Visualizing big data via a mixture of PARAMAP and Isomap

Abstract

Dimension reduction strives to represent higher dimensional data by a lower-dimensional structure. A famous approach by Carroll called Parametric Mapping or PARAMAP (Shepard & Carroll, 1966) works by iterative minimization of a loss function measuring the smoothness or continuity of the mapping from the lower dimensional representation to the original data. The algorithm was revitalized with essential modifications (Akkucuk & Carroll, 2006). Even though the algorithm was modified, it still needed to make a large number of randomly generated starts. In this paper we discuss the use of a variant of the Isomap method (Tenenbaum et al., 2000) to obtain a starting framework to replace the random starts. The core set of landmark points are selected by a special procedure akin to selection of seeds for the k-means algorithm. These core set of landmark points are used to create a rational start for running the PARAMAP algorithm only once but effectively reach a global minimum. Since Isomap is faster and less inclined to local optimum problems than PARAMAP, and the iterative process involved in adding new points to the configuration will be less time consuming (since only one starting configuration is used), we believe the resulting method should be better suited to deal with large data sets, and more prone to obtain an acceptable solution in realistic time.