Quaternion-based uncertain knowledge graph embedding

doi:10.11772/j.issn.1001-9081.2025081025

Journal of Computer Applications ›› 2026, Vol. 46 ›› Issue (9): 2809-2819.DOI: 10.11772/j.issn.1001-9081.2025081025

• Artificial intelligence • Previous Articles

Quaternion-based uncertain knowledge graph embedding

Renrui YIN, Guanfeng LI(), Shizhuo WANG, Yaya JIANG

School of Information Engineering，Ningxia University，Yinchuan Ningxia 750021，China

Received:2025-09-08 Revised:2025-11-13 Accepted:2025-11-14 Online:2025-11-21 Published:2026-09-10
Contact: Guanfeng LI
About author:YIN Renrui， born in 1999， M. S. candidate. Her research interests include uncertain knowledge graph embedding and reasoning， uncertainty detection in knowledge graphs.
LI Guanfeng， born in 1979， Ph. D.， associate professor. His research interests include knowledge engineering， intelligent computing.
WANG Shizhuo， born in 2002， M. S. candidate. His research interests include knowledge graph reasoning， reinforcement learning.
JIANG Yaya， born in 2002， M. S. candidate. Her research interests include uncertain knowledge graph reasoning， rule mining.
Supported by:
National Natural Science Foundation of China(62066038);Ningxia Natural Science Foundation(2024AAC03098);Ningxia High-level Talents Scientific Research Start-up Project(2023BSB03066)

基于四元数的不确定知识图谱嵌入

尹仁蕊, 李贯峰(), 王世卓, 蒋娅娅

宁夏大学信息工程学院，银川 750021

通讯作者: 李贯峰
作者简介:尹仁蕊（1999—），女，宁夏中卫人，硕士研究生，CCF会员，主要研究方向：不确定知识图谱嵌入和推理、知识图谱不确定性检测
李贯峰（1979—），男，宁夏固原人，副教授，博士，CCF会员，主要研究方向：知识工程、智能计算
王世卓（2002—），男，河南郑州人，硕士研究生，CCF会员，主要研究方向：知识图谱推理、强化学习
蒋娅娅（2002—），女，甘肃甘谷人，硕士研究生，CCF会员，主要研究方向：不确定知识图谱推理、规则挖掘。
基金资助:
国家自然科学基金资助项目(62066038);宁夏自然科学基金资助项目(2024AAC03098);宁夏高层次人才科研启动项目(2023BSB03066)

Abstract

Abstract:

The existing methods have made preliminary progress in Uncertain Knowledge Graph Embedding （UKGE）， but still have multiple limitations， particularly in modeling complex relations. The UKGE framework can only model simple relation patterns such as symmetry. Although the complex-plane rotation model RotatE is simple and efficient， and can model multi-relation facts by complex rotations， its expressiveness is constrained by the two-dimensional complex space， making it difficult to model high-dimensional rotations， and it is prone to the “gimbal lock” problem， which limits further modeling of complex relations. To address these limitations， an uncertain knowledge graph embedding model based on quaternions， termed QUKGE， was proposed. In QUKGE， entities were represented as four-dimensional quaternion vectors， and relations were modeled as rotation operators in quaternion space. At the scoring function level， quaternion rotation was realized by utilizing the Hamilton product of entity and relation embeddings， thereby representing a variety of complex relation patterns systematically， including symmetry， asymmetry， inverse relations， commutativity， and non-commutativity. The three-dimensional rotation characteristics of quaternions effectively alleviate the “gimbal lock” problem in high-dimensional rotation and significantly enhance the capability to model complex relationships. To validate the effectiveness of QUKGE， five comparative experiments were conducted on three public uncertain knowledge graph datasets： CN15K， NL27K， and PPI5K. Experimental results demonstrate that on the NL27K dataset， QUKGE achieves a 2.9 percentage points improvement in F1 score and a 1.0 percentage point increase in accuracy compared to the MUKGE （Multiplex Uncertain Knowledge Graph Embedding） model on relational fact classification task， providing support for future research in UKGE.

Key words: Uncertain Knowledge Graph Embedding (UKGE), quaternion, complex relation modeling, rotation

