SlowGuess/NAACL_2022_OCR huggingface dataset:
Baselines. To the best of our knowledge, the framework with a dangling detection module proposed in Sun et al. (2021) is the only study on dangling-aware entity alignment. It includes three dangling detection techniques: (i) NN classification, (ii) marginal ranking (MR), and (iii) background ranking (BR). As the NN classification performs much worse than others, we choose MR and BR as baselines. For a fair comparison with (Sun et al., 2021), we use the same base alignment model MTransE (Chen et al., 2017). The results using AliNet (Sun et al., 2020b) as a base are presented in Appx. D. Note that our methods are model-agnostic and can be incorporated with any detection and alignment methods. The entity alignment models that consider side information are left for future work.
Model configuration. In MHP, aside from our proposed components as described in Sec. 3, we have a base alignment module (e.g., MTransE) and a base dangling entity loss (e.g., MR) as in Sun et al. (2021). For KG embeddings and model weights, we use Xavier initialization (Glorot and Bengio, 2010) and optimize them using Adam optimizer (Kingma and Ba, 2014). The number of hidden units in the dangling entity classifier is 128. The number of nearest targets  $k$  and nearest sources  $m$  are set as 5. The learning rate is set to 0.001 for all components except WGAN where the learning rate is 5e-5 for three objectives. To terminate training, early stopping is used based on the F1 score of two-step entity alignment on validation set. The computational environment and other configuration details are reported in Appx. B and C.
4.2 Consolidated evaluation
Dangling entity detection. According to the results in Tab. 1, no matter which base dangling detection loss we adopt, MHP consistently achieves better F1 scores compared with the corresponding baseline by Sun et al. (2021) without our proposed techniques. In terms of the recall, MHP also outperforms baselines with a large margin, which indicates that our framework has a better coverage to find more dangling entities. With better recalls, MHP has the same level or slightly worse precision
compared with baselines. But our higher recall and F1 scores in dangling detection imply that more predicted matchable source entities would enter two-step entity alignment, which can improve the final alignment performance. Comparing MHP + MR and MHP + BR, we can see that the MR variant is generally better than the BR variant. This is because MR considers the similarity between a source and its nearest neighbor, which can benefit the learning of local high-order proximity in MHP. In summary, MHP demonstrates superior effectiveness for detecting dangling entities.
Two-step entity alignment. The results of two-step alignment are shown in Tab. 2. In general, MHP again consistently offers better F1 scores than baseline methods. The relative improvement ranges from  $11\%$  to  $32\%$ . The improvement can be partly attributed to the more accurate dangling entity detection performance, and thus less error is propagated to the alignment process. In contrast, baselines may try to align many dangling entities, which leads to lower performance on two-step alignment. As MHP with MR outperforms  $\mathrm{MHP} + \mathrm{BR}$  in dangling detection,  $\mathrm{MHP} + \mathrm{MR}$  also achieves better performance in two-step alignment. This indicates that dangling entity detection is of importance on the dangling-aware entity alignment problem since it has strong effects on the performance of two-step alignment.
4.3 Relaxed evaluation
Tab. 3 shows the results of relaxed evaluation. This setting only considers matchable source entities in the test phase to investigate how our framework affects the alignment learning of these entities.
Generally, MHP offers better performance than baselines on all language pairs in terms of all metrics. This indicates that dangling awareness captured by MHP further helps with a more precise alignment. The improvement can also be partly attributed to the alleviated hubness problem by the NCA loss which we investigate more in Sec. 4.4. Comparing two variants of MHP, we can see that  $\mathrm{MHP} + \mathrm{MR}$  usually outperforms the BR variants on most language pairs except for FR-EN. The reason could be that FR-EN has more entities and only with sufficient data BR can effectively separate dangling entities from randomly sampled target entities, while MR is not sensitive to data volume.
<sup>5</sup>AliNet performs worse than MTransE on dangling-aware entity alignment as found by Sun et al. (2021).
