A Relation in Knowledge Graph Is a Matching Matrix

2026-01-11 · Source: Agus’s Substack · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Data Science & Analytics · Depth: Advanced, long

Summary

This post, part of the "Knowledge Graphs as Geometry" series, introduces the bilinear family of knowledge graph embedding models, which score facts by measuring agreement, contrasting with TransE's distance-based approach. It details RESCAL, using a full relation matrix Mᵣ, and DistMult, a constrained version with a diagonal Mᵣ, which reduces parameters but forces all relations to be symmetric. DistMult scores `(Ann, parentOf, Carol)` and its reverse identically (both 9.002). ComplEx resolves this by using complex numbers for coordinates and conjugating the tail, enabling it to differentiate directions, scoring `(Ann, parentOf, Carol)` at 11.63 and the reverse at 0.91. This "carrier" change resolves the symmetry limitation without increasing operator complexity. The article also compares matching (ComplEx) and distance scoring profiles on the UMLS dataset, showing ComplEx's high and flat Hits@1 (0.80) profile versus DistMult's lower scores (0.59). It positions DistMult, RESCAL, and TuckER as a lineage based on constraints on a shared core tensor.

Key takeaway

For Machine Learning Engineers building knowledge graph applications, if you encounter issues with models like DistMult failing on directed relations due to forced symmetry, consider ComplEx. By shifting to complex number coordinates, ComplEx effectively represents directional facts, yielding significantly better scores for true directions (e.g., 11.63 vs. 0.91). Evaluate your model's prediction profile (Hits@K) to ensure it provides a single, reliable answer for agents, rather than a cluster of candidates.

Key insights

Changing the number system (carrier) for knowledge graph embeddings can resolve model limitations more effectively than increasing operator complexity.

Principles

• Bilinear models score by agreement, not distance.
• Relation matrices encode feature compatibility.
• Carrier choice (ℝ vs. ℂ) impacts representational power.

In practice

• Use ComplEx for directed relations.
• DistMult suits symmetric relations like `spouse`.
• Evaluate models by Hits profile shape, not just MRR.

Topics

• Knowledge Graph Embeddings
• Bilinear Models
• ComplEx
• DistMult
• Relation Symmetry
• Complex Numbers

Code references

• asudjianto-xml/Knowledge-Graph-Geometry
• asudjianto-xml/Knowledge-Graph-Geometry

Best for: AI Scientist, Machine Learning Engineer, AI Architect

Related on AIssential

• A Relation Is an Operation — Knowledge graphs represent facts as geometric operations, enabling link prediction through spatial transformations.
• A Relation Is a Rotation — RotatE models knowledge graph relations as rotations in the complex plane, resolving symmetry issues and enabling efficient inference.
• Format-Constraint Coupling in Knowledge Graph Construction from Statistical Tables — Knowledge graph fidelity from statistical tables critically depends on aligning serialization format with extraction schema, as mismatches cause catastrophic failures.
• Dimensionality Reduction Meets Network Science: Sensemaking on UMAP's kNN Graph — UMAP's internal kNN graph offers a more faithful representation of high-dimensional data manifolds than its 2D projection.
• Knowledge Graphs, Part II — Knowledge graphs are architectures defined by component configuration, not finite products.
• Model Graph Inductive Learning for Knowledge Graph Completion — MGIL improves knowledge graph link prediction by leveraging global structural patterns through a novel model graph and GNN approach.

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Editorial summary, takeaway, and curation by AIssential. Original article published by Agus’s Substack.