Dimensionality Reduction in Hyperbolic Data Spaces: Bounding Reconstructed-Information Loss

Abstract

We have started to see non-Euclidean geometries used in many applications, including visualization, network measurement, and geometric routing. Therefore, we need new mechanisms for managing and exploring non-Euclidean data. In this paper, we specifically address the dimensionality reduction problem for Hyperbolic data that are represented by the Poincare disk model. A desirable property of our solution is its reconstructed boundedness. In other words, we can reconstruct the data from its dimension-reduced version to obtain an approximation whose deviation from the original data is bounded and independent of the boundary measure of the original data space. We also propose a similarity search technique as an application of our reduction approach. We provide both theoretical and simulation analyses to substantiate the findings of our work.