Stereographic Projection for Embedding Hierarchical Structures in Hyperbolic Space

Shangyu Chen, Xiaohao Yang, Pengfei Fang, Mehrtash Tafazzoli Harandi, Dinh Q. Phung, Jianfei Cai

Abstract: Hyperbolic geometry has emerged as a promising tool in diverse domains in deep learning. In this study, we concentrate on a key component of hyperbolic neural networks-the mapping from Euclidean space to hyperbolic space. We explore the problems and drawbacks of existing practices in this mapping, such as exponential mapping and projection methods constrained within the Poincaré ball. We emphasize that these methods rely entirely on supervised relationship data to capture hierarchical structure in hyperbolic space. The exponential mapping, which does not involve learning any parameters, functions more like a predefined activation function. This type of mapping does not convey any hierarchical structure information, making the computational cost of this mapping unnecessary. We propose a novel approach called Stereographic Projection Transition Mapping (SPTM). Leveraging the intrinsic properties of hyperbolic space, SPTM explicitly represents hierarchical structures present in the Euclidean space. By analytical mapping relationships in the Euclidean space, SPTM offers a more efficient and interpretable way to represent hierarchical structures in the Poincaré ball without the need for excessive supervision.