Lie Group Knowledge Graph Embeddings

Updated 19 July 2025

Knowledge graph embedding on a Lie group is a geometric method that maps entities and relations to smooth manifold structures with inherent group properties.
It leverages Lie groups' differentiability and boundedness to naturally encode translation, rotation, and non-commutative relational patterns.
Applications include improved link prediction and relation reasoning with competitive accuracy while reducing the need for explicit normalization.

A knowledge graph embedding on a Lie group refers to a class of geometric representation learning approaches wherein entities and relations in a knowledge graph are mapped onto spaces that possess both manifold and group structure, specifically those of Lie groups. Lie groups are smooth manifolds equipped with a compatible group operation, enabling continuous symmetries and differentiable transformations. The core motivation for such embedding schemes derives from the algebraic and geometric congruence between relational structures within knowledge graphs and the mathematical framework of Lie groups; this congruence allows embeddings to naturally encode compositionality, symmetries, inversion, and complex relation patterns, while also addressing known limitations of simpler vector space models.

1. Algebraic and Geometric Foundations

Knowledge graphs represent entities and relationships as labeled directed triples (head, relation, tail). Early embedding models such as TransE mapped entities and relations to Euclidean vectors, enforcing a translation principle (e.g., $h + r \approx t$ in $\mathbb{R}^n$ ). However, Euclidean spaces lack intrinsic mechanisms for capturing group-theoretic properties such as invertibility, closure under composition, and non-commutative relation patterns, which are prevalent in real-world knowledge graphs.

Lie groups, being differentiable manifolds with a compatible group operation, offer a mathematically rigorous substrate for modeling these behaviors. Formally, an $n$ -dimensional torus $T^n = \mathbb{R}^n/\mathbb{Z}^n$ is a compact Abelian Lie group with addition modulo 1 in each coordinate. More generally, groups such as $SO(3)$ (three-dimensional rotations), $SU(2)$ (special unitary group), and their higher-dimensional analogs provide non-Abelian (non-commutative) structures suitable for encoding asymmetric and compositional relations.

2. Translation Principle on Lie Groups

The translation principle—the central notion behind translation-based embedding models—naturally generalizes to Lie groups. For a triple $(h, r, t)$ , embeddings are learned such that the group operation satisfies

$[h] + [r] = [t]$

if the group is Abelian (as in the torus case), or

$O_r(e_1) = e_2$

for a general group action $O_r$ acting on the entity representation in the non-Abelian setting (Ebisu et al., 2017, Yang et al., 2020).

A key advantage arises from the compactness of groups such as the torus: all embeddings remain bounded without the need for explicit normalization or regularization, thus avoiding the tension observed in Euclidean models that require embeddings to be projected onto a sphere, thereby distorting the translation principle (Ebisu et al., 2017).

3. Specific Model Instantiations

TorusE

TorusE is a prototypical model embedding entities and relations on the $n$ -torus, $T^n = \mathbb{R}^n/\mathbb{Z}^n$ (Ebisu et al., 2017). This allows addition and translation operations to be performed without divergence or external normalization, as the embedding space is compact and naturally bounded. To measure the satisfaction of the translation principle, distance functions such as

$d_{L_1}([x], [y]) = \sum_i \min(|\pi_{\text{frac}}(x_i) - \pi_{\text{frac}}(y_i)|, 1 - |\pi_{\text{frac}}(x_i) - \pi_{\text{frac}}(y_i)|)$

are used, where $\pi_{\text{frac}}(x)$ denotes the fractional component of $x$ .

TorusE exhibits advantages over classical translation models by avoiding the need for normalization, faithfully preserving additive group structure, and achieving up to $11\times$ computational speedups for high-dimensional embeddings while delivering competitive or superior performance on accuracy metrics such as Mean Reciprocal Rank (MRR) and Hits@1 (Ebisu et al., 2017).

Non-Abelian Models (SO3E, SU2E, 3H-TH)

Non-Abelian Lie groups, such as $SO(3)$ and $SU(2)$ , provide operations that capture composition order sensitivity (non-commutativity) inherent in many relationship types (Yang et al., 2020, Zhu et al., 2023). In the SO3E and SU2E models, relations are parameterized as rotation matrices generated by Euler angles (for $SO(3)$ ) or as unitary matrices generated from angular parameters (for $SU(2)$ ). Entities are embedded in representation spaces— $\mathbb{R}^3$ or $\mathbb{C}^2$ —on which the group action applies.

In hyperbolic models such as 3H-TH (Zhu et al., 2023), quaternion-based 3D rotations (forming a non-commutative Lie group) are composed with hyperbolic translations. This design allows simultaneous modeling of symmetries, inversion, non-commutative composition, hierarchy, and multiplicity, as the rotation operation provides non-commutative transformation and the hyperbolic structure supports hierarchy in low dimensions.

