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**QuatE** (Quaternion Embeddings) is a **knowledge graph embedding model that extends RotatE from 2D complex rotations to 4D quaternion space** — representing each relation as a quaternion rotation operator, leveraging the non-commutativity of quaternion multiplication to capture rich, asymmetric relational patterns that cannot be fully expressed in the complex plane.
**What Is QuatE?**
- **Definition**: An embedding model where entities and relations are represented as d-dimensional quaternion vectors, with triple scoring based on the Hamilton product between the head entity and normalized relation quaternion, measuring proximity to the tail entity in quaternion space.
- **Quaternion Algebra**: Quaternions extend complex numbers to 4D: q = a + bi + cj + dk, where i, j, k are imaginary units satisfying i² = j² = k² = ijk = -1 and the non-commutative multiplication rule ij = k but ji = -k.
- **Zhang et al. (2019)**: QuatE demonstrated that 4D rotation spaces capture richer relational semantics than 2D rotations, achieving state-of-the-art performance on WN18RR and FB15k-237.
- **Geometric Interpretation**: Each relation applies a 4D rotation (parameterized by 4 numbers) to the head entity — more degrees of freedom than RotatE's 2D rotations means more expressive relation representations.
**Why QuatE Matters**
- **Higher Expressiveness**: 4D quaternion rotations can represent any 3D rotation plus additional transformations — more degrees of freedom capture subtler relational distinctions.
- **Non-Commutativity**: Quaternion multiplication is non-commutative (q1 × q2 ≠ q2 × q1) — this inherently captures ordered, directional relations without special constraints.
- **State-of-the-Art Performance**: QuatE consistently achieves higher MRR and Hits@K than ComplEx and RotatE on standard benchmarks — the additional geometric expressiveness translates to empirical gains.
- **Disentangled Representations**: Quaternion components may disentangle different aspects of relational semantics (scale, rotation axes, angles) — richer structural representations.
- **Covers All Patterns**: Like RotatE, QuatE models symmetry, antisymmetry, inversion, and composition — but with richer parameterization.
**Quaternion Mathematics for KGE**
**Quaternion Representation**:
- Entity h: h = (h_0, h_1, h_2, h_3) where each component is a d/4-dimensional real vector.
- Relation r: normalized to unit quaternion — |r| = 1 (analogous to RotatE's unit modulus constraint).
- Hamilton Product: h ⊗ r = (h_0r_0 - h_1r_1 - h_2r_2 - h_3r_3) + (h_0r_1 + h_1r_0 + h_2r_3 - h_3r_2)i + ...
**Scoring Function**:
- Score(h, r, t) = (h ⊗ r) · t — inner product between the rotated head and the tail entity.
- Normalization: relation quaternion r normalized to |r| = 1 before computing Hamilton product.
**Non-Commutativity Advantage**:
- h ⊗ r ≠ r ⊗ h — applying relation then checking tail differs from applying relation to tail.
- Naturally encodes directional asymmetry without explicit constraints.
**QuatE vs. RotatE vs. ComplEx**
| Aspect | ComplEx | RotatE | QuatE |
|--------|---------|--------|-------|
| **Embedding Space** | Complex (2D) | Complex (2D, unit) | Quaternion (4D, unit) |
| **Parameters/Entity** | 2d | 2d | 4d |
| **Relation DoF** | 2 per dim | 1 per dim (angle) | 3 per dim (3 angles) |
| **Commutative** | Yes | Yes | No |
| **Composition** | Limited | Yes | Yes |
**Benchmark Performance**
| Dataset | MRR | Hits@1 | Hits@10 |
|---------|-----|--------|---------|
| **FB15k-237** | 0.348 | 0.248 | 0.550 |
| **WN18RR** | 0.488 | 0.438 | 0.582 |
| **FB15k** | 0.833 | 0.800 | 0.900 |
**QuatE Extensions**
- **DualE**: Dual quaternion embeddings — extends QuatE with dual quaternions encoding both rotation and translation in one algebraic structure.
- **BiQUEE**: Biquaternion embeddings combining two quaternion components — further extends expressiveness.
- **OctonionE**: Extension to 8D octonion space — maximum geometric expressiveness at significant computational cost.
**Implementation**
- **PyKEEN**: QuatEModel with Hamilton product implemented efficiently using real-valued tensors.
- **Manual PyTorch**: Implement Hamilton product explicitly — compute four real vector products, combine per quaternion multiplication rules.
- **Memory**: 4x parameters compared to real-valued models — ensure sufficient GPU memory for large entity sets.
QuatE is **high-dimensional geometric reasoning** — harnessing the rich algebra of 4D quaternion rotations to encode the full complexity of real-world relational patterns, pushing knowledge graph embedding expressiveness beyond what 2D complex rotations can achieve.