Fast Multidimensional Scaling Through Sampling,Springs and Interpolation

Abstract

The term ‘proximity data’ refers to data sets within which it is possible to assess the similarity of pairs of objects. Multidimensional scaling (MDS) is applied to such data and attempts to map high-dimensional objects onto low-dimensional space through the preservation of these similarity relations. Standard MDS techniques have in the past suffered from high computational complexity and, as such, could not feasibly be applied to data sets over a few thousand objects in size. Through a novel hybrid approach based upon stochastic sampling, interpolation and spring models, we have designed an algorithm running in O(N√N). Using Chalmers’ 1996 O(N²) spring model as a benchmark for the evaluation of our technique, we compare layout quality and run times using sets of synthetic and real data. Our algorithm executes significantly faster than Chalmers’ 1996 algorithm, while producing superior layouts. In reducing complexity and run time, we allow the visualisation of data sets of previously infeasible size. Our results indicate that our method is a solid foundation for interactive and visual exploration of data.