Local Multidimensional Scaling for Nonlinear Dimension Reduction, Graph Drawing and Proximity Analysis

In the past decade there has been a resurgence of interest in nonlinear dimension reduction. Among new proposals are “Local Linear Embedding” (LLE, Roweis and Saul 2000), “Isomap” (Tenenbaum et al. 2000) and Kernel PCA (KPCA, Schölkopf, Smola and Müller 1998), which all construct global lowdimensional embeddings from local affine or metric information. We introduce a competing method called “Local Multidimensional Scaling” (LMDS). Like LLE, Isomap and KPCA, LMDS constructs its global embedding from local information, but it uses instead a combination of MDS and “force-directed” graph drawing. We apply the force paradigm to create localized versions of MDS stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces. We solve the problem of tuning parameter selection with a meta-criterion that measures how well the sets of K-nearest neighbors agree between the data