# Partition function for multidimensional scaling energy

Let $D_{ij}$ a random matrix with i.i.d positive coefficients. One can take for instance $D_{ij}$ uniformly distributed in [0,1]. We consider the following energy function $H(x)$ defined for $x=(x_i)_1^n$, with each $x_i\in \mathbb{R}^k$, where $n>k$ are two positive integers:

$$H(x) = \sum_{i,j} \left(\|x_i-x_j\|^2 - D_{ij}\right)^2$$

I would like to find the expectation of $$H^*:=\inf_{x \in (\mathbb{R^k})^n} H(x)$$

To this end, I was thinking of using a statistical mechanics approach and estimate the partition function associated with $H$. I know the definition but I don't know how to work it out... Anyone can help ?

Thanks

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A matrix whose matrix entries are uniformly distributed in an interval is an extremely unnatural animal. A Gaussian distribution would be much more natural and give better results. With your problems, you are calculating various distances and other geometric properties of $n^2$-dimensional cubes with all of their faces, edges etc. The complications in this problem are purely mathematical and, I would also say, self-inflicted wounds because you chose such an unnatural distribution. –  Luboš Motl Mar 1 '13 at 12:50
Hi, thanks for the comment. I would also be interested in gaussian distribution. Can you do it ? –  mellow Mar 3 '13 at 10:54