JournalsjemsVol. 16, No. 4pp. 823–834

On the singular values of random matrices

  • Shahar Mendelson

    Technion - Israel Institute of Technology, Haifa, Israel
  • Grigoris Paouris

    Texas A&M University, College Station, USA
On the singular values of random matrices cover

Abstract

We present an approach that allows one to bound the largest and smallest singular values of an N×nN \times n random matrix with iid rows, distributed according to a measure on Rn\mathbb R^n that is supported in a relatively small ball and linear functionals are uniformly bounded in LpL_p for some p>8p>8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1±cn/N1 \pm c\sqrt{n/N} not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure of iid random variables with "heavy tails".

Cite this article

Shahar Mendelson, Grigoris Paouris, On the singular values of random matrices. J. Eur. Math. Soc. 16 (2014), no. 4, pp. 823–834

DOI 10.4171/JEMS/448