[Sur la plus petite valeur singulière de matrices aléatoires avec des colonnes indépendantes]
On étudie la plus petite valeur singulière d'une matrice carrée aléatoire dont les colonnes sont des vecteurs aléatoires i.i.d. suivant une loi à densité log-concave isotrope. On démontre une inégalité de déviation en fonction de la constante d'isotropie.
We study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic symmetric log-concave distribution. We prove a deviation inequality in terms of the isotropic constant of the distribution.
Accepté le :
Publié le :
@article{CRMATH_2008__346_15-16_853_0,
author = {Adamczak, Rados{\l}aw and Gu\'edon, Olivier and Litvak, Alexander and Pajor, Alain and Tomczak-Jaegermann, Nicole},
title = {Smallest singular value of random matrices with independent columns},
journal = {Comptes Rendus. Math\'ematique},
pages = {853--856},
publisher = {Elsevier},
volume = {346},
number = {15-16},
year = {2008},
doi = {10.1016/j.crma.2008.07.011},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2008.07.011/}
}
TY - JOUR AU - Adamczak, Radosław AU - Guédon, Olivier AU - Litvak, Alexander AU - Pajor, Alain AU - Tomczak-Jaegermann, Nicole TI - Smallest singular value of random matrices with independent columns JO - Comptes Rendus. Mathématique PY - 2008 SP - 853 EP - 856 VL - 346 IS - 15-16 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2008.07.011/ DO - 10.1016/j.crma.2008.07.011 LA - en ID - CRMATH_2008__346_15-16_853_0 ER -
%0 Journal Article %A Adamczak, Radosław %A Guédon, Olivier %A Litvak, Alexander %A Pajor, Alain %A Tomczak-Jaegermann, Nicole %T Smallest singular value of random matrices with independent columns %J Comptes Rendus. Mathématique %D 2008 %P 853-856 %V 346 %N 15-16 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2008.07.011/ %R 10.1016/j.crma.2008.07.011 %G en %F CRMATH_2008__346_15-16_853_0
Adamczak, Radosław; Guédon, Olivier; Litvak, Alexander; Pajor, Alain; Tomczak-Jaegermann, Nicole. Smallest singular value of random matrices with independent columns. Comptes Rendus. Mathématique, Tome 346 (2008) no. 15-16, pp. 853-856. doi : 10.1016/j.crma.2008.07.011. http://archive.numdam.org/articles/10.1016/j.crma.2008.07.011/
[1] Sampling convex bodies: a random matrix approach, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 1293-1303 (electronic)
[2] Geometric probability and random cotype 2, GAFA, Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 123-138
[3] -moments of random vectors via majorizing measures, Adv. Math., Volume 208 (2007), pp. 798-823
[4] Volume estimates for log-concave densities with application to iterated convolutions, Pacific J. Math., Volume 169 (1995), pp. 107-133
[5] Smallest singular value of random matrices and geometry of random polytopes, Adv. Math., Volume 195 (2005), pp. 491-523
[6] On singular values of matrices with independent rows, Bernoulli, Volume 12 (2006), pp. 761-773
[7] G. Paouris, personal communication
[8] Invertibility of random matrices: norm of the inverse, Ann. of Math., Volume 168 (2008), pp. 575-600
[9] The Littlewood–Offord problem and invertibility of random matrices, Adv. Math., Volume 218 (2008), pp. 600-633
[10] T. Tao, V. Vu, Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math., in press
Cité par Sources :