Methods	ZH-EN	EN-ZH	JA-EN	EN-JA	FR-EN	EN-FR
Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1
MR	.781	.702	.740	.866	.675	.759	.799	.708	.751	.864	.653	.744	.482	.575	.524	.639	.613	.625
BR	.811	.728	.767	.892	.700	.785	.816	.733	.772	.888	.731	.801	.539	.686	.604	.692	.735	.713
MHP + MR	.784	.831	.807	.858	.861	.859	.815	.791	.803	.865	.852	.858	.580	.724	.644	.707	.749	.727
MHP + BR	.758	.815	.785	.832	.847	.839	.783	.785	.784	.834	.848	.841	.569	.706	.635	.685	.747	.714
Table 1: Dangling entity detection results on DBP2.0. MR refers to marginal ranking and BR refers to the background ranking. The base alignment model is MTransE. More results based on AliNet are in Appx. Tab. 7.
Methods	ZH-EN	EN-ZH	JA-EN	EN-JA	FR-EN	EN-FR
Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1	Prec.	Rec.	F1
MR	.302	.349	.324	.231	.362	.282	.313	.367	.338	.227	.366	.280	.260	.220	.238	.213	.224	.218
BR	.312	.362	.335	.241	.376	.294	.314	.363	.336	.251	.358	.295	.265	.208	.233	.231	.213	.222
MHP + MR	.400	.363	.381	.375	.372	.373	.378	.394	.386	.371	.384	.377	.310	.249	.276	.266	.260	.263
MHP + BR	.393	.347	.368	.347	.331	.339	.374	.372	.373	.359	.344	.352	.290	.235	.259	.269	.239	.253
Table 2: Two-step entity alignment results on DBP2.0. The base alignment model is MTransE.
Methods	ZH-EN	EN-ZH	JA-EN	EN-JA	FR-EN	EN-FR
H@1	H@10	MRR	H@1	H@10	MRR	H@1	H@10	MRR	H@1	H@10	MRR	H@1	H@10	MRR	H@1	H@10	MRR
MTransE	.358	.675	.463	.353	.670	.461	.348	.661	.453	.342	.670	.452	.245	.524	.338	.247	.531	.342
w/ MR	.378	.693	.487	.383	.699	.491	.373	.686	.476	.374	.707	.485	.259	.541	.348	.265	.553	.360
w/ BR	.360	.678	.468	.357	.675	.465	.344	.660	.451	.346	.675	.456	.251	.525	.342	.249	.531	.343
MHP + MR	.418	.727	.523	.404	.724	.513	.408	.730	.517	.410	.747	.524	.274	.568	.371	.274	.566	.370
MHP + BR	.412	.718	.517	.396	.714	.505	.400	.727	.511	.400	.728	.511	.278	.574	.376	.272	.569	.370
Table 3: Entity alignment results in the relaxed setting on DBP2.0.
Methods	Dangling detection	Two-step alignment
F1	Δ	F1	Δ
MHP	.807	0	.381	0
- Dangling cls.	.752	-.055	.339	-.042
- OT	.789	-.018	.369	-.012
- NCA	.803	-.004	.361	-.020
Table 4: Ablation study in the consolidated setting on ZH-EN. We remove each technique and report the performance decline  $\Delta$  compared with the full MHP.
Figure 3: The number of total occurrences of most frequently aligned target entities on ZH-EN.
To investigate the effectiveness of each module in MHP, we conduct an ablation study on the consolidated setting and show the results in Tab. 4. Compared with the full version MHP, removing any component causes the degraded performance. Specifically, by removing the dangling classifier, the F1 score of dangling detection drops 0.055, which also leads to a large performance drop on two-step alignment. This indicates that the local high-order proximity is useful for dangling detection. Removing OT decreases the F1 scores on both detection and two-step alignment, showing
the effectiveness of globally aligning distributions. Lastly, leaving the NCA loss out makes the F1 score of two-step alignment decrease 0.02 compared with MHP, because using the NCA loss reduces the extent of hubness, as discussed below.
