PaperlayerView exact revision on arXiv

Source-linked AI summary

Biquaternionic Space with Complex-valued Attention for Temporal Knowledge Graph Completion

Rushan Geng, Cuicui Luo

arXiv:2609.14279v1cs.LGcs.AI

TL;DR

Temporal knowledge graph completion must infer missing facts, while existing models may use a single geometric space or static entity representations. BSCA combines biquaternionic transformations with context-dependent attention; across five benchmarks it reports its largest improvement on GDELT.

  • Problem

    Existing temporal knowledge graph models may use a single geometric space that limits representation of diverse relational patterns, or leave entity representations static.

    Source: §1 · p. 1

  • Method

    BSCA combines circular and hyperbolic transformations in biquaternionic space and uses complex-valued attention to fuse time- and relation-conditioned entity representations.

    Source: §1 · p. 1

  • Results

    Across five datasets, BSCA achieves the highest reported scores on all four metrics for four datasets; on GDELT, its MRR is 52.1% versus 38.1% for TuckERTNT.

    Source: §5.5 · p. 4

  • Takeaways & Limitations

    The findings support further investigation of hypercomplex representations for temporal knowledge graph completion.

    Source: §7 · p. 6

    The Figure 1(c) visualization is one qualitative example of a symmetric relation at one timestamp and does not establish a general condition for symmetry.

    Source: §5.7 · p. 4

Abstract

from arXiv · show

Temporal knowledge graph embedding (TKGE) models infer missing facts in knowledge graphs that evolve over time. Many existing models use a single geometric space, which can limit their ability to represent diverse relational patterns, or treat entity representations as static. We propose Biquaternionic Space with Complex-valued Attention (BSCA), a TKGE model that combines circular and hyperbolic rotations within a unified biquaternionic framework. A complex-valued attention mechanism adaptively fuses time-conditioned and relation-conditioned entity representations, allowing them to vary with temporal and relational context. Experiments on five benchmark datasets show competitive performance across datasets, with the largest improvement on GDELT: BSCA achieves an MRR of 52.1\%, compared with 38.1\% for the strongest baseline in our comparison.

1 Introduction

Temporal knowledge graphs are incomplete, and existing models may use limited geometric transformations or static entity representations. BSCA combines biquaternionic transformations with context-dependent entity representations, reporting its largest relative improvement on GDELT.

1 Introduction

  • Complex-valued attention fuses time-conditioned and relation-conditioned entity representations so their contributions vary with temporal and relational context.This addresses the limitation of approaches that incorporate time into relations while leaving entity representations static.Source: §1 · p. 1
  • On GDELT, BSCA improves MRR by approximately 37% relative to the strongest baseline in the comparison.The comparison is reported across five benchmarks.Source: §1 · p. 1
  • BSCA uses the non-commutative Hamilton product to combine circular and hyperbolic rotations in a unified biquaternionic framework.The model also uses scaling transformations.Source: §1 · p. 1

2 Related Work

Prior work spans geometric transformations for static knowledge graphs and time-aware representations for temporal completion. BSCA differs by using full non-commutative biquaternionic algebra with complex-valued attention.

2.1 Static Knowledge Graph Completion

  • Static KGE methods include translations, complex tensor factorization, and geometric transformations in complex, quaternion, dual-quaternion, and biquaternion spaces.BiQUE uses biquaternionic representations for static KGE, while DGS models interactions between two geometric spaces.Source: §2.1 · p. 1

2.2 Temporal Knowledge Graph Completion

  • Temporal KGE methods incorporate time through translations, hyperplanes, rotations, multivectors, temporal logic, and other time-aware representations.BSCA instead combines full non-commutative biquaternionic algebra with complex-valued attention to model changes in entity representations.Source: §2.2 · p. 2

3 Background

The background formulates temporal completion as ranking candidate entities for timestamped facts and introduces biquaternion algebra and four relation-pattern definitions used in the paper.

