The average singular value of a complex random matrix decreases with dimension

Abstract

We obtain a recurrence relation in $d$ for the average singular value $% α(d)$ of a complex valued $d\times d$\ matrix $\frac{1}{\sqrt{d}}X$ with random i.i.d., N( 0,1) entries, and use it to show that $α(d)$ decreases monotonically with $d$ to the limit given by the Marchenko-Pastur distribution.\ The monotonicity of $α(d)$ has been recently conjectured by Bandeira, Kennedy and Singer in their study of the Little Grothendieck problem over the unitary group $\mathcal{U}_{d}$ \cite{BKS}, a combinatorial optimization problem. The result implies sharp global estimates for $α(d)$, new bounds for the expected minimum and maximum singular values, and a lower bound for the ratio of the expected maximum and the expected minimum singular value. The proof is based on a connection with the theory of Turán determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.