Using Support Vector Machines for Facet Partitioning in Multidimensional Scaling

Abstract

In this article we focus on interpreting multidimensional scaling (MDS) configurations using facet theory. The facet theory approach is attempting to partition a representational space, facet by facet, into regions with certain simplifying constraints on the regions’ boundaries (e.g., concentric circular sub-spaces). A long-standing problem has been the lack of computational methods for optimal facet-based partitioning. We propose using support vector machines (SVM) to perform this task. SVM is highly attractive for this purpose as they allow for linear as well as nonlinear classification boundaries in any dimensionality. Using various classical examples from the facet theory literature we elaborate on the combined use of MDS and SVM for facet-based partitioning. Different types of MDS are discussed, and options for SVM kernel specification, tuning, and performance evaluation are illustrated.

KEYWORDS:

• Multidimensional scaling
• support vector machines
• facet theory

Acknowledgments

The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions is not intended and should not be inferred.

Notes

1 This solution was computed using the default tuning parameter setup from the e1071 package (Meyer et al., 2021). That is, no parameter tuning is involved. This is typically not recommended, but for this first example we keep the setup as simple as possible.

2 We introduce SVM in a non-technical way in the spirit of James et al. (2013) and Boehmke and Greenwell (2020). For a mathematically more rigorous SVM presentation see Vapnik (2000) and Steinwart and Christmann (2008).

3 The computational complexity of SMACOF is $\mathcal{O}(n^2);$ the complexity of SVM is between $\mathcal{O}(n^2)$ and $\mathcal{O}(n^3).$