Isoperimetric Inequalities for Real-Valued Functions with Applications to Monotonicity Testing

TL;DR

This paper extends isoperimetric inequalities to real-valued functions on the hypercube, leading to improved algorithms for testing and approximating monotonicity with near-optimal query complexities.

Contribution

It introduces a new Boolean decomposition for real-valued functions and applies it to enhance monotonicity testing and distance approximation algorithms.

Findings

Monotonicity tester with $ ilde{O}( ext{min}(r rac{ ext{sqrt}(d)}{d}))$ queries

Matching lower bounds for nonadaptive, 1-sided testers

Near-optimal nonadaptive distance approximation within $O( ext{sqrt}(d ext{log} d))$ factor

Abstract

We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra~(SICOMP 2018) for Boolean functions to the case of real-valued functions $f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real-valued function $f$ over an arbitrary partially ordered domain as a collection of Boolean functions over the same domain, roughly capturing the distance of $f$ to monotonicity and the structure of violations of $f$ to monotonicity. We apply our generalized isoperimetric inequality to improve algorithms for testing monotonicity and approximating the distance to monotonicity for real-valued functions. Our tester for monotonicity has query complexity $\widetilde{O}(\min(r \sqrt{d},d))$, where $r$ is the size of the image of the input function. (The best previously known tester, by Chakrabarty and Seshadhri (STOC 2013), makes $O(d)$ queries.) Our tester is nonadaptive and has 1-sided error. We show a matching lower bound for nonadaptive, 1-sided error testers for monotonicity. We also show that the distance to monotonicity of real-valued functions that are $\alpha$-far from monotone can be approximated nonadaptively within a factor of $O(\sqrt{d\log d})$ with query complexity polynomial in $1/\alpha$ and the dimension $d$. This query complexity is known to be nearly optimal for nonadaptive algorithms even for the special case of Boolean functions. (The best previously known distance approximation algorithm for real-valued functions, by Fattal and Ron (TALG 2010) achieves $O(d\log r)$-approximation.)