The Annals of Applied Probability

Order statistics of vectors with dependent coordinates, and the Karhunen–Loève basis

Alexander E. Litvak and Konstantin Tikhomirov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not identically distributed coordinates and let $T$ be an orthogonal transformation of $\mathbb{R}^{n}$. We show that the random vector $Y=T(X)$ satisfies \begin{equation*}\mathbb{E}\sum_{j=1}^{k}j\mbox{-}\min_{i\leq n}{X_{i}}^{2}\leq C\mathbb{E}\sum_{j=1}^{k}j\mbox{-}\min_{i\leq n}{Y_{i}}^{2}\end{equation*} for all $k\leq n$, where “$j\mbox{-}\min$” denotes the $j$th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S. Mallat and O. Zeitouni regarding optimality of the Karhunen–Loève basis for the nonlinear signal approximation. As a by-product, we obtain some relations for order statistics of random vectors (not only Gaussian) which are of independent interest.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2083-2104.

Dates
Received: December 2016
Revised: May 2017
First available in Project Euclid: 9 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1533780268

Digital Object Identifier
doi:10.1214/17-AAP1321

Mathematical Reviews number (MathSciNet)
MR3843824

Zentralblatt MATH identifier
06974746

Subjects
Primary: 62G30: Order statistics; empirical distribution functions 60E15: Inequalities; stochastic orderings 60G15: Gaussian processes 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 94A08: Image processing (compression, reconstruction, etc.) [See also 68U10]

Keywords
Order statistics Karhunen–Loève basis nonlinear approximation INID case

Citation

Litvak, Alexander E.; Tikhomirov, Konstantin. Order statistics of vectors with dependent coordinates, and the Karhunen–Loève basis. Ann. Appl. Probab. 28 (2018), no. 4, 2083--2104. doi:10.1214/17-AAP1321. https://projecteuclid.org/euclid.aoap/1533780268


Export citation

References

  • [1] Adamczak, R., Latała, R., Litvak, A. E., Pajor, A. and Tomczak-Jaegermann, N. (2014). Tail estimates for norms of sums of log-concave random vectors. Proc. Lond. Math. Soc. (3) 108 600–637.
  • [2] David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd ed. Wiley-Interscience, Hoboken, NJ.
  • [3] Gluskin, E. D. (1989). Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Mat. Sb. 64 85–96.
  • [4] Gordon, Y., Litvak, A., Schütt, C. and Werner, E. (2002). Orlicz norms of sequences of random variables. Ann. Probab. 30 1833–1853.
  • [5] Gordon, Y., Litvak, A., Schütt, C. and Werner, E. (2002). Geometry of spaces between polytopes and related zonotopes. Bull. Sci. Math. 126 733–762.
  • [6] Gordon, Y., Litvak, A., Schütt, C. and Werner, E. (2005). Minima of sequences of Gaussian random variables. C. R. Math. Acad. Sci. Paris 340 445–448.
  • [7] Gordon, Y., Litvak, A. E., Schütt, C. and Werner, E. (2006). On the minimum of several random variables. Proc. Amer. Math. Soc. 134 3665–3675.
  • [8] Gordon, Y., Litvak, A. E., Schütt, C. and Werner, E. (2012). Uniform estimates for order statistics and Orlicz functions. Positivity 16 1–28.
  • [9] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities, 2nd ed. The Univ. Press, Cambridge.
  • [10] Latała, R. (2011). Order statistics and concentration of $l_{r}$ norms for log-concave vectors. J. Funct. Anal. 261 681–696.
  • [11] Lechner, R., Passenbrunner, M. and Prochno, J. (2015). Uniform estimates for averages of order statistics of matrices. Electron. Commun. Probab. 20 no. 27.
  • [12] Litvak, A. E., Pajor, A. and Tomczak-Jaegermann, N. (2006). Diameters of sections and coverings of convex bodies. J. Funct. Anal. 231 438–457.
  • [13] Litvak, A. E. and Tomczak-Jaegermann, N. (2000). Random aspects of high-dimensional convex bodies. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1745 169–190. Springer, Berlin.
  • [14] Mallat, S. (2009). A Wavelet Tour of Signal Processing: The Sparse Way, 3rd ed. Elsevier/Academic Press, Amsterdam.
  • [15] Mallat, S. and Zeitouni, O. A conjecture concerning optimality of the Karhunen–Loeve basis in nonlinear reconstruction. Available at arXiv:1109.0489.
  • [16] Marshall, A. W. and Proschan, F. (1965). An inequality for convex functions involving majorization. J. Math. Anal. Appl. 12 87–90.
  • [17] Montgomery-Smith, S. (2002). Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables. Israel J. Math. 131 51–60.
  • [18] Šidák, Z. (1967). Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62 626–633.
  • [19] Zeitouni, O. (2006). A correlation inequality for nonlinear reconstruction. In Workshop on the Mathematical Foundations of Learning Theory, Paris.