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Dimensionality of Hamming Metrics and Rademacher Type

Author Alexandros Eskenazis

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Alexandros Eskenazis
  • CNRS, Institut de Mathématiques de Jussieu, Sorbonne Université, France
  • Trinity College, University of Cambridge, UK

Acknowledgements

I wish to thank Florent Baudier, Paata Ivanisvili and Assaf Naor for their constructive feedback on this work.

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Alexandros Eskenazis. Dimensionality of Hamming Metrics and Rademacher Type. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.55

Abstract

Let X be a finite-dimensional normed space. We prove that if the Hamming cube {-1,1}ⁿ embeds into X with bi-Lipschitz distortion at most D ≥ 1, then dim(X) ≳ sup_{p ∈ [1,2]} n^p/(D^p 𝖳_p(X)^p), where 𝖳_p(X) is the Rademacher type p constant of X. This estimate yields a mutual refinement of distortion lower bounds which follow from works of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020). The proof relies on a combination of semigroup techniques on the biased hypercube with the Borsuk-Ulam theorem from algebraic topology.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Functional analysis
  • Theory of computation → Random projections and metric embeddings
Keywords
  • Hamming cube
  • Rademacher type
  • metric embeddings
  • Borsuk-Ulam theorem

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References

  1. Alexandr Andoni, Moses S. Charikar, Ofer Neiman, and Huy L. Nguyen. Near linear lower bound for dimension reduction in 𝓁₁. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science - FOCS 2011, pages 315-323. IEEE Computer Soc., Los Alamitos, CA, 2011. URL: https://doi.org/10.1109/FOCS.2011.87.
  2. Alexandr Andoni, Piotr Indyk, and Ilya Razenshteyn. Approximate nearest neighbor search in high dimensions. In Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018. Vol. IV. Invited lectures, pages 3287-3318. World Sci. Publ., Hackensack, NJ, 2018. Google Scholar
  3. Juan Arias-de-Reyna and Luis Rodríguez-Piazza. Finite metric spaces needing high dimension for Lipschitz embeddings in Banach spaces. Israel J. Math., 79(1):103-111, 1992. URL: https://doi.org/10.1007/BF02764804.
  4. Keith Ball, Eric A. Carlen, and Elliott H. Lieb. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math., 115(3):463-482, 1994. URL: https://doi.org/10.1007/BF01231769.
  5. Imre Bárány and Victor S. Grinberg. On some combinatorial questions in finite-dimensional spaces. Linear Algebra Appl., 41:1-9, 1981. URL: https://doi.org/10.1016/0024-3795(81)90085-9.
  6. Yair Bartal, Lee-Ad Gottlieb, and Ofer Neiman. On the impossibility of dimension reduction for doubling subsets of 𝓁_p. In Computational geometry (SoCG'14), pages 60-66. ACM, New York, 2014. Google Scholar
  7. Yair Bartal and Manor Mendel. Dimension reduction for ultrametrics. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 664-665. ACM, New York, 2004. Google Scholar
  8. Florent Baudier, Krzysztof Swieçicki, and Andrew Swift. No dimension reduction for doubling subsets of 𝓁_q when q > 2 revisited. J. Math. Anal. Appl., 504(2):Paper No. 125407, 18, 2021. URL: https://doi.org/10.1016/j.jmaa.2021.125407.
  9. Jean Bourgain, Vitali Milman, and Haim Wolfson. On type of metric spaces. Trans. Amer. Math. Soc., 294(1):295-317, 1986. URL: https://doi.org/10.2307/2000132.
  10. Bo Brinkman and Moses Charikar. On the impossibility of dimension reduction in l₁. J. ACM, 52(5):766-788, 2005. URL: https://doi.org/10.1145/1089023.1089026.
  11. Bo Brinkman, Adriana Karagiozova, and James R. Lee. Vertex cuts, random walks, and dimension reduction in series-parallel graphs. In STOC'07 - Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 621-630. ACM, New York, 2007. URL: https://doi.org/10.1145/1250790.1250882.
