Non-asymptotic bounds for the average singular value of a complex Gaussian matrix

Abstract

Let $G_{d}$ be a $d \times d$ matrix with independent standard complex Gaussian entries, let $α_{\mathbb{C}}(d)$ be the expected average singular value of $G_{d}/\sqrt{d}$, and set $Δ_d := α_{\mathbb{C}}(d)-α_{\mathbb{C}}(d+1)$. The statistic $α_{\mathbb{C}}(d)$ admits a variational representation as an expected normalized maximum over the unitary group and governs approximation guarantees for the little Grothendieck problem over the unitary group and related unitary registration problems. We obtain a strictly positive lower bound and an upper bound for $Δ_d$, both valid in every dimension, together with corresponding bounds for $α_{\mathbb{C}}(d)$ around the Marchenko--Pastur limit. These bounds match the sharp leading behavior of complete asymptotic expansions for both quantities, whose coefficients are explicitly computable. The proof combines a three-term recurrence for $Y_d = d^{3/2}α_{\mathbb{C}}(d)$, obtained from its continuous dual Hahn representation, along with singularity analysis of the underlying Laguerre moment generating function.