Representation Tradeoffs for Hyperbolic Embeddings
This work provides efficient hyperbolic embeddings for hierarchical data, benefiting applications like natural language processing and knowledge graphs, though it is incremental with novel combinatorial and optimization methods.
The authors tackled the problem of embedding hierarchical data structures into hyperbolic space with low distortion and few dimensions, achieving a mean-average-precision of 0.989 on WordNet using only two dimensions, compared to 0.87 with 200 dimensions in prior work.
Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.