3.1 Problem Formulation

  • A temporal knowledge graph consists of timestamped subject–relation–object quadruples, and completion ranks candidate entities for missing subjects or objects.A useful scoring function assigns higher scores to true facts than to incorrect candidates.Source: §3.1 · p. 2

3.2 Biquaternions

  • A biquaternion has four components over the complex numbers, and its Hamilton product is generally non-commutative.Its Hermitian norm is non-negative and reduces to the standard quaternion norm when all four components are real.Source: §3.2 · p. 2
  • Biquaternion addition and subtraction operate componentwise, while multiplication follows the rules for the imaginary units i, j, and k.The Hamilton product is obtained by distributivity while respecting those multiplication rules.Source: §3.2 · p. 2

3.3 Relation Patterns

  • The paper considers symmetric, asymmetric, inverse, and temporally evolving relations as four relation patterns.The definitions specify whether relations hold in both directions, only one direction, across paired relations, or across timestamps.Source: §3.3 · p. 2

4 Methodology

BSCA represents temporal facts in biquaternionic space, fuses time- and relation-conditioned entity representations, and trains with link-prediction and temporal-smoothness objectives. Its stated representational capabilities cover four relation patterns.

4.1 BSCA Model

  • BSCA parameterizes entities, relations, and timestamps in biquaternionic space, with multiplicative components for transformations and additive components for translations and offsets.The time-conditioned relation transformation supports circular rotations, hyperbolic rotations, and scaling; attention fuses time- and relation-conditioned entity representations before scoring against the object embedding.Source: §4.1 · pp. 2–3
  • Complex-valued attention fuses time-conditioned and relation-conditioned subject representations, and BSCA scores facts using the inner product of the transformed subject and object embeddings.The number of parameters grows linearly with embedding dimension k for fixed numbers of entities, relations, and timestamps.Source: §4.1 · p. 3

4.2 Optimization

  • BSCA uses multiclass log-softmax training with N3 regularization and reciprocal learning, so subject and object prediction both contribute to training.The regularizer is applied to complex-valued embedding components.Source: §4.2 · p. 3

4.3 Temporal Regularization

  • Temporal regularization penalizes differences between embeddings at adjacent timestamps to discourage abrupt changes.The weight λ_τ balances the link-prediction objective and temporal-smoothness penalty.Source: §4.3 · p. 3

4.4 Modeling Various Relation Patterns

  • The stated propositions establish BSCA's representation capabilities for symmetric, asymmetric, inverse, and temporally evolving relations.The proofs are provided in Appendix A.Source: §4.4 · p. 3

5 Experiments

BSCA is evaluated on five temporal knowledge graph benchmarks, with the strongest reported gain on GDELT. Ablations, hyperparameter studies, and attention analyses examine its components, training behavior, and performance across relations and entities.

5.1 Datasets

  • The five benchmarks include ICEWS14, ICEWS05-15, GDELT, YAGO11k, and Wikidata12k; YAGO11k and Wikidata12k use coarser time intervals after preprocessing to reduce temporal sparsity.ICEWS14 and ICEWS05-15 cover 2014 and 2005–2015, while GDELT covers April 2015 to March 2016.Source: §5.1 · p. 3, Table 1 · p. 4

5.3 Baselines

  • BSCA is compared with a broad set of TKGC baselines; experiments use validation-set grid search, dimension 2000, and up to 150 epochs, with results reported from one representative run.The study also checks stability across multiple random seeds.Source: §5.3 · p. 4, §5.4 · p. 4

5.8 Hyperparameter Analysis

  • On ICEWS14, the best reported regularization weights are λμ=0.008 and λτ=0.01, and performance varies more with temporal regularization.Across ICEWS14 and YAGO11k, performance improves up to dimension 4000, with smaller gains beyond 2000; the main experiments use 2000 to balance performance and cost.Source: §5.8 · p. 4

5.5 Main Results

  • 52.1% MRR on GDELT versus 38.1% for TuckERTNT; BSCA leads all four metrics on ICEWS05-15, ICEWS14, GDELT, and Wikidata12k, and Hits@3 and Hits@10 on YAGO11k.CEC-BD has higher MRR and Hits@1 on YAGO11k.Source: §5.5 · p. 4, Table 2 · p. 5

5.6 Ablation Studies

  • Removing relation- and time-specific translations or attention lowers MRR across four datasets, although some ablated variants match or exceed the full model on individual Hits metrics.Several ablated variants also remain competitive with or outperform TComplEx and TNTComplEx.Source: §5.6 · p. 4, Table 3 · p. 5