  12. Per Enflo. On a problem of Smirnov. Ark. Mat., 8:107-109, 1969. URL: https://doi.org/10.1007/BF02589550.
  13. Per Enflo. On the nonexistence of uniform homeomorphisms between L_p-spaces. Ark. Mat., 8:103-105, 1969. URL: https://doi.org/10.1007/BF02589549.
  14. Per Enflo. On infinite-dimensional topological groups. In Séminaire sur la Géométrie des Espaces de Banach (1977-1978), pages Exp. No. 10-11, 11. École Polytech., Palaiseau, 1978. Google Scholar
  15. Alexandros Eskenazis. On Pisier’s inequality for UMD targets. Canad. Math. Bull., 64(2):282-291, 2021. URL: https://doi.org/10.4153/S0008439520000442.
  16. Alexandros Eskenazis. Some geometric applications of the discrete heat flow. Preprint available at https://arxiv.org/abs/2310.01868, 2023.
  17. Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings., pages 534-543, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238226.
  18. Tuomas Hytönen and Assaf Naor. Pisier’s inequality revisited. Studia Math., 215(3):221-235, 2013. URL: https://doi.org/10.4064/sm215-3-2.
  19. Piotr Indyk. Algorithmic applications of low-distortion geometric embeddings. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), pages 10-33. IEEE Computer Soc., Los Alamitos, CA, 2001. Google Scholar
  20. Paata Ivanisvili, Ramon van Handel, and Alexander Volberg. Rademacher type and Enflo type coincide. Ann. of Math. (2), 192(2):665-678, 2020. URL: https://doi.org/10.4007/annals.2020.192.2.8.
  21. Robert C. James. Nonreflexive spaces of type 2. Israel J. Math., 30(1-2):1-13, 1978. URL: https://doi.org/10.1007/BF02760825.
  22. William B. Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Conn., 1982), volume 26 of Contemp. Math., pages 189-206. Amer. Math. Soc., Providence, RI, 1984. URL: https://doi.org/10.1090/conm/026/737400.
  23. William B. Johnson, Joram Lindenstrauss, and Gideon Schechtman. On Lipschitz embedding of finite metric spaces in low-dimensional normed spaces. In Geometrical aspects of functional analysis (1985/86), volume 1267 of Lecture Notes in Math., pages 177-184. Springer, Berlin, 1987. URL: https://doi.org/10.1007/BFb0078145.
  24. William B. Johnson and Gideon Schechtman. Diamond graphs and super-reflexivity. J. Topol. Anal., 1(2):177-189, 2009. URL: https://doi.org/10.1142/S1793525309000114.
  25. Vincent Lafforgue and Assaf Naor. A doubling subset of L_p for p > 2 that is inherently infinite dimensional. Geom. Dedicata, 172:387-398, 2014. URL: https://doi.org/10.1007/s10711-013-9924-4.
  26. Michel Ledoux and Michel Talagrand. Probability in Banach spaces, volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1991. Isoperimetry and processes. URL: https://doi.org/10.1007/978-3-642-20212-4.
  27. James R. Lee, Manor Mendel, and Assaf Naor. Metric structures in L₁: dimension, snowflakes, and average distortion. European J. Combin., 26(8):1180-1190, 2005. URL: https://doi.org/10.1016/j.ejc.2004.07.002.
  28. James R. Lee and Assaf Naor. Embedding the diamond graph in L_p and dimension reduction in L₁. Geom. Funct. Anal., 14(4):745-747, 2004. URL: https://doi.org/10.1007/s00039-004-0473-8.
  29. James R. Lee, Assaf Naor, and Yuval Peres. Trees and Markov convexity. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1028-1037. ACM, New York, 2006. URL: https://doi.org/10.1145/1109557.1109671.
  30. David A. Levin and Yuval Peres. Markov chains and mixing times. American Mathematical Society, Providence, RI, second edition, 2017. With contributions by Elizabeth L. Wilmer, With a chapter on "Coupling from the past" by James G. Propp and David B. Wilson. URL: https://doi.org/10.1090/mbk/107.
