ComplEx (Complex Embeddings for Simple Link Prediction) is a knowledge graph embedding model that extends bilinear factorization into the complex number domain — using complex-valued entity and relation vectors to elegantly model both symmetric and antisymmetric relations simultaneously, achieving state-of-the-art link prediction by exploiting the asymmetry inherent in complex conjugation.

What Is ComplEx?

• Definition: A bilinear KGE model where entities and relations are represented as complex-valued vectors (each dimension has a real and imaginary part), scored by the real part of the trilinear Hermitian product: Score(h, r, t) = Re(sum of h_i × r_i × conjugate(t_i)).
• Key Insight: Complex conjugation breaks symmetry — Score(h, r, t) uses conjugate(t) but Score(t, r, h) uses conjugate(h), so the two scores are different for asymmetric relations.
• Trouillon et al. (2016): The original paper demonstrated that this simple extension of DistMult to complex numbers enables modeling the full range of relation types.
• Relation to DistMult: When imaginary parts are zero, ComplEx reduces exactly to DistMult — it is a strict generalization, adding expressive power at 2x memory cost.

Why ComplEx Matters

• Full Relational Expressiveness: ComplEx can model symmetric (MarriedTo), antisymmetric (FatherOf), inverse (ChildOf is inverse of ParentOf), and composition patterns — the four fundamental relation types in knowledge graphs.
• Elegant Mathematics: Complex numbers provide a natural geometric framework — symmetric relations correspond to real-valued relation vectors; antisymmetric relations require imaginary components.
• State-of-the-Art: For years, ComplEx held top positions on FB15k-237 and WN18RR benchmarks — demonstrating that the complex extension is practically significant, not just theoretically elegant.
• Efficient: Same O(N × d) complexity as DistMult (treating complex d-dimensional as real 2d-dimensional) — no quadratic parameter growth unlike full bilinear RESCAL.
• Theoretical Completeness: Proven to be a universal approximator of binary relations — given sufficient dimensions, ComplEx can represent any relational pattern.

Mathematical Foundation

Complex Number Representation:

• Each entity embedding: h = h_real + i × h_imag (two real vectors of dimension d/2).
• Each relation embedding: r = r_real + i × r_imag.
• Score: Re(h · r · conj(t)) = h_real · (r_real · t_real + r_imag · t_imag) + h_imag · (r_real · t_imag - r_imag · t_real).

Relation Pattern Modeling:

• Symmetric: When r_imag = 0, Score(h, r, t) = Score(t, r, h) — symmetric relations have zero imaginary part.
• Antisymmetric: r_real = 0 — Score(h, r, t) = -Score(t, r, h), perfectly antisymmetric.
• Inverse: For relation r and its inverse r', set r'_real = r_real and r'_imag = -r_imag — the complex conjugate.
• General: Any combination of real and imaginary components models intermediate symmetry levels.

ComplEx vs. Competing Models

Benchmark Performance

Extensions of ComplEx

• TComplEx: Temporal extension — time-dependent ComplEx for facts valid only in certain periods.
• ComplEx-N3: ComplEx with nuclear 3-norm regularization — dramatically improves performance with proper regularization.
• RotatE: Constrains relation vectors to unit complex numbers — rotation model that provably subsumes TransE.
• Duality-Induced Regularization: Theoretical analysis showing ComplEx's duality with tensor decompositions.

Implementation

• PyKEEN: ComplExModel with full evaluation pipeline, loss functions, and regularization.
• AmpliGraph: ComplEx with optimized negative sampling and batch training.
• Manual PyTorch: Define complex embeddings as (N, 2d) tensors; implement Hermitian product in 5 lines.

ComplEx is logic in the imaginary plane — a mathematically principled extension of bilinear models into complex space that elegantly handles the full spectrum of relational semantics through the geometry of complex conjugation.