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Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions

Author Renato Ferreira Pinto Jr.



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Renato Ferreira Pinto Jr.
  • University of Waterloo, Canada

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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References

  1. Dominique Bakry, Ivan Gentil, and Michel Ledoux. Analysis and geometry of Markov diffusion operators, volume 103. Springer, 2014. Google Scholar
  2. Aleksandrs Belovs. Adaptive lower bound for testing monotonicity on the line. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  3. Colin Bennett and Robert C Sharpley. Interpolation of operators. Academic press, 1988. Google Scholar
  4. Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L^p-testing. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 164-173, 2014. Google Scholar
  5. Hadley Black, Deeparnab Chakrabarty, and C Seshadhri. Directed isoperimetric theorems for boolean functions on the hypergrid and an Õ(n√d) monotonicity tester. arXiv preprint, 2022. URL: https://arxiv.org/abs/2211.05281.
  6. Hadley Black, Deeparnab Chakrabarty, and C Seshadhri. A d^1/2 + o(1) monotonicity tester for boolean functions on d-dimensional hypergrids. arXiv preprint, 2023. URL: https://arxiv.org/abs/2304.01416.
  7. Hadley Black, Deeparnab Chakrabarty, and Comandur Seshadhri. A o(d) ⋅ polylog n monotonicity tester for boolean functions over the hypergrid [n]^d. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2133-2151. SIAM, 2018. Google Scholar
  8. Hadley Black, Deeparnab Chakrabarty, and Comandur Seshadhri. Domain reduction for monotonicity testing: A o(d) tester for boolean functions in d-dimensions. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1975-1994. SIAM, 2020. Google Scholar
  9. Hadley Black, Iden Kalemaj, and Sofya Raskhodnikova. Isoperimetric inequalities for real-valued functions with applications to monotonicity testing. arXiv preprint, 2020. URL: https://arxiv.org/abs/2011.09441.
  10. Eric Blais, Joshua Brody, and Kevin Matulef. Property testing lower bounds via communication complexity. computational complexity, 21:311-358, 2012. Google Scholar
  11. Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 309-320. IEEE, 2014. Google Scholar
  12. Sergey Bobkov and Michel Ledoux. Poincaré’s inequalities and talagrand’s concentration phenomenon for the exponential distribution. Probability Theory and Related Fields, 107:383-400, 1997. Google Scholar
  13. Sergey G Bobkov and Christian Houdré. Isoperimetric constants for product probability measures. The Annals of Probability, pages 184-205, 1997. Google Scholar
  14. Stephen P Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. Google Scholar
  15. Lorenzo Brasco and Filippo Santambrogio. A note on some poincaré inequalities on convex sets by optimal transport methods. In Geometric Properties for Parabolic and Elliptic PDE’s: GPPEPDEs, Palinuro, Italy, May 2015 4, pages 49-63. Springer, 2016. Google Scholar
  16. Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer. Improved monotonicity testers via hypercube embeddings. arXiv preprint, 2022. URL: https://arxiv.org/abs/2211.09229.
  17. D. Chakrabarty and C. Seshadhri. An o(n) monotonicity tester for boolean functions over the hypercube. SIAM Journal on Computing, 45(2):461-472, January 2016. URL: https://doi.org/10.1137/13092770X.
  18. Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and lipschitz testing over hypercubes and hypergrids. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 419-428, 2013. Google Scholar
  19. Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory Of Computing, 10(17):453-464, 2014. Google Scholar
  20. Xi Chen, Rocco A. Servedio, and Li-Yang Tan. New algorithms and lower bounds for monotonicity testing. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 286-295, October 2014. URL: https://doi.org/10.1109/FOCS.2014.38.
  21. Shuyu Cheng, Guoqiang Wu, and Jun Zhu. On the convergence of prior-guided zeroth-order optimization algorithms. Advances in Neural Information Processing Systems, 34:14620-14631, 2021. Google Scholar
  22. Michael G Crandall and Luc Tartar. Some relations between nonexpansive and order preserving mappings. Proceedings of the American Mathematical Society, 78(3):385-390, 1980. Google Scholar
  23. Funda Ergün, Sampath Kannan, S Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 259-268, 1998. Google Scholar
  24. Renato Ferreira Pinto Jr. Directed poincaré inequalities and L¹ monotonicity testing of lipschitz functions. arXiv preprint, 2023. URL: https://arxiv.org/abs/2307.02193.
  25. Eldar Fischer. On the strength of comparisons in property testing. Information and Computation, 189(1):107-116, 2004. Google Scholar
  26. Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 474-483, 2002. Google Scholar
  27. Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 3(20):301-337, 2000. Google Scholar
  28. Nathaniel Harms and Yuichi Yoshida. Downsampling for testing and learning in product distributions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022. Google Scholar
  29. Bernhard Kawohl. Rearrangements and convexity of level sets in PDE, volume 1150 of Lecture Notes in Mathematics. Springer-Verlag Berlin, Heidelberg, 1985. Google Scholar
  30. Subhash Khot, Dor Minzer, and Muli Safra. On monotonicity testing and boolean isoperimetric-type theorems. SIAM Journal on Computing, 47(6):2238-2276, 2018. Google Scholar
  31. Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. Google Scholar
  32. Ramesh Krishnan S Pallavoor, Sofya Raskhodnikova, and Erik Waingarten. Approximating the distance to monotonicity of boolean functions. Random Structures & Algorithms, 60(2):233-260, 2022. Google Scholar
  33. Henri Poincaré. Sur les équations aux dérivées partielles de la physique mathématique. American Journal of Mathematics, pages 211-294, 1890. Google Scholar
  34. Michel Talagrand. Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem. Geometric & Functional Analysis GAFA, 3(3):295-314, 1993. Google Scholar
  35. Rüdiger Verfürth. A note on polynomial approximation in sobolev spaces. ESAIM: Mathematical Modelling and Numerical Analysis, 33(4):715-719, 1999. Google Scholar
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