Hubness problem. To examine whether the NCA loss reduces the hubness problem, we list a set of most frequently aligned (target) entities, and observe how frequently they appear as the nearest neighbor of other entities in the embedding space. We compare MHP with the MTransE + MR variant used by Sun et al. (2021). As shown in Fig. 3,
the most frequently aligned target entity (i.e., top 1) appears over 200 times as the nearest neighbor using the baseline, whereas it only appears around 100 times using MHP. A similar phenomenon is also observed for the top 3, top 5, and top 10 frequently aligned target entities. This indicates that the hubness problem is mitigated by using NCA.
4.5 Case study
To further investigate the superiority of MHP, we provide a case study on ZH-EN comparing MHP with the previous method. Fig. 4 shows that, the previous method predicts some dangling source entities as matchable based on their high cosine similarities (i.e.,  $>0.7$ ) to their nearest target entities. Each dangling entity and its corresponding nearest target entity are different but share similar meanings (e.g., are both war events in ancient China or locations). However, the nearest targets prefer other source entities with higher similarities. Using this second-order proximity information, MHP correctly detects these dangling entities with high probability scores (i.e.,  $>0.9$ ).
Additionally, Tab. 5 demonstrates more dangling entities which are not correctly detected by the previous method (Sun et al., 2021). Most of the dangling entities are aligned to some similar counterparts sharing the same attribute. For example, the dangling entity and its nearest target entity are both colleges, theoretical physicists, or political parties. However, from the view of the nearest target entity, it prefers other nearest neighbors on source KG. Such second-order proximity information cannot be captured by the previous method, which causes those dangling entities not able to be detected. In contrast, MHP can successfully detect those dangling entities with high probabilities. This shows the effectiveness of MHP and the informativeness of the second-order proximity.
Entity alignment. Embedding-based entity alignment methods seek to find identical entities between KGs in their embedding spaces. Such a method encodes each KG into an embedding space and capture entity alignment by learning a linear mapping between embedding spaces (Chen et al., 2017) or directly infer the embedding proximity in a shared space (Sun et al., 2017). Existing studies mainly fall into two lines of improving the embedding representations. The first line exploits bet
Figure 4: Case study on ZH-EN where some dangling entities wrongly predicted as matchable by the previous first-order method can be correctly predicted as dangling with high probabilities via MHP. Arrows point from an entity to its NN in the other KG. The scores above arrows denote cosine similarities and those beside dangling ent. are probabilities of dangling by MHP.
ter graph encoders to improve embedding learning (Sun et al., 2018; Wang et al., 2018; Cao et al., 2019; Sun et al., 2020a,b; Fey et al., 2020). The second group considers the side information of entities (Chen et al., 2018b; Trisedya et al., 2019; Zhang et al., 2019; Xu et al., 2019b; Wang et al., 2020; Tang et al., 2020; Wu et al., 2019; Yang et al., 2019; Liu et al., 2020, 2021). Interested readers can refer to the recent surveys (Sun et al., 2020c; Zeng et al., 2021). Note that prior methods nearly all assume one-to-one perfect match exists between two KGs, without considering dangling entities.
Recently, Sun et al. (2021) have proposed a new problem setting, i.e., dangling-aware entity alignment, which is more practical as dangling entities naturally exist in real-world KGs. This problem setting requests a model to both detect dangling entities and align matchable ones. As the pioneering work, Sun et al. (2021) propose three baseline methods (i.e., marginal ranking, background ranking, and nearest neighbor classification) based on the nearest neighbor of source entities. Thus, these methods only rely on the first-order proximity, which is the major difference with MHP.
