Large-deviation asymptotics of condition numbers of random matrices. (December 2021) - Singull, Martin
Abstract: Let $\mathbf{X}$ be a $p\times n$ random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k ( p, n ) of $\mathbf{X}$ in terms of large deviations for large n, with p being fixed or $p=p(n)\rightarrow\infty$ with $p(n)=o(n)$ . We propose two main ingredients: (i) to relate the large-deviation probabilities of k ( p, n ) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k ( p, n ) using the upper tails of ratios of two independent $\chi^2$ random variables, which enables us to establish an application in statistical inference.