Knowledge Graph Embedding (KGE) is a powerful technique for predicting missing links in Knowledge Graphs (KGs) by learning the entities and relations. Hyperbolic space has emerged as a promising embedding space for KGs due to its ability to represent hierarchical data. Nevertheless, most existing hyperbolic KGE methods rely on tangent approximation and are not fully hyperbolic, resulting in distortions and inaccuracies. To overcome this limitation, we propose LorentzKG, a fully hyperbolic KGE method that represents entities as points in the Lorentz model and represents relations as the intrinsic transformation—the Lorentz transformations between entities. We demonstrate that the Lorentz transformation, which can be decomposed into Lorentz rotation/reflection and Lorentz boost, captures various types of relations including hierarchical structures. Experimental results show that our LorentzKG achieves state-of-the-art performance.

Motivated by the widespread interest in the cross-lingual transfer of NLP models from high resource to low resource languages, research on Cross-lingual word embeddings (CLWEs) has gained much popularity over the years. Among the most successful and attractive CLWE models are the unsupervised CLWE models. These unsupervised CLWE models pose the alignment task as a Wasserstein-Procrustes problem aiming to estimate a permutation matrix and an orthogonal matrix jointly. Most existing unsupervised CLWE models resort to Optimal Transport (OT) based methods to estimate the permutation matrix. However, linear programming algorithms and approximate OT solvers via Sinkhorn for computing the permutation matrix scale cubically and quadratically, respectively, in the input size. This makes it impractical and infeasible to compute OT distances exactly for larger sample size, resulting in a poor approximation quality of the permutation matrix and subsequently a less robust learned transfer function or mapper. This paper proposes an unsupervised projection-based CLWE model called quantized Wasserstein Procrustes (qWP) that jointly estimates a permutation matrix and an orthogonal matrix. qWP relies on a quantization step to estimate the permutation matrix between two probability distributions or measures. This approach substantially improves the approximation quality of empirical OT solvers given fixed computational cost. We demonstrate that qWP achieves state-of-the-art results on the Bilingual lexicon Induction (BLI) task.