Electronic Journal of Probability

Estimating the covariance of random matrices

Pierre Youssef

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We extend to the matrix setting a recent result of Srivastava-Vershynin about estimating the covariance matrix of a random vector. The result can be interpreted as a quantified version of the law of large numbers for  positive semi-definite matrices which verify some regularity assumption. Beside giving examples, we discuss the notion of log-concave matrices and give estimates on the smallest and largest eigenvalues of a sum of such matrices.

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Electron. J. Probab., Volume 18 (2013), paper no. 107, 26 pp.

Accepted: 19 December 2013
First available in Project Euclid: 4 June 2016

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Youssef, Pierre. Estimating the covariance of random matrices. Electron. J. Probab. 18 (2013), paper no. 107, 26 pp. doi:10.1214/EJP.v18-2579. https://projecteuclid.org/euclid.ejp/1465064332

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  • Adamczak, Radosław; Guédon, Olivier; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole. Moment estimates for convex measures. Electron. J. Probab. 17 (2012), no. 101, 19 pp.
  • R. Adamczak, A. Litvak, A. Pajor, and N. Tomczak-Jaegermann. Tail estimates for norms of sums of log-concave random vectors. arXiv:1107.4070, July 2011.
  • Adamczak, Radosław; Litvak, Alexander E.; Pajor, Alain; Tomczak-Jaegermann, Nicole. Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Amer. Math. Soc. 23 (2010), no. 2, 535–561.
  • Adamczak, Radosław; Litvak, Alexander E.; Pajor, Alain; Tomczak-Jaegermann, Nicole. Sharp bounds on the rate of convergence of the empirical covariance matrix. C. R. Math. Acad. Sci. Paris 349 (2011), no. 3-4, 195–200.
  • Ahlswede, Rudolf; Winter, Andreas. Strong converse for identification via quantum channels. IEEE Trans. Inform. Theory 48 (2002), no. 3, 569–579.
  • Aubrun, Guillaume. Sampling convex bodies: a random matrix approach. Proc. Amer. Math. Soc. 135 (2007), no. 5, 1293–1303 (electronic).
  • Batson, Joshua D.; Spielman, Daniel A.; Srivastava, Nikhil. Twice-Ramanujan sparsifiers. STOC'09-Proceedings of the 2009 ACM International Symposium on Theory of Computing, 255–262, ACM, New York, 2009.
  • Bhatia, Rajendra. Matrix analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997. xii+347 pp. ISBN: 0-387-94846-5
  • Chafaï, Djalil; Guédon, Olivier; Lecué, Guillaume; Pajor, Alain. Interactions between compressed sensing random matrices and high dimensional geometry. Panoramas et Synthèses [Panoramas and Syntheses], 37. Société Mathématique de France, Paris, 2012. 181 pp. ISBN: 978-2-85629-370-6
  • M. De Carli Silva, N. Harvey, and c. Sato. Sparse Sums of Positive Semidefinite Matrices. arXiv:1107.0088v2, July 2011.
  • M. Fradelizi, O. Guédon, and A. Pajor. Spherical thin-shell concentration for convex measures. Available at arXiv:1306.6794.
  • Guédon, Olivier; Milman, Emanuel. Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21 (2011), no. 5, 1043–1068.
  • Hoeffding, Wassily. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 1963 13–30.
  • Kannan, Ravi; Lovász, László; Simonovits, Miklós. Random walks and an $O^ *(n^ 5)$ volume algorithm for convex bodies. Random Structures Algorithms 11 (1997), no. 1, 1–50.
  • Kolchin, Valentin F.; Sevast'yanov, Boris A.; Chistyakov, Vladimir P. Random allocations. Translated from the Russian. Translation edited by A. V. Balakrishnan. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.; distributed by Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. xi+262 pp. ISBN: 0-470-99394-4
  • Lewis, A. S. The convex analysis of unitarily invariant matrix functions. J. Convex Anal. 2 (1995), no. 1-2, 173–183.
  • L. Mackey, M. Jordan, R. Chen, B. Farrell, and J. Tropp. Matrix concentration inequalities via the method of exchangeable pairs. Available at arXiv:1201.6002.
  • Oliveira, Roberto Imbuzeiro. Sums of random Hermitian matrices and an inequality by Rudelson. Electron. Commun. Probab. 15 (2010), 203–212.
  • Paouris, G. Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), no. 5, 1021–1049.
  • Paouris, Grigoris. Small ball probability estimates for log-concave measures. Trans. Amer. Math. Soc. 364 (2012), no. 1, 287–308.
  • Rosenthal, Haskell P. On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8 1970 273–303.
  • Rudelson, M. Random vectors in the isotropic position. J. Funct. Anal. 164 (1999), no. 1, 60–72.
  • Srivastava, Nikhil. Spectral sparsification and restricted invertibility. Thesis (Ph.D.) - Yale University. ProQuest LLC, Ann Arbor, MI, 2010. 98 pp. ISBN: 978-1124-09198-3
  • Srivastava, Nikhil; Vershynin, Roman. Covariance estimation for distributions with $2+\varepsilon$ moments. Ann. Probab. 41 (2013), no. 5, 3081–3111.
  • Tropp, Joel A. User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12 (2012), no. 4, 389–434.