Suppose we have a countable set of objects $\{x_i|i \in [1..m]\}$ in a metric space $(\mathbb R^n,d_n)$ and a map ($F$) mapping the objects to objects in a metric space $(\mathbb R^1,d_1)$. For each pair of objects we can define an error:
$$ e_{ij} = \frac{d_m(x_i,x_j)}{d_1(F(x_i),F(x_j))} $$
Let $e_{ii}$ be 1.
It is obvious that we can define the countable set in a way that any projection has an error ($e_{ij}\ne 1$) at least for one pair of objects.
But for given set of objects we can define the map trying to minimize the errors.
I had met this problem developing a web site's rating system based on several criteria. At the moment I use the following algorithm for calculating F (images of objects):
- select random $\{y_i|i \in [1..m]\}$ set in space $(\mathbb R^1,d_1)$ and think that $y_i=F(x_i)$ where F is defining map
- for each dot calculate $\delta_i=\sum\limits_{j}k\frac{1}{2}e_{ij}(y_i-y_j)$ where $k$ is a parameter ($0 \le k \le 1$) and them change items in the set $y_i \to y_i + \delta_i$
I repeat the step two over and over several times, but at each step I make the parameter $k$ smaller.
I infer this algorithm from physics intuition: if two objects are too close they push away, if they are too far away they attract and $k$ is a resistance of an environment.
I want to get deeper knowledge about this problem, so could you provide me some links to articles and literature about it or the official name of the problem.