Locally multidimensional scaling by creating neighborhoods in diffusion maps

Abstract

This paper analyzes and improves an advanced multidimensional scaling method, known as locally multidimensional scaling, which assumes that high-dimensional data lie on a low-dimensional manifold. The method preserves local distances in the manifold by using classical scaling on a set of clusters in the high-dimensional data. These clusters are called neighborhoods, and the success of the method depends on the proper selection of these neighborhoods. At present, a neighborhood set is difficult to tune, and even if done well, the method may not function properly in dealing with noisy data. Our proposal utilizes clustering in a diffusion map, and thereby improves the original method in two ways. First, neighborhood selection is easier to tune, and second, the neighborhoods chosen enable the improved method to work under noisy data conditions. Our experiments demonstrate better tuning and robustness-to-noise results compared with the original method and some other existing multidimensional scaling methods on synthetic and real data sets.