Geometric MDS Performance for Large Data Dimensionality Reduction and Visualization

Abstract

Multidimensional scaling (MDS) is a widely used technique for mapping data from a high-dimensional to a lower-dimensional space and for visualizing data. Recently, a new method, known as Geometric MDS, has been developed to minimize the MDS stress function by an iterative procedure, where coordinates of a particular point of the projected space are moved to the new position defined analytically. Such a change in position is easily interpreted geometrically. Moreover, the coordinates of points of the projected space may be recalculated simultaneously, i.e. in parallel, independently of each other. This paper has several objectives. Two implementations of Geometric MDS are suggested and analysed experimentally. The parallel implementation of Geometric MDS is developed for multithreaded multi-core processors. The sequential implementation is optimized for computational speed, enabling it to solve large data problems. It is compared with the SMACOF version of MDS. Python codes for both Geometric MDS and SMACOF are presented to highlight the differences between the two implementations. The comparison was carried out on several aspects: the comparative performance of Geometric MDS and SMACOF depending on the projection dimension, data size and computation time. Geometric MDS usually finds lower stress when the dimensionality of the projected space is smaller.