We settle a conjecture of Bandeira, Kennedy and Singer, arising from their analysis of approximation ratios for the little Grothendieck problem over the unitary group, by proving that the average singular value of a normalized square complex Gaussian random matrix strictly decreases with the dimension. The starting point is a recurrence relation of Abreu, derived from the Christoffel--Darboux formula and a Tur\'{a}n determinant for Laguerre polynomials. This recurrence reduces the problem to the estimation of a mixed Laguerre integral. We transform this estimate into an explicit finite inequality by expanding the relevant integrals in the orthogonal basis associated with the weight $x^{1/2}\,\mathrm{e}^{-x}$. A telescoping identity then converts the resulting inequality into a positive form. The final positivity argument combines explicit estimates for central binomial coefficients, a logarithmic lower bound, and a finite verification of small dimensions. This completes the proof that the average singular value strictly decreases with the dimension in the square complex Gaussian case. As a consequence, the same monotonicity is obtained for $N\times(N+\lambda)$ complex Gaussian matrices with fixed rectangularity $\lambda=0,1,2,\dots$.
We settle the real half of a conjecture of Bandeira, Kennedy and Singer on the dimension dependence of the Gaussian constant governing the little Grothendieck problem over the orthogonal group. For an $N\times N$ standard real Gaussian matrix $G_N$, the average singular value $\alpha_{\mathbb R}(N)=N^{-3/2}\,\mathbb E\|G_N\|_*$ satisfies the quantitative estimate \[ \alpha_{\mathbb R}(N+1)-\alpha_{\mathbb R}(N)>\frac{1}{1000N^2}, \qquad N\ge1. \] Thus, the real constants increase strictly to the Marchenko--Pastur limit $8/(3\pi)$. The proof is finite-dimensional and exposes a mechanism not visible in the limiting spectral law. We decompose the Laguerre-orthogonal mean into its Laguerre-unitary counterpart and an explicit correction, then complete the resulting finite Laguerre sums to infinite diagonal tails. A bivariate generating function yields a positive diagonal kernel with a dimension-monotone remainder. This puts consecutive orthogonal corrections in common positive coordinates, where the nearest diagonal alone supplies an $N^{-2}$ reserve that dominates the unitary one-step term. The required unitary estimate is derived directly from Abreu's recurrence, and the first five dimensions are handled by exact closed forms.
The main theorem shows that consistency holds for all monotone set functions if and only if two structural conditions are satisfied: monotonicity with respect to contexts and a reduction property excluding positive localized support outside $B$.
O. Hutník, Natália Puškárová· Knowledge-Based Systems· 0 citations