Lambda: Learning Matchable Prior For Entity Alignment with Unlabeled Dangling Cases
Authors: Hang Yin
Liyao Xiang
Dong Ding
Yuheng He
Yihan Wu
Pengzhi Chu
Xinbing Wang
Chenghu Zhou
Venue: NeurIPS 2024 poster (2024)
Keywords: Knowledge Graph
Entity Alignment
Positive-Unlabeled Learning
Dangling Cases
Primary area: graph_neural_networks
TL;DR: entity alignment with unlabeled dangling entity pineer work using novel PU learning algorithm

Abstract: We investigate the entity alignment (EA) problem with unlabeled dangling cases, meaning that partial entities have no counterparts in the other knowledge graph (KG), yet these entities are unlabeled. The problem arises when the source and target graphs are of different scales, and it is much cheaper to label the matchable pairs than the dangling entities. To address this challenge, we propose the framework \textit{Lambda} for dangling detection and entity alignment. Lambda features a GNN-based encoder called KEESA with a spectral contrastive learning loss for EA and a positive-unlabeled learning algorithm called iPULE for dangling detection. Our dangling detection module offers theoretical guarantees of unbiasedness, uniform deviation bounds, and convergence. Experimental results demonstrate that each component contributes to overall performances that are superior to baselines, even when baselines additionally exploit 30\% of dangling entities labeled for training.

LAMBDA: Learning Matchable Prior For Entity Alignment with Unlabeled Dangling Cases
Hang Yin, Liyao Xiang"
Dong Ding Shanghai Jiao Tong University Shanghai, China
Shanghai Jiao Tong University
Shanghai, China
Xyinhang_SJTV ciangliyao088)$sjtee edu.
18916162516dsj wat.edd.ccom cn
Yuheng He, Yihan Wu, Pengzhi Chu, Xinbing Wang Shanghai Jiao Tong University
Chenghu Zhou Chinese Academy of Sciences
Shanghai, China
Beijing, China
(heyuheng, caracalla, pzchu, xwang8) ju. edu. cn
chouchsjtu@gmen com
Abstract
We investigate the entity alignment (EA) problem with unlabeled dangling cases, meaning that partial entities have counterparts in the other knowledge graph (KG), yet these entities are unlabeled. The problem arises when the source and tar- get graphs are different scales, and it much cheaper label the matchable pairs than the dangling entities. To address this challenge, we propose the framework Lambda for dangling detection and entity aligmment. Lambda features GNN- based encoder called KEESA with spectral contrastive learning loss for EA and a positive-unlabeles learning algorithm called iPULE for dangling detection. Our dangling detection module offers theoretical guarantees of unbiasedness, uniform deviation bounds, and convergence. Experimental results demonstrate that each component contributes to overall performances that are superior to baselines, even when baselines additionally exploit 30% of dangling entities labeled for training.
Introduction
Entity alignment is problem that seeks entities referring to the same real-world identity across different knowledge graphs (KGs), and is widely deployed in fields such knowledge fusion, question-answertt
web mining, etc. To address the issue, embedding- -baeed methods have been proposed to capture entity similarity in the embedding space through translation-based (24 42 20
or graph neural network (GNN)-based [41 37 27 26 35 models. Particularly, if the entities do not have counterparts another KG, the entities referred to dangling entities, as shown in Fig.
In many real-world scenarios, the labels for the dangling entities on KGs are often missing, as those labels are much harder to acquire. For example, in KG plagiarism detection, it is relatively easy to align entity pairs that both exist KGs, but one would have to traverse all possible pairs to conclude an entity is not paired. Hence EA with unlabeled dangling entities is hard but realistic problem. The problem even worsens in EA on KGs of different scales where the dangling entities take large proportion of all nodes.
Despite that many works have been investigating the EA problem with dangling entities, few have focused on EA with unlabeled dangling cases. We list closely related works in Table [ The work of 3 extends the conventional EA methods MTransE 5 and AliNet [7] to the case with dangling
CCorresponding author: Liyao Xiang, xiangliyao08 @sjtu.educnn
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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entities, and thus require portion of labeled dangling entities for training. Based on their works, MHP [19 has improved performance with additional knowledge, i... the high-order proximities information for alignment. Both UED [24] and SoTea [23 are unsupervised schemes that rely on side information such as entity name/attribute as global alignment information. Different from prior works, we consider a stricter case where neither side information nor any labeled dangling entities are known, as side information often leads to name-bias [21 while labels for dangling entities are hard to obtain. matchable entity in target KG
Method
Side Info
Dangling Labels
33
w/ MTransE
X
3
w/ AliNet
X
UED
24
X
SoTea 23
V
X
MHP 19
X
high-order info
dangling entity In source KG
dangling entity in target KG
Our Work
X
X
Figure 1: Examples of dangling entities.
Table 1: Different EA models with dangling cases.
The EA with unlabeled dangling entities faces unique challenges: first, the unlabeled dangling entities would cause erroncous information to propagate through neighborhood aggregation if applying conventional GNN-based embedding methods, negatively affecting the dangling detection and alignment of matchable entities. Second, the absence of labeled dangling entities makes its feature distribution non-observable, requiring the model to distinguish potential dangling entities while learning the representation of matchable entities. Hence the EA problem has be solved with mere positive (matchable entities with labels) and unlabeled samples, yet without any prior knowledge of the distribution of the nodes.
We tackle the first challenge by proposing a novel GNN-based EA framework. To eliminate the 'pollution' of dangling entities, the adaptive dangling indicator has been applied globally for selective aggregation. Relation projection attention is designed to combine both entity and relation information for more comprehensive representation. The designed spectral contrastive learning loss disentangles the matchable entities from dangling ones while portraying a unified embedding space for entity alignment.
As to the second challenge, we first derive an unbiased risk estimator and a tighter uniform deviation bound for the positive-unlabeled (PU) learning loss. However, such an estimator still requires prior knowledge of the proportion of positive entities among all nodes. Thus we develop an iterative strategy to estimate such prior knowledge while training the classifier with PU learning loss. The prior estimation could also be seen as dangling entity detection; if too few entities are determined to be matchable, one can stop all subsequent procedures and decide the two KGs cannot be aligned.
Our framework Lambda is provided in Fig. 2 where there are two phases corresponding to two trained models dangling detection and entity aligmment. Both phases share one GNN- encoder and the spectral contrastive learning loss. The dangling detection additionally uses a positive- unlabeled learning loss. The GNN- based encoder contains both the intra-graph and the cross graph representation learning modules. After the dangling detection, the estimated proportion of matchable entities is figured for judging whether two KGs could be aligned. Only aligned KGs are sent for EA model training, and then only first-phase predicted matchable entity embeddings are obtained from the EA model for inference. Finally, we select pairs of entities that are mutually nearest by their embeddings as aligned pairs.
Highlights of our contributions are as follows: we raise the challenging problem of EA with unlabeled dangling entities for the first time. To resolve the issue, we propose the framework Lambda featured by GNN-based encoder called KEESA with spectral contrastive learning for EA and a positive- unlabeled learning algorithm for dangling detection called iPULE. We provide theoretical analysis of PU learning on the unbiasedness, uniform deviation bound, and convergence. Experiments on variety of real-world datasets have demonstrated our alignment performance is superior to baselines, even the baselines with 30% labeled dangling entities. Our code is available on githut?
https://github.co hwwp://hwww...oo//oooooooooooooo 12358/Neur1P5 2024 Leermi_nannnng-ee Matchable Baichaabbe--Prior-P-e Frior-For-Entiee Unlabeled-Dangee Dangling Cases
with

