Bounding the Smallest Singular Value of a Random Matrix Without Concentration

Vladimir Koltchinskii, Shahar Mendelson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

83 Citations (Scopus)

Abstract

Given a random vector X in Rn, set X1,·, XN to be independent copies of X and let Γ = 1/√N∑i=1N (Xi, ) ei be the matrix whose rows are X1√ N,·, XN√N. We obtain new probabilistic lower bounds on the smallest singular value λmin(Γ) in a rather general situation, and in particular, under the assumption that X is an isotropic random vector for which ⊃t\in Sn-1P t, X|u L/u2+η 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|ℓ2n. 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. These estimates follow from general lower bounds on the infimum of the quadratic empirical process f →N-1i=1N f2(Xi) which are of independent interest.

Original languageEnglish
Pages (from-to)12991-13008
Number of pages18
JournalInternational Mathematics Research Notices
Volume2015
Issue number23
DOIs
Publication statusPublished - 2015
Externally publishedYes

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