5.7 Analysis of Relation Embeddings

  • Figure 1’s histograms show near-zero components for the learned transformation in one symmetric Consult fact at one timestamp, a qualitative example rather than a general symmetry rule.The analyzed fact is (Afghanistan, Consult, Iran, 2014-03-29).Source: §5.7 · p. 4, Figure 1 · p. 5

5.8 Hyperparameter Analysis

  • The 300-epoch training curves generally stabilize after approximately 70 epochs on ICEWS14 and YAGO11k; ICEWS14 fluctuates around epochs 60 and 150 before returning to a similar performance level.Source: §5.8 · pp. 4–5, Figure 2 · p. 6

5.9 Attention Analysis

  • Attention improves MRR for four of six selected relations and all five selected entities, but its effect is slightly negative for “Reject” and “Discuss by telephone”.The relation-level variation is consistent with the modest aggregate gains reported in the ablation study.Source: §5.9 · p. 5, Table 4 · p. 6

6 Model Efficiency Analysis

BSCA’s reported efficiency comparisons show both computational costs and savings against different baselines, indicating a balance between predictive performance and cost in the tested setting.

6 Model Efficiency Analysis

  • BSCA uses 38.41 million parameters and takes 9.5 minutes to train in the reported ICEWS14 setting.Its biquaternionic transformations and attention add cost relative to TComplEx and TNTComplEx.Source: §6 · p. 6
  • Compared with TeLM, BSCA uses approximately 40% fewer parameters and cuts training time from 17.1 to 9.5 minutes.The authors describe this as a useful balance between predictive performance and computational cost in the tested setting.Source: §6 · p. 6
  • BSCA uses fewer parameters than TLT-KGE(C), but training takes longer: 9.5 versus 8.2 minutes.Source: §6 · p. 6

7 Conclusion

The conclusion presents BSCA as a biquaternionic TKGE model with complex-valued attention, reports strong link prediction performance across five benchmarks, and supports further investigation of hypercomplex representations.

7 Conclusion

  • BSCA combines circular and hyperbolic transformations in biquaternionic space with complex-valued attention for context-dependent entity representations.Source: §7 · p. 6
  • Across five benchmarks, BSCA achieves strong link prediction performance, with its largest MRR gain on GDELT.Source: §7 · p. 6
  • Ablation studies show consistent MRR benefits from additive components and attention, while the efficiency comparison indicates moderate computational cost.Source: §7 · p. 6
  • The findings support further investigation of hypercomplex representations for temporal knowledge graph completion.Source: §7 · p. 6

A Proofs of Propositions

The appendix defines the notation used for two biquaternion operations: the Hamilton product and the inner product.

A Proofs of Propositions

A.1 Proof of Proposition 1

The proof expands the scoring function for a symmetric relation, factors a shared term, and gives an invertibility condition under which BSCA can model symmetric patterns.

A.1 Proof of Proposition 1

  • The proof expands the scoring function and then factors out the common term e_r^τ.Source: §A.1 · p. 8
  • For a symmetric relation, the proof uses equality of the scores for (s,o,τ) and (o,s,τ).Source: §A.1 · p. 8
  • When e_r^τ is invertible, the condition e_s◦e_o = e_o◦e_s is sufficient for BSCA to model symmetric patterns.The stated condition is satisfied if e_s◦e_o is symmetric.Source: §A.1 · p. 8

A.2 Proof of Proposition 2

The proofs characterize when BSCA can represent asymmetric, inverse, and temporal-evolution patterns through conditions on its relation transformations.

A.2 Proof of Proposition 2

  • BSCA can represent asymmetric relations when the time-conditioned relation transformation is non-invertible.Source: §A.2 · p. 8

A.3 Proof of Proposition 3

  • BSCA can model inverse patterns when the two time-conditioned relation transformations are inverses of each other.Source: §A.3 · p. 8

A.4 Proof of Proposition 4

  • BSCA can model temporal evolution patterns when the transformations for the two relations at their respective timestamps are inverses.Source: §A.4 · p. 8
Loading 2609.14279v1…