Embedding Knowledge Graph in Function Spaces
Louis Mozart Kamdem Teyou, Caglar Demir, Axel-Cyrille Ngonga Ngomo
TL;DR
This paper reframes knowledge graph embedding by representing entities and relations as functions in a finite-dimensional function space $\mathcal{F}$ rather than as static vectors in $\mathbb{R}^d$. It introduces three functional embedding families—polynomial-based $\textsc{FMult}_n$, trig/complex-based $\textsc{FMult}^i_n$, and neural-network-based $\textsc{FMult}$—and derives their scoring via $\mathcal{L}^2(\Omega)$ inner products, with exact polynomial formulas and neural-network approximations. Empirical results across eight benchmarks show complementary strengths: polynomial embeddings excel on symmetric or hierarchical data, while neural-network embeddings capture complex, non-linear relations (especially in UMLS and NELL datasets); trig-based variants remain an avenue for future exploration. The work demonstrates that functional representations can outperform traditional static embeddings on several KG completion tasks, offering a principled path to more expressive, compositional, and interpretable KG embeddings, aided by public code and detailed derivations.
Abstract
We introduce a novel embedding method diverging from conventional approaches by operating within function spaces of finite dimension rather than finite vector space, thus departing significantly from standard knowledge graph embedding techniques. Initially employing polynomial functions to compute embeddings, we progress to more intricate representations using neural networks with varying layer complexities. We argue that employing functions for embedding computation enhances expressiveness and allows for more degrees of freedom, enabling operations such as composition, derivatives and primitive of entities representation. Additionally, we meticulously outline the step-by-step construction of our approach and provide code for reproducibility, thereby facilitating further exploration and application in the field.