Knowledge Graph Completion with Mixed Geometry Tensor Factorization

TL;DR

This work introduces MIG-TF, a parameter-efficient mixed-geometry tensor factorization for knowledge graph completion that couples a pretrained Euclidean Tucker model with a low-parameter hyperbolic interaction term based on Lorentz geometry. The hyperbolic component is implemented via a differentiable tetrahedron-inspired score, while the Euclidean core remains fixed, resulting in a sum of Euclidean and hyperbolic contributions to the link-prediction score. Empirically, MIG-TF achieves state-of-the-art results on FB15k-237, YAGO3-10, and WN18RR with significantly fewer parameters than competing models, and shows robustness to noise and favorable performance on non-hierarchical graph structures. The framework highlights the benefits of combining Euclidean and hyperbolic representations, with opportunities for future extensions to spherical geometry and related optimizations.

Abstract

In this paper, we propose a new geometric approach for knowledge graph completion via low rank tensor approximation. We augment a pretrained and well-established Euclidean model based on a Tucker tensor decomposition with a novel hyperbolic interaction term. This correction enables more nuanced capturing of distributional properties in data better aligned with real-world knowledge graphs. By combining two geometries together, our approach improves expressivity of the resulting model achieving new state-of-the-art link prediction accuracy with a significantly lower number of parameters compared to the previous Euclidean and hyperbolic models.

Figures (12)

• Figure 1: Links distribution on three benchmark knowledge graphs considered in this work. The FB15k-237 dataset violates the power law the most, which is indicated by the longest platue in the leftmost part of the curve. Correspondingly, the top-performing model on this dataset among previous state-of-the-art turns out to be Euclidean. In contrast, our MIG-TF approach outperforms both Euclidean and hyperbolic models, see Table \ref{['table:results']}.
• Figure 2: The graphs at the top row demonstrate the top 20 vertices in terms of their number of connections in the FB15k-237 knowledge graph. The pink marks indicate vertices that were predicted in more than $50\%$ of cases by both the Euclidean (left), hyperbolic (center) and mixed geometry (right) models. The bar charts illustrate the hit rate (HR@10) of the predictions for links between vertices from each group. Vertices are arranged in descending order based on the number of incoming links. As can be observed, due to the high number of relations among active vertices, the hyperbolic model (TPTF ) significantly underperforms in predicting relations between active vertices compared to the Euclidean model(TuckER). However, the combination of Euclidean and hyperbolic models outperforms both of them.
• Figure 3: Left score function landscape corresponds to our score function \ref{['lpitf_scor']}, whilst right one corresponds to \ref{['eq:scorelor']}. As seen from the plots the landscapes of score functions for different distances are similar. For more score function landscapes see Appendix.
• Figure 4: Each embedding of a relation ($t_1, ..., t_n$) defines a cone that encompasses the embeddings of the entities associated with that relation.
• Figure 5: The proposed MIG-TF model architecture.