Extending metric multidimensional scaling with Bregman divergences

Abstract

Sum of weighted square distance errors has been a popular way of defining stress function for metric multidimensional scaling (MMDS) like the Sammon mapping. In this paper we generalise this popular MMDS with Bregman divergences, as an example we show that the Sammon mapping can be thought of as a truncated Bregman MMDS (BMMDS) and we show that the full BMMDS improves upon the Sammon mapping on some standard data sets and investigate the reasons underlying this improvement. We then extend a well known family of MMDS, that deploy a strategy of focusing on small distances, with BMMDS and investigate limitations of the strategy empirically. Then an opposite strategy is introduced to create another family of BMMDS that gives increasing mapping quality. A data preprocessing method and a distance matrix preprocessing are introduced.