Let $W_{N,N+\lambda}$ have the Laguerre unitary distribution with size $N$ and real shape $\lambda\ge0$. For $s>0$, we consider the normalized moment $$C_{s,\lambda}(N)=N^{-s-1}\mathbb{E}[\operatorname{Tr}W_{N,N+\lambda}^{s}]$$ and its dimension decrement $\Gamma_{N,s,\lambda}=C_{s,\lambda}(N)-C_{s,\lambda}(N+1)$. Iter...
We study the normalized half moment $\alpha_{\mathbb R}^{(\lambda)}(N)$ of the size-$N$ Laguerre orthogonal ensemble for real shape $\lambda\ge0$. At integer shape this is the expected average singular value of an $N\times(N+\lambda)$ real Gaussian matrix. The square mean increases with the dimension, whereas every rea...
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 poi...
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\...
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
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