Dimensionality Reduction with Reconstructed Boundedness in Hyperbolic Data Spaces

Euclidean geometry has been a de facto model for representing and processing data spaces. We have, however, started to see non-Euclidean geometries being used in many applications, including visualization, network measurement, and geometric routing. For these applications, Euclidean-based techniques may not be effective. As such, we need new mechanisms for managing and exploring non-Euclidean data. We focus on the dimensionality reduction problem in multidimensional Hyperbolic spaces. 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 of the original data, in which the accuracy deviation 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.