  31. Nathan Linial. Finite metric-spaces - Combinatorics, geometry and algorithms. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pages 573-586. Higher Ed. Press, Beijing, 2002. Google Scholar
  32. Nathan Linial, Avner Magen, and Assaf Naor. Girth and Euclidean distortion. Geom. Funct. Anal., 12(2):380-394, 2002. URL: https://doi.org/10.1007/s00039-002-8251-y.
  33. Jiří Matoušek. Note on bi-Lipschitz embeddings into normed spaces. Comment. Math. Univ. Carolin., 33(1):51-55, 1992. Google Scholar
  34. Jiří Matoušek. On the distortion required for embedding finite metric spaces into normed spaces. Israel J. Math., 93:333-344, 1996. URL: https://doi.org/10.1007/BF02761110.
  35. Jiří Matoušek. Using the Borsuk-Ulam theorem. Universitext. Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler. Google Scholar
  36. Manor Mendel and Assaf Naor. Metric cotype. Ann. of Math. (2), 168(1):247-298, 2008. URL: https://doi.org/10.4007/annals.2008.168.247.
  37. Assaf Naor. A spectral gap precludes low-dimensional embeddings. In 33rd International Symposium on Computational Geometry, volume 77 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 50, 16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  38. Assaf Naor. Metric dimension reduction: a snapshot of the Ribe program. In Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018. Vol. I. Plenary lectures, pages 759-837. World Sci. Publ., Hackensack, NJ, 2018. Google Scholar
  39. Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1):165-197, 2006. URL: https://doi.org/10.1215/S0012-7094-06-13415-4.
  40. Assaf Naor, Gilles Pisier, and Gideon Schechtman. Impossibility of dimension reduction in the nuclear norm [extended abstract]. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1345-1352. SIAM, Philadelphia, PA, 2018. URL: https://doi.org/10.1137/1.9781611975031.88.
  41. Assaf Naor and Gideon Schechtman. Remarks on non linear type and Pisier’s inequality. J. Reine Angew. Math., 552:213-236, 2002. URL: https://doi.org/10.1515/crll.2002.092.
  42. Assaf Naor and Robert Young. Foliated corona decompositions. Acta Math., 229(1):55-200, 2022. URL: https://doi.org/10.4310/acta.2022.v229.n1.a2.
  43. Krzysztof Oleszkiewicz. On a discrete version of the antipodal theorem. Fund. Math., 151(2):189-194, 1996. Google Scholar
  44. Gilles Pisier. Sur les espaces de Banach qui ne contiennent pas uniformément de l^1_n. C. R. Acad. Sci. Paris Sér. A-B, 277:A991-A994, 1973. Google Scholar
  45. Gilles Pisier. "Type" des espaces normés. In Séminaire Maurey-Schwartz 1973-1974: Espaces L^p, applications radonifiantes et géométrie des espaces de Banach, Exp. No. 3,, pages 12 pp. (errata, p. E.1). ,, 1974. Google Scholar
  46. Gilles Pisier. Martingales with values in uniformly convex spaces. Israel J. Math., 20(3-4):326-350, 1975. URL: https://doi.org/10.1007/BF02760337.
  47. Gilles Pisier. Un exemple concernant la super-réflexivité. In Séminaire Maurey-Schwartz 1974-1975: Espaces L^p applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2,, page 12. ,, 1975. Google Scholar
  48. Gilles Pisier. Probabilistic methods in the geometry of Banach spaces. In Probability and analysis (Varenna, 1985), volume 1206 of Lecture Notes in Math., pages 167-241. Springer, Berlin, 1986. URL: https://doi.org/10.1007/BFb0076302.
  49. Gilles Pisier and Quanhua Xu. Random series in the real interpolation spaces between the spaces v_p. In Geometrical aspects of functional analysis (1985/86), volume 1267 of Lecture Notes in Math., pages 185-209. Springer, Berlin, 1987. URL: https://doi.org/10.1007/BFb0078146.
  50. Santosh S. Vempala. The random projection method, volume 65 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence, RI, 2004. With a foreword by Christos H. Papadimitriou. URL: https://doi.org/10.1090/dimacs/065.
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