Condition number of a square matrix with i.i.d. columns drawn from a convex body

Abstract

We study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic log-concave distribution. An important example is obtained by sampling vectors uniformly distributed in an isotropic convex body. We deduce that the condition number of such matrices is of the order of the size of the matrix and give an estimate on its tail behavior.

Small-Ball Probabilities for the Volume of Random Convex Sets

“…2 and 9 for definitions). In the last several years, there have been many important results concerning random matrices generated by log-concave measures, see e.g., [1,2] and the references therein. In this important class we obtain more precise estimates, such as the following theorem.…”

Circular law for random matrices with unconditional log-concave distribution

“…In view of the results for Gaussian matrices [38] it is natural to conjecture that for ε ∈ (0, 1), P(s n (A n + M n ) ≤ εn −1/2 ) ≤ Cε. For a matrix A n with independent log-concave isotropic columns it is proven in [3] …”

Small deviations of the determinants of random matrices with Gaussian entries

“…Moreover, the problem itself seems to be of interest. Regarding the topic, there are papers devoted to the small deviations for the smallest singular values of random matrices (see Adamczak et al (2012) and references therein). As for results for determinants, Li and Weil (2008) obtained the distribution of the determinant for the i.i.d.…”