I have some arbitrary pairwise similarity metric for some objects, and I am considering trying to find the best way to position the objects onto a line segment such that the pairwise euclidean distances between points on the segment are best representative of the pairwise similarities between the objects. The application is for part of an experimental visualization technique.
As an optimization problem, maybe it could formulated something like this?
$$argmin_f \sum_{e_1, e_2 \in E} | \frac{ S(e_1,e_2) } { \max_{ a, b\in E }S(a,b) }-|f( e_1 ) - f(e_2)||$$ $$f: E \longrightarrow [0,1] $$
In short, if two points are close on the line segment, they should be similar. Is this an existing problem that people have studied? Anyone know anything about it?