Optimal transport. Optimal transport (OT) aims to find the plan with minimal transportation cost for changing one distribution to another distribution, which naturally provides a way to align two distributions. Arjovsky et al. (2017) use the Wasserstein distances to recast the learning of generative adversarial network (GAN) as a transportation problem. OT has been widely used in other applications like text generation (Chen et al., 2018a) and graph
Dangling entity	Cls. Prob.	The nearest target	Cosine Sim.	The nearest source	Cosine Sim.
哥伦比亚国际学院(Columbia International College)	0.95	University of Ottawa	0.67	渥太华大学(University of Ottawa)	0.80
丁肇中(Samuel C. C. Ting)	0.91	George Uhlenbeck	0.65	乔治·乌伦贝克(George Uhlenbeck)	0.78
美国国会地铁(Congressional Subway)	0.99	United States Congress	0.67	美国国会(United States Congress)	0.75
王豫元(Larry Wang)	1.00	Wu Den-yih	0.65	吴敦义(Wu Den-yih)	0.91
新生党(Japan Renewal Party)	1.00	Democratic Party of Japan	0.64	民主党(日本)(Democratic Party of Japan)	0.85
意大利裔澳洲人(Italian Australians)	0.99	Chinese Australians	0.64	澳大利亚华人(Chinese Australians)	0.71
新加坡发展部(Ministry of Development (Singapore))	0.93	Ministry of Transport (Singapore)	0.72	林瑞生(Lim Swee Say)	0.75
Table 5: Some dangling source entities wrongly predicted as matchable by the previous method, while MHP predicts them as dangling with high probabilities. Cls. Prob. denotes the probabilities of dangling generated by MHP. The fourth column denotes the cosine similarity between the dangling entity and its nearest target. The nearest source is the nearest neighbor of the nearest target on the source KG. The last column denotes the cosine cosine similarity between the nearest target and its nearest source.
matching (Xu et al., 2019a). Pei et al. (2019) formalize entity alignment as OT in the conventional setting, which however only considers one-to-one alignment between matchable entities. We instead leverage OT to identify dissimilar parts of embedding distributions to detect dangling entities, meanwhile using OT only as one of the three high-order measures for alignment.
In this paper, we propose a framework, MHP, with mixed high-order proximities for dangling-aware entity alignment. MHP captures the local high-order proximity via a dangling classifier based on both the first- and second-order proximities. Additionally, we propose a Optimal Transport based method considering the global high-order proximity to facilitate both dangling detection and entity alignment. Comprehensive experiments on two alignment settings show the effectiveness of utilizing mixed high-order proximities. Furthermore, our extensive ablation study demonstrates the effectiveness of each technique.
We appreciate the anonymous reviewers for their insightful comments and suggestions. Juncheng Liu and Xiaokui Xiao are supported by the Ministry of Education, Singapore (Grant No. MOE2018-T2-2-091). Muhao Chen is supported by the National Science Foundation of United States Grant IIS 2105329, and by the DARPA MCS program under Contract No. N660011924033 with the United States Office Of Naval Research. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agencies.
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We present the dataset statistics of DBP2.0 (Sun et al., 2021) in Tab. 6. DBP2.0 contains three crosslingual settings for dangling-aware entity alignment, i.e., Chinese-English (ZH-EN), Japanese-English (JA-EN) and French-English (FR-EN). Please note that FR-EN is much larger than ZH-EN and JA-EN, and our methods are scalable to such a large dataset.
Datasets	# Entities	# Danglings	# Rel.	# Triples	# Align.
ZH-EN	ZH	84,996	51,813	3,706	286,067	33,183
EN	118,996	85,813	3,402	586,868
JA-EN	JA	100,860	61,090	3,243	347,204	39,770
EN	139,304	99,534	3,396	668,341
FR-EN	FR	221,327	97,375	2,841	802,678	123,952
EN	278,411	154,459	4,598	1,287,231
Table 6: Dataset statistics of DBP2.0
B Computational Environment