# When is the result of multidimensional-scaling unique up to isometry?

What conditions on the ambient space and/or the given matrix of dissimilarities guarantee that all point configurations that minimise the error function of multidimensional scaling (MDS) are congruent, i.e. related to one another via an isometry of the space?

An example where it fails: The resultant point configuration is not determined up to isometry in general. Consider for example the dissimilarity matrix obtained from a discrete metric space on $$n$$ points, embedded via MDS into the Euclidean plane. It can be shown that the vertices of any regular $$n$$-gon (of an appropriate radius) give a local minimum, where the order of the vertices is determined by the initialisation alone (there are also other local minima, but these don't concern us here). However, since an arbitrary isometry of Euclidean space is determined by the images of three non co-linear points (this is "trilateration"), an isometry can not be found for every permutation of the vertices. Thus not all resultant point configurations are related by an isometry of the ambient.
An example where it does work: Consider dissimilarities given by a configuration $$X$$ of $$n$$ points in the Euclidean plane. Then any MDS point configuration $$Y$$ that matches these dissimilarities perfectly (i.e. have stress zero) is related to $$X$$ via an isometry (for $$n < 3$$ this is trivial; for $$n \geq 3$$ it is the side-side-side congruence theorem for triangles).