Suppose I have a random symmetric matrix W of size $n\times n$, with i.i.d. coefficients uniformly distributed in [0,1], and I set $W_{ii} = 0$.

Then I apply a Multidimensional Scaling of dimension $k$, which I define as minimizing the quantity: $H=n^{-2}\sum_{i,j} (W_{ij} - ||x_i-x_j||^2)^2$ with respect to the $n$ points $x_i \in R^k$. Say I denote $H^*$ the minimum.

The question is : what is the behavior of $\mathbb{E}[H^*]$ as a function of $k$ and $n$ ?

Is there a mathematical article out there dealing with this question ? or is it an open question ?



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