# Finding most Representative Sample in “Pair Statistics”

By "Pair Statistics" I understand statics that are based on values $\varphi:\mathcal{P}\times\mathcal{P}\ni(p,q)\mapsto y\in\mathbb{R}$ that can be observed for every pair $(p,q)$ of individuals of a population $\mathcal{P}$, where $\varphi(p,q)=\varphi(q,p)$

Question:

how can a discrete subset $P\subset\mathcal{P}\$of individuals be determined that represents the best "approximation" of the statistical properties of $\mathcal{P}$?

Background of my question is the idea to interpret point-distribution problems (e.g. on a manifold) as the problem of determining the set of $n$ points, that most closely resembles the distance statistics of the manifold.

The concrete idea would be to determine a set of statistical parameters, say, mean and standard deviation $\left(\mu(p),\sigma(p)\right):=\left(\text{mean}(dist(p,q)),\text{sdev}(dist(p,q))\right)$, for all distances from $p$ to all (other) points $q$ of the manifold.

Now, the same kind of statistical parameters can be calculated for each point $p_\Sigma$ of a sample, where the statistical are however calculated from the distances to the (other) points $q_\Sigma$ of the sample, yielding $\left(\mu(p_\Sigma),\sigma(p_\Sigma)\right)_\Sigma$.

The objective would then be to determine the optimal sample $\mathcal{P}_\Sigma^*\subset\mathcal{P}$ that minimizes some "double norm" of the differences $\|\ \big(\|\left(\mu(p_\Sigma),\sigma(p_\Sigma)\right)_\Sigma\ -\ \left(\mu(p_\Sigma),\sigma(p_\Sigma)\right)\|\big)\ \|$, where the "inner" norm measures the difference between the nominal and actual statistical parameter values of an individual point of the sample, and where the "outer" norm measures the entirety of deviations of the sample point's deviations from the nominal values of the statistical parameters of the individual distance statistics.

The notation $(\cdot,\cdot)_\Sigma$ denotes statistical parameters obtained from distances between sample points.