Majorizing Stress Formula Two

TL;DR

This work extends smacof-style multidimensional scaling to Kruskal's stress formula two by presenting a convergent majorization (MM) algorithm. By recasting the ratio $\sigma_2(X)=\sigma_R(X)/\eta_2^2(X)$ into a minimization of a difference via the Dinkelbach trick and deriving explicit update rules, the authors guarantee monotone convergence under a reasonable initialization and provide practical R implementations. The approach hinges on matrices $V$, $B(X)$, and $M(X)$ to form the majorization step, with a fixed-point condition linking the gradient to the majorization update. Empirical examples on Ekman and De Gruijter data illustrate convergence behavior and dataset-dependent differences between $\sigma_2$-minimization and raw-stress minimization, underscoring the method's robustness for non-metric MDS. The Appendix delivers a concrete R implementation (stress2.R) enabling practitioners to apply the method directly.

Abstract

Modifications of the smacof algorithm for multidimensional scaling are proposed that provide a convergent majorization algorithm for Kruskal's stress formula two.

Figures (4)

• Figure 1: Ekman Metric Stress 2 Solution
• Figure 2: Ekman Metric Raw Stress Solution
• Figure 3: Gruijter Metric Stress 2 Solution
• Figure 4: Gruijter Metric Raw Stress Solution

Theorems & Definitions (9)

• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Theorem 4.1
• proof
• Theorem 5.1