Abstract
We analyze the behavior of a random matrix with independent rows, each distributed according to the same probability measure on \({\mathbb R}^{n}\) or on ℓ2. We investigate the spectrum of such a matrix and the way the ellipsoid generated by it approximates the covariance structure of the underlying measure. As an application, we provide estimates on the deviation of the spectrum of Gram matrices from the spectrum of the integral operator.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alesker, S.: ψ 2 estimates for the Euclidean norm on a convex body in isotropic position. Operator Theory Adv. Appl. 77, 1–4 (1995)
Bourgain, J.: Random points in isotropic convex bodies, in Convex Geometric Analysis (Berkeley, CA, 1996). Math. Sci. Res. Inst. Publ. 34, 53–58 (1999)
Giannopoulos, A.A., Milman, V.D.: Concentration property on probability spaces. Adv. Math. 156, 77–106 (2000)
Herbrich, R.: Learning kernel classifiers. MIT Press, Cambridge (2002)
Johnson, W.B., Schechtman, G.: Finite dimensional subspaces of L p . In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 1, North-Holland, Amsterdam (2001)
Kannan, R., Lovász, L., Simonovits, M.: Random walks and O *(n 5) volume algorithm for convex bodies. Random structures and algorithms 2(1), 1–50 (1997)
Kato, T.: A short introduction to perturbation theory for linear operators. Springer, Heidelberg (1982)
Koltchinskii, V., Giné, E.: Random matrix approximation of spectra of integral operators. Bernoulli 6, 113–167 (2000)
Lust-Piquard, F., Pisier, G.: Non-commutative Khinchine and Paley inequalities. Ark. Mat. 29, 241–260 (1991)
Milman, V.D., Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In: Parisi-Presicce, F. (ed.) WADT 1997. Lecture notes in mathematics, vol. 1376, pp. 64–104. Springer, Heidelberg (1989)
Schölkopf, B., Smola, A.J.: Learning with kernels. MIT Press, Cambridge (2002)
Rudelson, M.: Random vectors in the isotropic position. Journal of Functional Analysis 164, 60–72 (1999)
Talagrand, M.: The generic chaining (forthcoming)
Van der Vaart, A.W., Wellner, J.A.: Weak convergence and Empirical Processes. Springer, Heidelberg (1996)
Zwald, L., Bousquet, O., Blanchard, G.: Statistical properties of kernel principal component analysis. In: Shawe-Taylor, J., Singer, Y. (eds.) COLT 2004. LNCS (LNAI), vol. 3120, pp. 594–608. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mendelson, S., Pajor, A. (2005). Ellipsoid Approximation Using Random Vectors. In: Auer, P., Meir, R. (eds) Learning Theory. COLT 2005. Lecture Notes in Computer Science(), vol 3559. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11503415_29
Download citation
DOI: https://doi.org/10.1007/11503415_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26556-6
Online ISBN: 978-3-540-31892-7
eBook Packages: Computer ScienceComputer Science (R0)