摘要：

现有方法虽然在不确定知识图谱嵌入（UKGE）上取得了初步进展，但仍存在诸多不足，尤其在对复杂关系的建模方面表现有限。UKGE框架仅能刻画对称等简单关系模式。尽管复数平面上的旋转模型RotatE能借助复数旋转刻画多关系事实，简洁而高效，但它的表达能力受限于二维复数空间，难以刻画高维旋转，且易出现“万向锁”现象，限制了对复杂关系的进一步刻画。针对上述局限，提出一种基于四元数的不确定知识图谱嵌入模型QUKGE，该模型将实体表示为四维四元数向量，并将关系建模为四元数空间中的旋转算子。在评分函数层面，利用实体与关系的汉密尔顿积实现四元数旋转，从而系统地表征对称、反对称、逆关系、可交换及不可交换等复杂关系模式。四元数的三维旋转特性有效缓解了高维旋转中的“万向锁”，并显著提升了对复杂关系的刻画能力。为验证QUKGE的有效性，在CN15K、NL27K和PPI5K这3个公开不确定知识图谱数据集上进行5项对比实验。实验结果表明，QUKGE在NL27K数据集上与MUKGE（Multiplex Uncertain Knowledge Graph Embedding）模型相比，在关系事实分类任务中的F1分数与准确率分别提升了2.9和1.0个百分点，为UKGE的进一步研究提供了支撑。

关键词: 不确定知识图谱嵌入, 四元数, 复杂关系建模, 旋转

CLC Number:

TP391.1

Renrui YIN, Guanfeng LI, Shizhuo WANG, Yaya JIANG. Quaternion-based uncertain knowledge graph embedding[J]. Journal of Computer Applications, 2026, 46(9): 2809-2819.

尹仁蕊, 李贯峰, 王世卓, 蒋娅娅. 基于四元数的不确定知识图谱嵌入[J]. 《计算机应用》唯一官方网站, 2026, 46(9): 2809-2819.

Figures/Tables 11

Fig. 1 Complex space spherical coordinate system

Fig. 2 QUKGE model architecture

Tab. 1 Basic information of datasets

数据集	实体数	关系数	三元组数			置信度
数据集	实体数	关系数	训练	验证	测试	平均值	标准值
CN15K	15 000	36	204 984	16 881	19 293	0.629	0.232
NL27K	27 221	404	149 100	12 278	14 034	0.797	0.242
PPI5K	4 999	7	230 929	19 017	21 720	0.415	0.213

Tab. 2 Relevant information about QUKGE variant models

模型	得分函数 $f (h, r, t)$	参数
QUKGE_TransE	$(Q h + W r) - Q t$	$Q h, W r, Q t ∈ R k$
QUKGE_HolE	$W r T Q h ⋆ Q t$	$Q h, W r, Q t ∈ R k$
QUKGE_ComplEx	$W r ⋅ (Q h ⊗ Q t)$	$Q h, W r, Q t ∈ C k$
QUKGE_RotatE	$Q h ∘ W r - Q t$	$Q h, W r, Q t ∈ C k, W r i = 1$

Tab. 2 Relevant information about QUKGE variant models

模型	得分函数 $f (h, r, t)$	参数
QUKGE_TransE	$(Q h + W r) - Q t$	$Q h, W r, Q t ∈ R k$
QUKGE_HolE	$W r T Q h ⋆ Q t$	$Q h, W r, Q t ∈ R k$
QUKGE_ComplEx	$W r ⋅ (Q h ⊗ Q t)$	$Q h, W r, Q t ∈ C k$
QUKGE_RotatE	$Q h ∘ W r - Q t$	$Q h, W r, Q t ∈ C k, W r i = 1$

Tab. 3 Confidence prediction results of different models

模型	CN15K		NL27K		PPI5K
模型	MSE	MAE	MSE	MAE	MSE	MAE
URGE	0.103 2	0.227 2	0.074 8	0.113 5	0.014 4	0.060 0
UKGE_rect	0.086 1	0.199 0	0.023 6	0.069 0	0.009 5	0.037 9
UKGE_logi	0.098 6	0.207 4	0.034 3	0.079 3	0.009 6	0.040 7
BEUrRE	0.074 9	0.198 8	0.020 1	0.068 9	—	—
MUKGE_rect	0.070 5	0.197 3	0.016 3	0.067 6	0.003 5	0.030 2
MUKGE_logi	0.099 9	0.204 2	0.021 1	0.061 2	0.003 6	0.025 0
QUKGE_TransE	0.127 3	0.252 6	0.073 3	0.107 2	0.008 1	0.048 4
QUKGE_HolE	0.110 3	0.224 4	0.051 2	0.095 9	0.009 0	0.047 8
QUKGE_ComplEx	0.109 0	0.209 4	0.026 7	0.075 5	0.005 9	0.040 6
QUKGE_RotatE	0.100 4	0.206 7	0.023 3	0.072 3	0.007 8	0.045 4
QUKGE	0.074 7	0.199 5	0.017 9	0.060 8	0.002 8	0.023 4

