H-MDS: a new approach for interactive visualization with multidimensional scaling in the hyperbolic space

Abstract

We introduce a novel projection-based visualization method for high-dimensional data sets by combining concepts from MDS and the geometry of the hyperbolic spaces. This approach hyperbolic multi-dimensional scaling (H-MDS) is a synthesis of two important concepts for explorative data analysis and visualization: (i) multi-dimensional scaling uses proximity or pair distance data to generate a low-dimensional, spatial presentation of the data; (ii) previous work on the “hyperbolic tree browser” demonstrated the extraordinary advantages for an interactive display of graph-like data in the two-dimensional hyperbolic space ( H 2) . In the new approach, H-MDS maps proximity data directly into the H 2 . This removes the restriction to “quasi-hierarchical”, graph-based data—a major limitation of (ii). Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data, this type of data can be considered the most general. We review important properties of the hyperbolic space and, in particular, the circular Poincaré model of the H 2 . It enables effective human–computer interaction: by mouse dragging the “focus”, the user can navigate in the data without loosing the context. In H 2 the “fish-eye” behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H 2 . Especially, the exponential growth of length and area of the underlying space makes the H 2 a prime target for mapping hierarchical and (now also) high-dimensional data. Several high-dimensional mapping examples including synthetic and real-world data are presented. Since high-dimensional data produce “ring”-shaped displays, we present methods to enhance the display by modulating the dissimilarity contrast. This is demonstrated for an application for unstructured text: i.e., by using multiple film critiques from news:rec.art.movies.reviews and www.imdb.com, each movie is placed within the H 2 —creating a “space of movies” for interactive exploration.