Supervised multidimensional scaling for visualization, classification, and bipartite ranking

摘要

Least squares multidimensional scaling (MDS) is a classical method for representing a n - n dissimilarity matrix D. One seeks a set of configuration points z_1,. .., z_n ∈ ?~S such that D is well approximated by the Euclidean distances between the configuration points: D_(ij) ≈ ∥z_i - z_j∥_2. Suppose that in addition to D, a vector of associated binary class labels ∈ {1,2}~n 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 z_1,. .., z_n ∈ ?~S such that D _(ij) ≈ ∥z_i-z_j∥_2, and such that z_(is) > z_(js) for s = 1,. .., S tends to occur when y_i > y_j. 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.