Theoretical Foundations of Ordinal Multidimensional Scaling, Including Internal and External Unfolding

TL;DR

This paper develops a comprehensive theoretical framework for various forms of ordinal multidimensional scaling, including internal and external unfolding, using both continuum and asymptotic discrete models to deepen understanding.

Contribution

It introduces a unified theoretical foundation for ordinal multidimensional scaling variants, bridging continuum and discrete models for the first time.

Findings

Provides a formal theory for internal and external unfolding methods.

Analyzes models in both continuum and asymptotic discrete settings.

Enhances understanding of ordinal MDS through rigorous mathematical insights.

Abstract

We provide a comprehensive theory of multiple variants of ordinal multidimensional scaling,including internal unfolding and external unfolding. We first follow Shepard (1966) and work in a continuum model to gain insight. We then follow Kleindessner and von Luxburg (2014) and work in an asymptotic discrete setting.