Directed Isoperimetry and Monotonicity Testing: A Dynamical Approach

TL;DR

This paper introduces a new approach linking isoperimetric inequalities and monotonicity testing for functions on the unit cube, providing a sublinear query complexity monotonicity tester and a novel directed Poincaré inequality using a PDE-based method.

Contribution

It develops a sublinear query complexity monotonicity tester for Lipschitz functions and establishes a directed Poincaré inequality via a PDE approach, connecting isoperimetry and monotonicity.

Findings

Established a $ ilde{O}(rac{ ext{sqrt}(d) M^2)}{ ext{epsilon}^2})$ query complexity for the monotonicity tester.

Proved a directed Poincaré inequality relating monotonicity violations to the directed gradient.

Introduced the directed heat equation as a PDE tool for analyzing monotonicity and isoperimetry.

Abstract

This paper explores the connection between classical isoperimetric inequalities, their directed analogues, and monotonicity testing. We study the setting of real-valued functions $f : [0,1]^d \to \mathbb{R}$ on the solid unit cube, where the goal is to test with respect to the $L^p$ distance. Our goals are twofold: to further understand the relationship between classical and directed isoperimetry, and to give a monotonicity tester with sublinear query complexity in this setting. Our main results are 1) an $L^2$ monotonicity tester for $M$-Lipschitz functions with query complexity $\widetilde O(\sqrt{d} M^2 / \epsilon^2)$ and, behind this result, 2) the directed Poincar\'e inequality $\mathsf{dist}^{\mathsf{mono}}_2(f)^2 \le C \mathbb{E}[|\nabla^- f|^2]$, where the "directed gradient" operator $\nabla^-$ measures the local violations of monotonicity of $f$. To prove the second result, we introduce a partial differential equation (PDE), the directed heat equation, which takes a one-dimensional function $f$ into a monotone function $f^*$ over time and enjoys many desirable analytic properties. We obtain the directed Poincar\'e inequality by combining convergence aspects of this PDE with the theory of optimal transport. Crucially for our conceptual motivation, this proof is in complete analogy with the mathematical physics perspective on the classical Poincar\'e inequality, namely as characterizing the convergence of the standard heat equation toward equilibrium.