lower bound, which matches (up to poly-logarithmic factors) the uniformity testing upper bound for general distributions (Canonne, Chen, Kamath, Levi, Waingarten~cite{CCKLW21}). Hence, we show that monotonicity does not help, beyond logarithmic factors, in testing uniformity of distributions with subcube conditional queries.

Problem

• Monotonicity testing in high-dimensional distributions
• Query complexity using subcube conditioning
• Uniformity testing of monotone distributions

Innovation

• Directed isoperimetric inequalities application
• Subcube conditioning query complexity analysis
• Generalization of Khot-Minzer-Safra inequality