More Voronoi Homogeneity Analysis (2026-06-04)

My brief introductory remarks for the April 10 workshop

https://deleeuwworkshop.stat.ucla.edu/#schedule

had the title “A Tale of Two Loss Functions”. The two loss functions were, of course, the Gifi meet-loss of homogeneity analysis (MCA) and the stress of smacof. Patrick’s presentation at the workshop uses the same two chapters to discuss my work in “dimension reduction”.

I am currently working on the paper “Voronoi Homogeneity Analysis”, latest version at

https://github.com/deleeuw/voronoi

In the introductory section there is a new derivation of the fact that homogeneity analysis is a special case of non-metric smacof (and that consequently the Gifi system can be embedded in smacof with restrictions). More specifically for m variables MCA consists of m linked nonmetric unfolding problems, which are linked because they have the same X but different Y_j. Thus it is now “A Tale of one Loss Function”.

Voronoi MCA relaxes the ordinal constraints on the disparities. As a consequence, as in unfolding, there are the usual degenerate solutions with perfect fit. But because of the weak ordinal constraints other perfect fit solutions exist, which are not necessarily degenerate (or only partly degenerate).

The paper has software and examples to illustrate the workings of the algorithm and the influence of various constraints (some of which emulate MCA properties). The appendix introduces MALS, which is Majorized Alternating Least Squares.