Hyperbolic Ordinal Embedding
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Proceedings of The Eleventh Asian Conference on Machine Learning, PMLR 101:10651080, 2019.
Abstract
Given ordinal relations such as the object $i$ is more similar to $j$ than $k$ is to $l$, ordinal embedding is to embed these objects into a lowdimensional space with all ordinal constraints preserved. Although existing approaches have preserved ordinal relations in Euclidean space, whether Euclidean space is compatible with true data structure is largely ignored, although it is essential to effective embedding. Since real data often exhibit hierarchical structure, it is hard for Euclidean space approaches to achieve effective embeddings in low dimensionality, which incurs high computational complexity or overfitting. In this paper we propose a novel hyperbolic ordinal embedding (HOE) method to embed objects in hyperbolic space. Due to the hierarchyfriendly property of hyperbolic space, HOE can effectively capture the hierarchy to achieve embeddings in an extremely lowdimensional space. We have not only theoretically proved the superiority of hyperbolic space and the limitations of Euclidean space for embedding hierarchical data, but also experimentally demonstrated that HOE significantly outperforms Euclideanbased methods.
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