Projection on an Acute Symmetrical Simplicial Closed Convex Cone and its Application to Star Graphs

Summary

Additive trees extend hierarchical representations, and a one-to-one correspondence has been set up between these trees and qua-dripolar semi-distances, i.e. semi-distances satisfying the four-point condition. Among these dissimilarities, star semi-distances are noteworthy. They verify : d(i,j) =a_i +a_j , for all distinct units i and j, and yield a special representation, called a star graph.

After exploring some geometrical properties of star semi-distances, the least squares approximation of a dissimilarity by such a semi-distance is examined. It corresponds to a least squares problem with positive constraints. This problem is shown to be related to the projection of a vector x on a very special cone, the closed convex hull of independent vectors which have the same acute angle.

It is proved that a coordinate of the solution is null when the associated coordinate of x is non-positive. Then a descent algorithm follows, and at each step, a very simple analytic expression yields the projection on the corresponding face. Moreover, when the coordinates of x are ranked by increasing order, the suitable face for the solution is directly obtained. This gives a second algorithm.

Finally, some specific properties of the least squares star sémi-distance are discussed.