Nonlinear Mapping Using a Hybrid of PARAMAP and Isomap Approaches

Abstract

Dimensionality reduction aims to represent higher dimensional data by a lower-dimensional structure. A well-known approach by Carroll, Parametric Mapping (PARAMAP) (Shepard and Carroll 1966) relies on iterative minimization of a loss function (called kappa or “κ”) measuring the smoothness or continuity of the mapping from the lower dimensional representation to the original data. The approach was resuscitated recently with important algorithmic modifications (Akkucuk 2004; Akkucuk and Carroll 2003, 2006). However improved, the approach still involved the need to make a large number of random starts. In this paper we discuss the use of a variant of the Isomap method (Tenenbaum et al. 2000) to obtain a starting framework, consisting of a core set of landmark points. These core set of landmark points are used to construct a rational start for running PARAMAP algorithm only once. Since Isomap is faster and less prone 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 sets of realistically based data, and more inclined to obtain a satisfactory solution in reasonable time.