Bounding the Smallest Singular Value of a Random Matrix Without Concentration

Abstract

Given X a random vector in R n , set X 1 , ..., X N to be independent copies of X and let Γ = 1 √ N N i=1 X i , • e i be the matrix whose rows are X1 √ N , . . ., XN √ N .We obtain sharp probabilistic lower bounds on the smallest singular value λ min (Γ) in a rather general situation, and in particular, under the assumption that X is an isotropic random vector for which sup t∈S n-1 E| t, X | 2+η ≤ L for some L, η > 0. Our results imply that a Bai-Yin type lower bound holds for η > 2, and, up to a log-factor, for η = 2 as well.The bounds hold without any additional assumptions on the Euclidean norm X ℓ n 2 .Moreover, we establish a nontrivial lower bound even without any higher moment assumptions (corresponding to the case η = 0), if the linear forms satisfy a weak 'small ball' property.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF