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On the quality of randomized approximations of Tukey's depth (2309.05657v2)

Published 11 Sep 2023 in stat.ML, cs.LG, and math.PR

Abstract: Tukey's depth (or halfspace depth) is a widely used measure of centrality for multivariate data. However, exact computation of Tukey's depth is known to be a hard problem in high dimensions. As a remedy, randomized approximations of Tukey's depth have been proposed. In this paper we explore when such randomized algorithms return a good approximation of Tukey's depth. We study the case when the data are sampled from a log-concave isotropic distribution. We prove that, if one requires that the algorithm runs in polynomial time in the dimension, the randomized algorithm correctly approximates the maximal depth $1/2$ and depths close to zero. On the other hand, for any point of intermediate depth, any good approximation requires exponential complexity.

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References (47)
  1. Greg Aloupis. Geometric measures of data depth. DIMACS series in discrete mathematics and theoretical computer science, 72:147, 2006.
  2. The complexity and approximability of finding maximum feasible subsystems of linear relations. Theoretical Computer Science, 147(1-2):181–210, 1995.
  3. Concentration inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, 2013.
  4. Half-space depth of log-concave probability measures, 2022. URL https://arxiv.org/abs/2201.11992.
  5. Output-sensitive algorithms for Tukey depth and related problems. Statistics and Computing, 18:259–266, 2008.
  6. Deterministic and randomized polynomial-time approximation of radii. Mathematika. A Journal of Pure and Applied Mathematics, 48(1-2):63–105, 2001.
  7. Victor-Emmanuel Brunel. Concentration of the empirical level sets of Tukey’s halfspace depth. Probability Theory and Related Fields, 173(3):1165–1196, 2019.
  8. Timothy M Chan. An optimal randomized algorithm for maximum Tukey depth. In SODA, volume 4, pages 430–436, 2004.
  9. Absolute approximation of Tukey depth: Theory and experiments. Computational Geometry, 46(5):566–573, 2013.
  10. Robust covariance and scatter matrix estimation under Huber’s contamination model. The Annals of Statistics, 46(5):1932–1960, 2018.
  11. The random Tukey depth. Computational Statistics & Data Analysis, 52(11):4979–4988, 2008.
  12. Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electronic Journal of Statistics, 4:254 – 270, 2010.
  13. David Donoho. Breakdown properties of multivariate location estimators. Technical report, Technical report, Harvard University, 1982.
  14. Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness. The Annals of Statistics, 20(4):1803 – 1827, 1992.
  15. Exact computation of the halfspace depth. Computational Statistics & Data Analysis, 98:19–30, 2016.
  16. Zonoid data depth: Theory and computation. In COMPSTAT: Proceedings in Computational Statistics, pages 235–240. Springer, 1996.
  17. Approximate computation of projection depths, 2020. URL https://arxiv.org/abs/2007.08016.
  18. Bounding the norm of a log-concave vector via thin-shell estimates. In Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2011-2013, pages 107–122. Springer, 2014.
  19. Paul Funk. Über eine geometrische Anwendung der Abelschen Integralgleichung. Mathematische Annalen, 77(1):129–135, 1915.
  20. Robert D Gordon. Values of Mills’ ratio of area to bounding ordinate and of the normal probability integral for large values of the argument. The Annals of Mathematical Statistics, 12(3):364–366, 1941.
  21. The densest hemisphere problem. Theoretical Computer Science, 6(1):93–107, 1978.
  22. B. Klartag. A central limit theorem for convex sets. Inventiones Mathematicae, 168(1):91–131, 2007a.
  23. B. Klartag. Power-law estimates for the central limit theorem for convex sets. Journal of Functional Analysis, 245(1):284–310, 2007b.
  24. Bourgain’s slicing problem and KLS isoperimetry up to polylog. Geometric and Functional Analysis, 32(5):1134–1159, 2022.
  25. Zonoid trimming for multivariate distributions. The Annals of Statistics, 25(5):1998–2017, 1997.
  26. M. Ledoux. The Concentration of Measure Phenomenon. American Mathematical Society, 2001.
  27. P. Lévy. Problèmes conrets d’analyse fonctionelle. Gauthier-Villars, 1951.
  28. Regina Y Liu. On a notion of simplicial depth. Proceedings of the National Academy of Sciences, 85(6):1732–1734, 1988.
  29. Regina Y Liu. On a notion of data depth based on random simplices. The Annals of Statistics, pages 405–414, 1990.
  30. Regina Y Liu. Data depth and multivariate rank tests. L1-statistical analysis and related methods, pages 279–294, 1992.
  31. Erwin Lutwak. Chapter 1.5 - selected affine isoperimetric inequalities. In P.M. GRUBER and J.M. WILLS, editors, Handbook of Convex Geometry, pages 151–176. North-Holland, Amsterdam, 1993.
  32. J. Matoušek. Lectures on Discrete Geometry. Springer, 2002.
  33. Karl Mosler. Multivariate dispersion, central regions, and depth: the lift zonoid approach, volume 165. Springer Science & Business Media, 2002.
  34. Choosing among notions of multivariate depth statistics, 2021.
  35. Data depth and floating body. Statistics Surveys, 13, 2019.
  36. Uniform convergence rates for the approximated halfspace and projection depth. Electronic Journal of Statistics, 14(2):3939–3975, 2020.
  37. A. Prékopa. On logarithmic concave measures and functions. Acta Sci. Math.(Szeged), 34:335–343, 1973.
  38. Richard J. Samworth. Recent Progress in Log-Concave Density Estimation. Statistical Science, 33(4):493 – 509, 2018.
  39. Log-concavity and strong log-concavity: a review. Statistics Surveys, 8:45, 2014.
  40. Erhard Schmidt. Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie. I. Mathematische Nachrichten, 1(2-3):81–157, 1948.
  41. Rolf Schneider. Functional equations connected with rotations and their geometric applications. Enseignenment Math.(2), 16:297–305, 1970.
  42. Employing the MCMC technique to compute the projection depth in high dimensions. Journal of Computational and Applied Mathematics, 411:114278, 2022.
  43. Werner A Stahel. Robuste schätzungen: infinitesimale optimalität und schätzungen von kovarianzmatrizen. PhD thesis, ETH Zürich, 1981.
  44. J. W. Tukey. Mathematics and the picturing of data. Proceedings of the International Congress of Mathematicians, Vancouver, 1975, 2:523–531, 1975. URL https://ci.nii.ac.jp/naid/10029477185/en/.
  45. Yijun Zuo. A new approach for the computation of halfspace depth in high dimensions. Communications in Statistics - Simulation and Computation, 48(3):900–921, 2019.
  46. General notions of statistical depth function. Annals of Statistics, pages 461–482, 2000a.
  47. On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. Journal of Statistical Planning and Inference, 84(1-2):55–79, 2000b.
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