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KG entity encoder with selective aggregation: 8
Spectral Contrastive Learning
seed alignment anchor
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Intra-graph
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Dangling Datection
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Merative Positive Unlabeled Learning
Algorithm flow
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M.A.A
Cold Start
MLP classiler
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Relation
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Entiyy
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Figure The illustration of our framework.
2 Preliminaries and Related Work
2.1 Entity Alignment
Embedding- baased entity alignment methods have evolved rapidly and are gradually becoming the
mainstream approach of EA in recent years due to their flcxibility and effectiveness [17], which aim
encode KGs into low-dimensional embedding space capture the similarities of entities [18] 10]
could be divided into translation- bassd [2 20 and GNN-based 41 37 27
26 35).
Previous EA methods mostly assume one-to-one correspondence between two KGs, but such an assumption does not always hold and thus leads to a performance drop in real-world cases (38]. Notably, Sun el al. [ as a pioneering work modeled it upon supervised setting, i.e., small set of aligned entities and labeled dangling entities. On this basis, MHP [99] employed more dangling information concerning high-order proximities in both training and inference. UED (24 and So [22] propose an unsupervised translation-bas method for joint entity alignment and dangling entity detection without labeled dangling entities, while the practical problem of matching cost is ignored.
2.2 Positive-Unlabled Learning
Let X e IRd d N, and Y E {+1} be the input and output random variables. We also define p(z, y) to be the joint probability density of (X, Y), Px() = p( y= +1), Pu (x) p(z y = - 1) to be the (Positive) and N (Negative) marginals (a.ka. flss-condiiionell densities), and pa (1: be the U (Unlabeled) marginal. The class-prion probability is expressed as R. p(y +1). which is assumed be known throughout the paper, and can be estimated from known datasets 13]
The PU learning problem setting is as follows: the positive and unlabeled data are sampled inde- pendently from (x) and pa pa( as X, {>r")"=
Pp((r) and X, {-:)) (x)),, and classifier is trained from X, and X, in contrast to learning a classifier telling negative samples apart from positive ones. The general assumption of the previous work is let the unlabeled distribution be equal to the overall data distribution, i.c., pz(z)= p(æ) since (z) cannot be obtained, but the assumption hardly holds in many real-world scenarios, for example transductive learning, making methods in
13 30 infeasible.
Selective Aggregation with Spectral Contrastive Learning
Notation: Source and target KG G. = (Fs,, R.: Ta), G. = (Fr, R4, T:) stored in triples <entity, relation, entity>: entities E, relations R, and triples T C E R E, E. D,UM,, E, D,UM,
3