Module-Based Extensions

The ModulE framework generalizes group-theoretic embedding spaces to modules over potentially non-commutative rings, incorporating Lie groups such as $U(1)$ (complex unit circle) and $UH(1)$ (quaternion unit sphere) as rotation groups (Chai et al., 2022). Entities are embedded as pairs of scalar and vector elements, each transformed by respective group actions, unifying earlier vector space and group embedding approaches under a broader module structure.

4. Theoretical and Practical Properties

Embedding on a Lie group enables models to satisfy essential relational algebraic axioms:

Closure: Relation compositions remain group elements (e.g., $r_1 \cdot r_2$ is a valid relation).
Identity and Inverses: Each relation has an inverse; group actions have identities, enabling natural modeling of invertible and symmetric relations.
Associativity: The group operation supports unambiguous multi-relation composition, crucial for complex path reasoning.
Non-commutativity: Non-Abelian Lie groups encode order sensitivity, allowing the capture of directed or asymmetric relationship phenomena.

Empirically, these properties manifest as improved generalization and representation of varied relation patterns (e.g., in FB15k-237 and WN18RR benchmarks). Models like SO3E and SU2E demonstrate state-of-the-art or competitive MRR and Hits@k in settings where relation composition and inversion dominate (Yang et al., 2020).

5. Optimization, Scalability, and Regularization

Lie group manifolds (including tori and rotation groups) are differentiable and thus compatible with standard optimization routines such as gradient descent. The compactness of groups like tori ensures that embeddings cannot diverge, eliminating the need for post-hoc regularization and sidestepping the deformation effect induced by sphere-based normalization in models like TransE (Ebisu et al., 2017).

For high-dimensional embeddings, group-based models have shown considerable gains in computational efficiency, notably by avoiding expensive normalization steps and, when using FFT-based algorithms for Lie group harmonics, accelerating computations involving group actions (Rosen et al., 2023).

A related development is the use of Lie group manifolds as a unifying substrate for reducing heterogeneity among factor tensors in temporal knowledge graph embedding via tensor decomposition. Mapping factors onto a smooth Lie group manifold (e.g., SO(2), using Givens rotations and logarithmic mapping to the Lie algebra) homogenizes tensor distributions, improves fusion efficiency, and enhances predictive performance without additional model parameters (Li et al., 14 Apr 2024).

6. Applications, Limitations, and Future Prospects

Lie group-based knowledge graph embedding models have broad applicability in link prediction, relation composition reasoning, and scenarios demanding faithful preservation of symmetry and hierarchy. For models employing group actions (multiplication, rotation, or vector fields on manifolds), the embedding space can be tailored to reflect known relational patterns or domain symmetries—for example, rotational invariance in biomedical or physical knowledge graphs, or non-commutative structure in scientific or social data.

Possible limitations include the increased mathematical and computational complexity inherent in parameterizing and optimizing embeddings over continuous group manifolds, particularly for non-compact or high-dimensional groups. Ensuring that the learned representations conform to the group's structure during stochastic optimization may require Riemannian optimization or careful regularization.

Future research directions involve:

Systematic investigation of alternative compact Lie groups and their representation theories as embedding spaces (Ebisu et al., 2017, Yang et al., 2020).
Expanding module-based approaches to richer non-commutative rings and their associated Lie groups (Chai et al., 2022).
Integrating group-invariant Laplacian operators for regularization and eigenfunction feature extraction in knowledge graphs with explicit symmetry (Rosen et al., 2023).
Theoretical exploration of the trade-off between expressivity and computational tractability as group structure complexity increases.
Development of end-to-end models that jointly learn group actions and entity representations in the presence of temporal or dynamic knowledge graphs, leveraging tensor decomposition and group homogenization techniques (Li et al., 14 Apr 2024).

7. Summary Table: Selected Lie-Group-Based Knowledge Graph Embedding Models

Model	Underlying Lie Group	Core Relation Modeling Mechanism	Notable Properties
TorusE	$T^n$ (n-torus)	Addition modulo 1	Boundedness, translation, no external normalization (Ebisu et al., 2017)
SO3E, SU2E	$SO(3)$ , $SU(2)$	Rotation matrix (non-Abelian) actions	Non-commutative composition, invertibility (Yang et al., 2020)
3H-TH	$\mathbb{H}^3$ + Quaternions	3D rotation (quaternion), hyperbolic add.	Hierarchy, symmetry, inversion, non-commutative composition (Zhu et al., 2023)
ModulE	$U(1)$ , $UH(1)$ , modules	Group/module-based scaling and rotation	Non-commutative algebra, module theory (Chai et al., 2022)
G-GL	General unitary Lie groups	G-invariant graph Laplacian construction	Symmetry adaptation, accelerated convergence (Rosen et al., 2023)

Embedding models based on Lie groups provide a mathematically principled, expressive, and empirically effective approach for knowledge graph representation learning, addressing structural, computational, and generalization limitations present in traditional vector-space-based models.