• Bernoulli
  • Volume 12, Number 5 (2006), 761-773.

On singular values of matrices with independent rows

Shahar Mendelson and Alain Pajor

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We present deviation inequalities of random operators of the form N - 1 i =1 NX iX i from the average operator E (XX) , where X i are independent random vectors distributed as X , which is a random vector in R n or in 2 . We use these inequalities to estimate the singular values of random matrices with independent rows (without assuming that the entries are independent).

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Bernoulli, Volume 12, Number 5 (2006), 761-773.

First available in Project Euclid: 23 October 2006

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random vectors in R^n singular values of integral operators


Mendelson, Shahar; Pajor, Alain. On singular values of matrices with independent rows. Bernoulli 12 (2006), no. 5, 761--773. doi:10.3150/bj/1161614945.

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  • [1] Alesker, S. (1995) l2 estimates for the Euclidean norm on a convex body in isotropic position. Oper. Theory Adv. Appl., 77, 1-4.
  • [2] Bárány, I. and Füredy, Z. (1988) Approximation of the sphere by polytopes having few vertices. Proc. Amer. Math. Soc., 102(3), 651-659.
  • [3] Bourgain, J. (1999) Random points in isotropic convex bodies. In K.M. Ball and V. Milman (eds) Convex Geometric Analysis, Math. Sci. Res. Inst. Publ. 34, pp. 53-58. Cambridge: Cambridge University Press.
  • [4] Carl, B. and Pajor, A. (1988) Gelfand numbers of operators with values in a Hilbert space. Invent. Math., 94, 479-504.
  • [5] Davidson, K. and Szarek, S. (2001) Local operator theory, random matrices and Banach spaces. In W.B. Johnson and J. Lindenstrauss (eds), Handbook of the Geometry of Banach Spaces, Vol. I, pp. 317-366. Amsterdam: Elsevier.
  • [6] de la Pena, V. and Giné, E. (1999) Decoupling. New York: Springer-Verlag.
  • [7] Giannopoulos, A.A. and Milman, V.D. (2000) Concentration property on probability spaces. Adv. Math., 156(1), 77-106.
  • [8] Gluskin, E.D. (1988) Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Mat. Sb. (N.S.) 136(178), 85-96; translation in Math. USSR Sb. 64(1), 85-96 (1989).
  • [9] Gohberg, I. and Goldberg, G. (1980) Basic Operator Theory. Basel: Birkhäuser.
  • [10] Kannan, R., Lovász, L. and Simonovits, M. (1997) Random walks and O*(n5) volume algorithm for convex bodies. Random Structures Algorithms, 2(1), 1-50.
  • [11] Koltchinskii, V. (1998) Asymptotics of spectral projections of some random matrices approximating integral operators. In E. Eberlein, M. Hahn and M. Talagrand (eds), High Dimensional Probability, Progr. Probab. 43, pp. 191-227. Basel: Birkhäuser.
  • [12] Koltchinskii, V. and Giné, E. (2000) Random matrix approximation of spectra of integral operators. Bernoulli, 6, 113-167.
  • [13] Ledoux, M. (2001) The Concentration of Measure Phenomenon, Math. Surveys Monogr. 89. Providence, RI: American Mathematical Society.
  • [14] Litvak, A., Pajor, A., Rudelson, M. and Tomczak-Jaegermann, N. (2005) Smallest singular value of random matrices and geometry of random polytopes. Adv. Math., 195, 491-523.
  • [15] Lust-Piquard, F. and Pisier, G. (1991) Non-commutative Khinchine and Paley inequalities. Ark. Mat., 29, 241-260.
  • [16] Mendelson, S. (2003) On the performance of kernel classes. J. Mach. Learn. Res., 4, 759-771.
  • [17] Milman, V.D. and Pajor, A. (1989) Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In J. Lindenstrauss and V.D. Milman (eds) Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1376, pp. 64-104. Berlin: Springer-Verlag.
  • [18] Milman, V.D. and Schechtman, G. (1986) Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200. Berlin: Springer-Verlag.
  • [19] Paouris, G. (2005) On the l2 behaviour of linear functionals on isotropic convex bodies. Studia Math., 168, 285-299.
  • [20] Pisier, G. (1983) Some applications of the metric entropy condition to harmonic analysis. In R.C. Blei and S.J. Sidney (eds), Banach Spaces, Harmonic Analysis, and Probability Theory, Lecture Notes in Math. 995, pp. 123-154. Berlin: Springer-Verlag.
  • [21] Rudelson, M. (1999) Random vectors in the isotropic position. J. Funct. Anal., 164, 60-72.
  • [22] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. New York: Springer-Verlag.