Function Repository Resource:

PairwiseMultidimensionalScaling

Source Notebook

Multidimensional scaling algorithm for embedding pairwise distances into a Cartesian space

Contributed by: Nikolay Murzin

ResourceFunction["PairwiseMultidimensionalScaling"][dm]

return a list of 2-dimensional coordinates representing embedding of a distance matrix dm.

ResourceFunction["PairwiseMultidimensionalScaling"][dm,dim]

embed into a given integer dimension dim.

Details

Usually the multidimensional scaling algorithm is used for a given list of vectors, and there is an intermediate step of computing distances between them. In some cases, only the distances are known and the task is to find a lower-dimensional embedding such that the resulting distances are approximately preserved.

DimensionReduce with Method → "MultidimensionalScaling"or ResourceFunction["MultidimensionalScaling"] can be used for computing Multidimensional scaling for a list of vectors.

ResourceFunction["PairwiseMultidimensionalScaling"] implements classical multidimensional scaling, which is an exact solution to a minimization of the strain function problem.

Examples

Basic Examples (2)

Compute 2-dimensional embedding from a distance matrix of three points:

In[1]:=

Out[1]=

Compute 3-dimensional embedding from a distance matrix of four points:

In[2]:=

ResourceFunction["PairwiseMultidimensionalScaling"][( {
{0., 1., 2., 3.},
{4., 0., 5., 6.},
{7., 8., 0., 9.},
{-1., -2., -3., 0.}
} ), 3]

Out[2]=

Applications (3)

Coordinatize an edge-weighted graph:

In[3]:=

$With[{g = Graph[{1 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 3, 3 \[UndirectedEdge] 1}, EdgeWeight -> {1 \[UndirectedEdge] 2 -> 1.9}]}, Graph[g, VertexCoordinates -> ResourceFunction["PairwiseMultidimensionalScaling"][ GraphDistanceMatrix[g]]] ]$