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Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions

Author Renato Ferreira Pinto Jr.

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Author Details

Renato Ferreira Pinto Jr.
  • University of Waterloo, Canada

Funding

  • Ferreira Pinto Jr., Renato: Partly funded by an NSERC Canada Graduate Scholarship Doctoral Award.

Acknowledgements

We thank Eric Blais for helpful discussions throughout the course of this project, and for comments and suggestions on preliminary versions of this paper.

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Renato Ferreira Pinto Jr.. Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.61

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Lower bounds and information complexity
  • Mathematics of computing → Mathematical analysis
Keywords
  • Monotonicity testing
  • property testing
  • isoperimetric inequalities
  • Poincaré inequalities

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