On the singular values of random matrices

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on R^n that is supported in a relatively small ball and linear functionals are uniformly bounded in L_p for some p>8 , in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c√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".