Electronic Journal of Probability

Moment inequalities for matrix-valued U-statistics of order 2

Stanislav Minsker and Xiaohan Wei

Full-text: Open access

Abstract

We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative Khintchine inequality for the spectral norm of the Rademacher chaos, could be of independent interest.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 133, 32 pp.

Dates
Received: 1 August 2018
Accepted: 7 November 2019
First available in Project Euclid: 13 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1573614085

Digital Object Identifier
doi:10.1214/19-EJP392

Subjects
Primary: 60E15: Inequalities; stochastic orderings

Keywords
U-statistics moment inequalities concentration inequalities Khintchine inequality

Rights
Creative Commons Attribution 4.0 International License.

Citation

Minsker, Stanislav; Wei, Xiaohan. Moment inequalities for matrix-valued U-statistics of order 2. Electron. J. Probab. 24 (2019), paper no. 133, 32 pp. doi:10.1214/19-EJP392. https://projecteuclid.org/euclid.ejp/1573614085


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References

  • [1] Radoslaw Adamczak, Moment inequalities for U-statistics, Ann. Probab. 34 (2006), no. 6, 2288–2314.
  • [2] Radoslaw Adamczak and Rafal Latala, The LIL for U-statistics in Hilbert spaces, Journal of Theoretical Probability 21 (2008), no. 3, 704–744.
  • [3] Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, Concentration inequalities: A nonasymptotic theory of independence, Oxford University Press, 2013.
  • [4] Jean-Christophe Bourin and Eun-Young Lee, Unitary orbits of hermitian operators with convex or concave functions, Bulletin of the London Mathematical Society 44 (2012), no. 6, 1085–1102.
  • [5] Richard Y. Chen, Alex Gittens, and Joel A. Tropp, The masked sample covariance estimator: an analysis using matrix concentration inequalities, Information and Inference (2012).
  • [6] Xiaohui Chen, On bootstrap approximations for high-dimensional U-statistics and random quadratic forms, arXiv preprint arXiv:1610.00032 (2016).
  • [7] V. de la Pena and E. Gine, Decoupling: From dependence to independence, Springer-Verlag, New York, 1999.
  • [8] V. de la Pena and S. J. Montgomery-Smith, Decoupling inequalities for the tail probabilities of multivariate U-statistics, Annals of Probability 23 (1995), no. 2, 806–816.
  • [9] Sjoerd Dirksen et al., Tail bounds via generic chaining, Electronic Journal of Probability 20 (2015).
  • [10] Simon Foucart and Holger Rauhut, A mathematical introduction to compressive sensing, vol. 1, Birkhäuser Basel, 2013.
  • [11] Evarist Giné, Stanislaw Kwapien, Rafal Latala, and Joel Zinn, The LIL for canonical U-statistics of order 2, Annals of Probability (2001), 520–557.
  • [12] Evarist Gine, Rafal Latala, and Joel Zinn, Exponential and moment inequalities for U-statistics, High Dimensional Probability II (2000), 13–38.
  • [13] Evarist Giné and Richard Nickl, Mathematical foundations of infinite-dimensional statistical models, vol. 40, Cambridge University Press, 2016.
  • [14] Evarist Giné and Joel Zinn, On Hoffmann-Jørgensen’s inequality for U-processes, Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference, Springer, 1992, pp. 80–91.
  • [15] Fang Han and Han Liu, Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution, Bernoulli: Official Journal of the Bernoulli Society for Mathematical Statistics and Probability 23 (2017), no. 1, 23.
  • [16] Wassily Hoeffding, A class of statistics with asymptotically normal distribution, The Annals of Mathematical Statistics (1948), 293–325.
  • [17] Christian Houdré and Patricia Reynaud-Bouret, Exponential inequalities, with constants, for U-statistics of order two, Stochastic inequalities and applications, Springer, 2003, pp. 55–69.
  • [18] Rustam Ibragimov and Sh. Sharakhmetov, Analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal inequalities for symmetric statistics, Scandinavian Journal of Statistics 26 (1999), no. 4, 621–633.
  • [19] Marius Junge, Qiang Zeng, et al., Noncommutative Bennett and Rosenthal inequalities, The Annals of Probability 41 (2013), no. 6, 4287–4316.
  • [20] Vladimir S. Korolyuk and Yu V. Borovskich, Theory of U-statistics, vol. 273, Springer Science & Business Media, 2013.
  • [21] Jeanne Kowalski and Xin M. Tu, Modern applied $U$-statistics, vol. 714, John Wiley & Sons, 2008.
  • [22] M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer-Verlag, Berlin, 2011.
  • [23] Françoise Lust-Piquard, Inégalités de Khintchine dans $c_{p}$ $(1< p<\infty )$, CR Acad. Sci. Paris 303 (1986), 289–292.
  • [24] Françoise Lust-Piquard and Gilles Pisier, Noncommutative Khintchine and Paley inequalities, Arkiv för Matematik 29 (1991), no. 1, 241–260.
  • [25] Lester Mackey, Michael I. Jordan, Richard Y. Chen, Brendan Farrell, Joel A. Tropp, et al., Matrix concentration inequalities via the method of exchangeable pairs, The Annals of Probability 42 (2014), no. 3, 906–945.
  • [26] Stanislav Minsker and Xiaohan Wei, Robust modifications of U-statistics and applications to covariance estimation problems, arXiv preprint arXiv:1801.05565 (2018).
  • [27] Gilles Pisier, Non-commutative vector-valued $l_{p}$ spaces and completely $p$-summing maps, Asterisque – Societe Mathematique de France 247 (1998).
  • [28] H. Rauhut, Compressive sensing and structured random matrices, Radon Series Comp. Appl. Math (2012), 1–94.
  • [29] Holger Rauhut, Circulant and Toeplitz matrices in compressed sensing, arXiv preprint arXiv:0902.4394 (2009).
  • [30] Robert J. Serfling, Approximation theorems of mathematical statistics, vol. 162, John Wiley & Sons, 2009.
  • [31] J. A. Tropp, An introduction to matrix concentration inequalities, Foundations and Trends in Machine Learning (2015), 1–230.
  • [32] Joel Tropp, On the conditioning of random subdictionaries, Applied and Computational Harmonic Analysis 25 (2008), no. 1, 1–24.
  • [33] Joel A. Tropp, User-friendly tail bounds for sums of random matrices, Foundations of Computational Mathematics 12 (2012), no. 4, 389–434.
  • [34] Joel A. Tropp, The expected norm of a sum of independent random matrices: an elementary approach, High Dimensional Probability VII, Springer, 2016, pp. 173–202.
  • [35] Joel A. Tropp, Second-order matrix concentration inequalities, Applied and Computational Harmonic Analysis (2016).
  • [36] R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, arXiv preprint arXiv:1011.3027 (2010).
  • [37] Marten Wegkamp and Yue Zhao, Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas, Bernoulli 22 (2016), no. 2, 1184–1226.