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.
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.
Accepted:
Published online:
Radosław Adamczak 1; Olivier Guédon 2; Alexander Litvak 3; Alain Pajor 4; Nicole Tomczak-Jaegermann 3
@article{CRMATH_2008__346_15-16_853_0,
author = {Rados{\l}aw Adamczak and Olivier Gu\'edon and Alexander Litvak and Alain Pajor and Nicole Tomczak-Jaegermann},
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},
}
TY - JOUR AU - Radosław Adamczak AU - Olivier Guédon AU - Alexander Litvak AU - Alain Pajor AU - Nicole Tomczak-Jaegermann 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 DO - 10.1016/j.crma.2008.07.011 LA - en ID - CRMATH_2008__346_15-16_853_0 ER -
%0 Journal Article %A Radosław Adamczak %A Olivier Guédon %A Alexander Litvak %A Alain Pajor %A Nicole Tomczak-Jaegermann %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 %R 10.1016/j.crma.2008.07.011 %G en %F CRMATH_2008__346_15-16_853_0
Radosław Adamczak; Olivier Guédon; Alexander Litvak; Alain Pajor; Nicole Tomczak-Jaegermann. Smallest singular value of random matrices with independent columns. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 853-856. doi : 10.1016/j.crma.2008.07.011. https://comptes-rendus.academie-sciences.fr/mathematique/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
Cited by Sources:
Comments - Policy