Supervised multidimensional scaling for visualization, classification, and bipartite ranking

Least squares multidimensional scaling (MDS) is a classical method for representing a nxn dissimilarity matrix . One seeks a set of configuration points such that is well approximated by the Euclidean distances between the configuration points: . Suppose that in addition to , a vector of associated binary class labels corresponding to the n observations is available. We propose an extension to MDS that incorporates this outcome vector. Our proposal, supervised multidimensional scaling (SMDS), seeks a set of configuration points such that , and such that zis>zjs for s=1,...,S tends to occur when yi>yj. This results in a new way to visualize the observations. In addition, we show that SMDS leads to a method for the classification of test observations, which can also be interpreted as a solution to the bipartite ranking problem. This method is explored in a simulation study, as well as on a prostate cancer gene expression data set and on a handwritten digits data set.