Representability of the sign-decomposition factorization by existing KGE scoring functions

Determine whether standard knowledge graph embedding models that score triples via dot products of the form φ(s,r,o) = h_{s,r}^T e_o can represent the sign-decomposition factorization Y = s(H E^T) described in Theorem 4.1, where Y ∈ {0,1}^{|E||R| × |E|} is the adjacency matrix, s(x) = 1 if x > 0 and 0 otherwise is applied element-wise, H ∈ R^{|E||R| × (2c+1)}, E ∈ R^{|E| × (2c+1)}, and c is the maximum out-degree across subject–relation pairs.

Background

The paper proves a sufficient dimensionality bound for exact sign reconstruction and, by extension, ranking reconstruction: any knowledge graph with maximum out-degree c admits a decomposition of its adjacency matrix Y as Y = s(H ET) with embeddings of dimension 2c+1. This result extends sign-rank bounds to rectangular bipartite matrices relevant for KGEs.

While the bound is constructive, the authors note that current KGEs typically use scoring functions linear in the object embedding (e.g., φ(s,r,o) = h_{s,r}T e_o) and raise the practical question of whether such models can exactly represent the proposed sign-decomposition factorization under realistic parameterizations.

References

Note that two practical questions remain. First, can existing KGEs, with their specific scoring functions, actually represent the factorisation technique described in Theorem \ref{thm:exact_sign_decomposition_kge}? Second, even if they can, can this solution be efficiently learned through gradient-based optimisation in practice? We leave these for future work.

— Breaking Rank Bottlenecks in Knowledge Graph Embeddings (2506.22271 - Badreddine et al., 27 Jun 2025) in Section 4, A sufficient bound for sign and ranking reconstruction (paragraph beginning “Note that two practical questions remain.”)