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Monotonicity Testing of High-Dimensional Distributions with Subcube Conditioning

Deeparnab Chakrabarty, Xi Chen, Simeon Ristic, C. Seshadhri, Erik Waingarten

TL;DR

This work resolves the query complexity of monotonicity testing for high-dimensional distributions under subcube conditioning, showing a near-tight ${\tildeΘ(n/\varepsilon^2)}$ bound and introducing a real-valued directed isoperimetric framework to analyze distribution testers. The core method centers on an edge tester that leverages 1-dimensional subcube conditioning and a novel $(L^1,\ell^2)$-Talagrand inequality to relate distance to monotonicity to detectable edge biases. The authors also establish a matching lower bound via moment-matching constructions on product distributions, and extend the analysis to uniformity testing under the monotone promise, proving a ${\tildeΩ(\sqrt{n}/\varepsilon^2)}$ bound. Overall, the results demonstrate that monotonicity does not materially improve uniformity testing in this model, and they introduce tools that may be of independent interest for high-dimensional distribution testing.

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 $\tildeΘ(n/\varepsilon^2)$, 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 $\tildeΘ(\sqrt{n}/\varepsilon^2)$. 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.

Monotonicity Testing of High-Dimensional Distributions with Subcube Conditioning

TL;DR

This work resolves the query complexity of monotonicity testing for high-dimensional distributions under subcube conditioning, showing a near-tight bound and introducing a real-valued directed isoperimetric framework to analyze distribution testers. The core method centers on an edge tester that leverages 1-dimensional subcube conditioning and a novel -Talagrand inequality to relate distance to monotonicity to detectable edge biases. The authors also establish a matching lower bound via moment-matching constructions on product distributions, and extend the analysis to uniformity testing under the monotone promise, proving a bound. Overall, the results demonstrate that monotonicity does not materially improve uniformity testing in this model, and they introduce tools that may be of independent interest for high-dimensional distribution testing.

Abstract

We study monotonicity testing of high-dimensional distributions on 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 (Rubinfeld, Vasilian~\cite{RV20}, and Aliakbarpour, Gouleakis, Peebles, Rubinfeld, Yodpinyanee~\cite{AGPRY19}). We show that the subcube \emph{query complexity} is , 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 . 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 . 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.

Paper Structure

This paper contains 19 sections, 22 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Testing Monotonicity.
  4. Main Ideas
  5. Related Work
  6. Monotonicity Testing of Distributions.
  7. Distribution Testing Beyond Sample-Complexity.
  8. Testing Monotonicity
  9. The Real-Valued Directed Talagrand Inequality
  10. The proof of \ref{['thm:l1-talagrand']}
  11. Lower Bound for Testing Monotonicity
  12. Preliminaries
  13. The one-dimensional mean distributions
  14. The distributions $\mathcal{D}_{yes}$ and $\mathcal{D}_{no}$
  15. Indistinguishability of $\mathcal{D}_{\text{yes}}$ and $\mathcal{D}_{\text{no}}$
  16. ...and 4 more sections

Key Result

Theorem 1

There is an algorithm for testing monotonicity of distributions over $\{-1,1\}^n$ that uses $\tilde{O}(n/\varepsilon^2)$ subcube conditioning queries. The algorithm works in the weaker coordinate oracle model BCSV23, where queries are only made on one-dimensional subcubes.

Figures (1)

  • Figure 1: Algorithm for Testing Monotonicity of Distributions

Theorems & Definitions (28)

  • Theorem 1: Monotonicity Testing Upper Bound
  • Theorem 2: Monotonicity Testing Lower Bound
  • Theorem 3
  • Theorem 4: Testing Uniformity of Monotone Distributions
  • Theorem 5
  • Claim 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 6: KMS18PRW22
  • Corollary 2.4
  • ...and 18 more