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

TL;DR

This paper proves that the average singular value of a complex random matrix decreases with dimension, confirming a recent conjecture and providing new bounds and estimates relevant to optimization and spectral analysis.

Contribution

It establishes the monotonic decrease of the average singular value with dimension and links it to orthogonal polynomial theory, addressing a conjecture in random matrix theory.

Findings

Average singular value decreases monotonically with dimension

Provides sharp bounds for expected singular values

Establishes a lower bound for the ratio of maximum to minimum singular values

Abstract

We obtain a recurrence relation in $d$ for the average singular value $% \alpha (d)$ 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 $\alpha (d)$ decreases monotonically with $d$ to the limit given by the Marchenko-Pastur distribution.\ The monotonicity of $\alpha (d)$ 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 $\alpha (d)$, 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 Tur\'{a}n determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.