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Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 495–522 | Cite as

On Optimal Matching of Gaussian Samples

  • M. LedouxEmail author
Article
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Let X1, . . .,Xn be independent random variables having as common distribution the standard Gaussian measure μ on ℝ2 and let \( {\mu}_n=\frac{1}{n}\sum \limits_{i=1}^n{\delta}_{X_i} \) be the associated empirical measure. We show that

\( \frac{1}{C}\frac{\log n}{n}\le \) 𝔼 \( \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} \)

for some numerical constant C > 0, where W2 is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan.

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References

  1. 1.
    M. Ajtai, J. Komlós, and G. Tusnády, “On optimal matchings,” Combinatorica, 4, 259–264 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    L. Ambrosio, N. Gigli, and G. Savaré, “Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds,” Ann. Probab., 43, 339–404 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L. Ambrosio, A. Mondino, and G. Savaré, “Nonlinear diffusion equations and curvature conditions in metric measure spaces” (2015).Google Scholar
  4. 4.
    L. Ambrosio, F. Stra, and D. Trevisan, “A PDE approach to a 2-dimensional matching problem,” (2016). To appear in Probab. Theory Related Fields.Google Scholar
  5. 5.
    D. Bakry, “Étude des transformations de Riesz sur les variétés riemanniennes à courbure de Ricci minorée,” in: Séminaire de Probabilités XXI, Lect. Notes Math., 1247 (1987), pp. 137–172.Google Scholar
  6. 6.
    D. Bakry, I. Gentil, and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der mathematischen Wissenschaften, 348, Springer (2014).Google Scholar
  7. 7.
    F. Barthe and C. Bordenave, “Combinatorial optimization over two random point sets,” in: Séminaire de Probabilités XLV, Lect. Notes Math., 2078 (2013), pp. 483–535.Google Scholar
  8. 8.
    S. Bobkov and M. Ledoux, One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances (2016). To appear in Memoirs Amer. Math. Soc.Google Scholar
  9. 9.
    E. Boissard and T. Le Gouic, “On the mean speed of convergence of empirical and occupation measures in Wasserstein distance,” Ann. Inst. Henri Poincaré Probab. Stat., 50, 539–563 (2014).Google Scholar
  10. 10.
    F. Bolley, A. Guillin, and C. Villani, “Quantitative concentration inequalities for empirical measures on noncompact spaces,” Probab. Theory Related Fields, 137, 541–593 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J. Boutet de Monvel and O. Martin, “Almost sure convergence of the minimum bipartite matching functional in Euclidean space,” Combinatorica, 22, 523–530 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    S. Caracciolo, C. Lucibello, G. Parisi, and G. Sicuro, “Scaling hypothesis for the Euclidean bipartite matching problem,” Physical Review E, 90, 012118 (2014).Google Scholar
  13. 13.
    I. Chavel, Riemannian Geometry. A Modern Introduction, 2nd ed., Cambridge Studies in Advanced Mathematics, 98, Cambridge Univ. Press (2006).Google Scholar
  14. 14.
    E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge Univ. Press (1989).Google Scholar
  15. 15.
    S. Dereich, M. Scheutzow, and R. Schottstedt, “Constructive quantization: approximation by empirical measures,” Ann. Inst. Henri Poincaré Probab. Stat., 49, 1183–1203 (2013).Google Scholar
  16. 16.
    V. Dobrić and J. Yukich, “Asymptotics for transportation cost in high dimensions,” J. Theoret. Probab., 8, 97–118 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    R. Dudley, “The speed of mean Glivenko–Cantelli convergence,” Ann. Math. Statist., 40, 40–50 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Erbar, K. Kuwada, and K.-T. Sturm, “On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces,” Invent. Math., 201, 993–1071 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, Amer. Math. Soc. (1998).Google Scholar
  20. 20.
    N. Fournier and A. Guillin, “On the rate of convergence in Wasserstein distance of the empirical measure,” Probab. Theory Related Fields, 162, 707–738 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Hahn and Y. Shao, “An exposition of Talagrand’s mini-course on matching theorems.” in: Probability in Banach spaces, 8, Progr. Probab., 30, Birkh¨auser (1992), pp. 3–38.Google Scholar
  22. 22.
    N. Holden, Y. Peres, and A. Zhai, “Gravitational allocation for uniform points on the sphere” (2017).Google Scholar
  23. 23.
    W. Johnson, G. Schechtman, and J. Zinn, “Best constants in moment inequalities for linear combinations of independent and exchangeable random variables,” Ann. Probab., 13, 234–253 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    K. Kuwada, “Duality on gradient estimates and Wasserstein controls,” J. Funct. Anal., 258, 3758–3774 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    T. Leighton and P. Shor, ‘Tight bounds for minimax grid matching with applications to the average case analysis of algorithms,” Combinatorica, 9, 161–187 (1989).Google Scholar
  26. 26.
    P.-A. Meyer, “Transformations de Riesz pour les lois gaussiennes,” in: Séminaire de Probabilités XV, Lect. Notes Math., 1059 (1984), pp. 179–193.Google Scholar
  27. 27.
    F. Otto and C. Villani, “Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality,” J. Funct. Anal., 173, 361–400 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    H. P. Rosenthal, “On the subspaces of L p (p > 2) spanned by sequences of independent random variables,” Israel J. Math., 8, 273–303 (1970).Google Scholar
  29. 29.
    P. Shor, “How to pack better than Best Fit: Tight bounds for average-case on-line bin packing,” in: Proc. 32nd Annual Symposium on Foundations of Computer Sciences (1991), pp. 752–759.Google Scholar
  30. 30.
    P. Shor and J. Yukich, “Minimax grid matching and empirical measures,” Ann. Probab., 19, 1338–1348 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    M. Talagrand, “Matching random samples in many dimensions,” Ann. App. Probab., 2, 846–856 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    M. Talagrand, “The transportation cost from the uniform measure to the empirical measure in dimension 3,” Ann. Probab., 22, 919–959 (1994).Google Scholar
  33. 33.
    M. Talagrand, “Matching theorems and discrepency computations using majorising measures,” J. Amer. Math. Soc., 17, 455–537 (1994).Google Scholar
  34. 34.
    M. Talagrand, Upper and Lower Bounds of Stochastic Processes, Modern Surveys in Mathematics, 60, Springer-Verlag (2014).Google Scholar
  35. 35.
    N. Varopoulos, “Hardy–Littlewood theory for semi-groups,” J. Funct. Anal., 11, 240–260 (1985).Google Scholar
  36. 36.
    C. Villani, Optimal Transport. Old and New, Grundlehren der mathematischen Wissenschaften, 338, Springer (2009).Google Scholar
  37. 37.
    F.-Y. Wang, Analysis for Diffusion Processes on Riemannian Manifolds, World Scientific (2014).Google Scholar
  38. 38.
    J. Yukich, “Some generalizations of the Euclidean two-sample matching problem,” in: Probability in Banach spaces, 8, Progr. Probab., 30, Birkhäuser (1992), pp. 55–66.Google Scholar
  39. 39.
    J. Yukich, “Probability theory of classical Euclidean optimization problems,” Lect. Notes Math., 1675 (1998).Google Scholar

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Authors and Affiliations

  1. 1.Université de Toulouse–Paul-SabatierToulouseFrance
  2. 2.Institut Universitaire de FranceParisFrance

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