|
|||||
|
|
Smallest singular value of random matrices and geometry of random polytopes |
|
| Authors: | AE Litvak M Rudelson N Tomczak-Jaegermann |
| Institution: | a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1 b Equipe d’Analyse et Mathématiques Appliquées, Université de Marne-la-Vallée, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée Cedex 2, France c Department of Mathematics, 202 Mathematical Sciences Building, University of Missouri, Columbia, MO 65211, USA |
| Abstract: | We study the behaviour of the smallest singular value of a rectangular random matrix, i.e., matrix whose entries are independent random variables satisfying some additional conditions. We prove a deviation inequality and show that such a matrix is a “good” isomorphism on its image. Then, we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes (absolutely convex hulls of rows of random matrices). All our results hold with high probability, that is, with probability exponentially (in dimension) close to 1. |
| Keywords: | Random matrices Random polytopes Singular values Deviation inequalities |
| 本文献已被 ScienceDirect 等数据库收录！ | |