On Interactive Visualization of High-dimensional Data using the Hyperbolic Plane

摘要

We propose a novel projection based visualization method for high-dimensional datasets by combining concepts from MDS, and the geometry of the hyperbolic spaces. Our approach Hyperbolic Multi-Dimensional Scaling (H-MDS) extends earlier work [7] using hyperbolic spaces for visualization of tree structures data ("hyperbolic tree browser"). By borrowing concepts from multi-dimensional scaling we map proximity data directly into the 2-dimensional hyperbolic space (H2). This removes the restriction to "quasi-hierarchical", graph-based data - limiting previous work. 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 used the circular Poincare model of the H2 which allows effective human-computer interaction: by moving the "focus" via mouse the user can navigate in the data without loosing the "context". In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data. We present several high-dimensional mapping examples including synthetic and real world data and a successful application for unstructured text. By analyzing and integrating multiple film critiques from news:rec.art.movies.reviews and the internet movie database, each movie becomes placed within the H2. Here the idea is, that related films share more words in their reviews than unrelated. Their semantic proximity leads to a closer arrangement. The result is a kind of high-level content structured display allowing the user to explore the "space of movies".