Tab. 4 Ranking results of relational facts by different models

模型	CN15K		NL27K		PPI5K
模型	Linear	Exp	Linear	Exp	Linear	Exp
TransE	0.601	0.591	0.730	0.722	0.710	0.700
DistMult	0.689	0.677	0.911	0.897	0.894	0.880
ComplEx	0.723	0.712	0.921	0.913	0.896	0.881
URGE	0.572	0.570	0.593	0.593	0.726	0.723
UKGE_rect	0.773	0.775	0.939	0.942	0.946	0.946
UKGE_logi	0.789	0.788	0.955	0.956	0.970	0.969
BEUrRE	0.796	0.795	0.942	0.942	0.951	0.952
MUKGE_rect	0.832	0.835	0.877	0.882	0.959	0.958
MUKGE_logi	0.849	0.850	0.945	0.947	0.991	0.990
QUKGE_TransE	0.804	0.796	0.893	0.896	0.986	0.993
QUKGE_HolE	0.811	0.808	0.882	0.925	0.980	0.992
QUKGE_ComplEx	0.819	0.806	0.947	0.953	0.983	0.992
QUKGE_RotatE	0.834	0.820	0.942	0.942	0.987	0.994
QUKGE	0.845	0.844	0.967	0.968	0.997	0.995

Tab. 5 Classification results of relational facts by different models

模型	CN15K		NL27K		PPI5K
模型	F1	Acc	F1	Acc	F1	Acc
TransE	23.1	67.9	65.1	53.4	83.2	98.5
DistMult	27.9	71.1	72.1	70.1	86.9	97.1
ComplEx	18.9	73.2	63.3	53.4	83.2	98.9
URGE	21.2	86.0	83.6	88.7	85.2	98.6
UKGE_rect	28.8	90.4	92.3	95.2	95.1	99.4
UKGE_logi	25.9	90.1	88.4	93.0	94.5	98.7
BEUrRE	28.7	89.9	92.4	95.6	95.0	99.4
MUKGE_rect	32.0	90.4	87.5	92.5	94.7	99.3
MUKGE_logi	28.3	90.2	91.5	94.4	95.3	99.7
QUKGE_TransE	22.6	85.8	89.1	90.4	88.4	96.9
QUKGE_HolE	23.4	85.7	90.3	91.9	83.3	94.2
QUKGE_ComplEx	25.6	87.6	84.8	86.5	92.7	98.0
QUKGE_RotatE	29.2	89.4	86.0	86.9	82.3	95.7
QUKGE	30.1	90.8	94.4	95.4	97.8	99.4

Tab. 6 Comparison of MRR between QUKGE and QUKGERotatE on NL27K dataset

关系	MRR		三元组数	关系类型
关系	QUKGE	QUKGE_RotatE	三元组数	关系类型
city located in geopolitical location	1.000	0.767	5	1-1
headquartered in	0.938	0.938	8	1-1
organization lead by person	0.831	0.805	44	1-1
proxy for	0.753	0.510	29	1-1
inverse of arthropod and other arthropods	0.453	0.352	6	1-n
companies headquartered here	0.264	0.301	11	1-n
agent collaborates with agent	0.443	0.390	187	1-n
city television station	0.407	0.365	14	1-n
animal that feeds on insects	0.045	0.046	7	1-n
language of university	0.096	0.080	2	n-1

Tab. 7 Comparison of efficiencies of different models

对比模型	单次训练耗时/s	全程平均每轮耗时/s	收敛迭代次数	实际训练轮数
QUKGE_TransE	0.318	0.281	2 180	2 380
QUKGE_HolE	0.329	0.290	1 500	1 700
QUKGE_ComplEx	0.467	0.431	1 260	1 460
QUKGE_RotatE	0.459	0.424	2 000	2 200
QUKGE_TF	0.634	0.455	1 420	1 620
QUKGE	0.420	0.392	1 320	1 520

Tab. 8 Impact of different learning rates on performance on PPI5K dataset

lr	MSE	MAE	Linear	Exp	F1/%	Acc/%
0.000 2	0.79	6.18	0.981	0.991	77.4	94.5
0.000 5	0.30	2.92	0.996	0.989	93.6	98.2
0.001 0	0.28	2.34	0.998	0.997	97.8	99.4
0.005 0	0.30	2.53	0.993	0.992	96.4	99.0
0.010 0	0.32	2.60	0.998	0.997	96.8	99.1

Tab. 9 Impact of different embedding dimensions on performance on PPI5K dataset

嵌入维度	MSE	MAE	Linear	Exp	F1/%	Acc/%
8	0.27	2.65	0.997	0.996	96.4	99.0
16	0.28	2.34	0.998	0.997	97.8	99.4
32	0.29	2.35	0.998	0.997	97.7	99.4
64	0.34	2.48	0.997	0.996	96.4	99.3
128	0.36	2.53	0.996	0.995	96.5	99.2
256	0.38	2.57	0.996	0.995	96.3	99.2

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Quaternion-based uncertain knowledge graph embedding

基于四元数的不确定知识图谱嵌入

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