2026-07-24T17:54:48Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18886 2024-03-06T11:02:50Z ddc:004 open_access

Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions Ferreira Pinto Jr., Renato Monotonicity testing property testing isoperimetric inequalities Poincaré inequalities We study the connection between directed isoperimetric inequalities and monotonicity testing. In recent years, this connection has unlocked breakthroughs for testing monotonicity of functions defined on discrete domains. Inspired the rich history of isoperimetric inequalities in continuous settings, we propose that studying the relationship between directed isoperimetry and monotonicity in such settings is essential for understanding the full scope of this connection. Hence, we ask whether directed isoperimetric inequalities hold for functions f:[0,1]ⁿ → R, and whether this question has implications for monotonicity testing. We answer both questions affirmatively. For Lipschitz functions f:[0,1]ⁿ → ℝ, we show the inequality d^mono₁(f) ≲ 𝔼 [‖∇^- f‖₁], which upper bounds the L¹ distance to monotonicity of f by a measure of its "directed gradient". A key ingredient in our proof is the monotone rearrangement of f, which generalizes the classical "sorting operator" to continuous settings. We use this inequality to give an L¹ monotonicity tester for Lipschitz functions f:[0,1]ⁿ → ℝ, and this framework also implies similar results for testing real-valued functions on the hypergrid. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Renato Ferreira Pinto Jr. 2023 Is Part Of LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2023.61 urn:nbn:de:0030-drops-188867 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.61 eng https://creativecommons.org/licenses/by/4.0/legalcode