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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 wi

Author

Pinto, Renato Ferreira

Published

2024

Language

EN