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where D denotes dangling and M denotes matchable. A set of pre-aligned anchor node pairs are S= ((u,v))u v)lu M,, Mc u = v). (see ( more details).
We start by introducing the KEESA (KG Entity Encoder with Selective Aggregation).
3.1 KG Entity Encoder with Selective Aggregation
Adaptive Dangling Indicator & Relation Projection Attention. Real-world EA tasks often involve graphs with dangling distortion [3 3]. Conventional GNN aggregation will 'pollute" matchable entities' embeddings with dangling. However, hard dangling indicator for the entity is confident as only approximate results can be obtained without labels. Incorrect indicators may lead to inappropriate aggregation and thus destruction of the KG structure. Instead, apply a learnable scalar weight Pey for each e neighboring message:
1+1 =
Σ
tanh(re,)
0u, W
U(E.)
adaptive dangling indicator
where tanh serves to normalize Tey to the range [-1, 1]. The initialization of Tay is critical, please the implementation details for more.
As compressed feature of Ej - Tey plain scalar, we link relation rı 's embedding hr, to the associated entity by hrg for capturing more comprehensive attention. A matrix W, E Rdxd with orthogonal regularizer I,, is applied to hr, perform projection while preserving its norm for better convergence:
Tej Wr and L. -|W,T W, Laxd
The attention coefficient obtained by the following equation, where T is the attention vector:
exp(v T hrz )
Cijk
Eem Kon PET,UUT exp(v h -m
(2)
Intra- & Cross-Graph Representation Learning. Based on the above, we can express the embed- ding of e, the + 1)-th layer h.+1 as Eq. 1 where W/+1 is specified as W1+1 Taxd 2h,, h by the triplet e,, Tkk ej > inclusive relation embedding hy. We adopt the tanh(-) the activation function. The Householder transformation W++1 is applied the last layer embedding h', to restore the useful relative positions of KG entities at each layer recursively.
Overall, the intra- intra-graph representation he, of e; is obtained by concatenating embeddings from all layers. Its cross-gres poss-graph representation h prozy can be described by he, and proxy q,. where the latter is generated by Proxy Matching Attention Layer [25] to align the embeddings across two graphs. With Sp representing the set of proxy vectors, and sim(-) denoting the cosine similarity, we have:
exp(sim(he, q;)) (he, 9,).
he; = [ 1 ||..... [ and
h
prozy
Σ
q, En.es, exp(sim qx))
Finally, we employ gating mechanism 31 to integrate both intra-graph representation he, and cross-graph representation hprozy into h!
oz. sigmoid W hy hprozy + b).
h ((ee, he, + (1 Oe,) (rrooy)|lre.l
where W, and are the gate weight and gate bias, respectively. The learnable weight of e, is also attached to the embedding. It is worth noticing that for each entity either G, or Gr, they encoded by one shared KEESA with below spectral contrastive learning for a unified representation space.
3.2 Spectral Contrastive Learning
In this part, we propose the spectral contrastive learning loss Linto with high-quality negative sample mining, which serves both tasks (entity alignment and dangling detection) at the same time.

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N Specifically, given a pre-aligned matchable entity E Ap. let there be paired positive sample entity ex E Xp. such that (es,e4) S, and N sampled entity } as negative samples (e,, ez) & S. The spectral contrastive learning loss is one specific form of alignment loss H()):
N
Linto
Σ ELEAP
log
1+
Σ j
exp(A H(en, ((,n))
(3)
Unified Representation for Entity Alignment. We expect unified feature space where the distance between aligned anchor node pairs is as close as possible, while the unaligned is on the contrary. To satisfy this intuition, we introduce an alignment loss:
H(e,, e e/, [sim(es, e²) sim(e,, e4. + 7
(4)
where [x]] represents max(0, 2) and sim(-) indicates L2-norm distance between the embeddings. A soft margin r is involved to discourage trivial solutions, e.g., sim(e;, et)) (e, et 0.
Discrimination for Dangling Detection. For proposed dangling detection, the vital task is to discriminate the dangling from the matchable ones with unlabeled dangling entities. Hence unsupervised method of spectral clustering is exploited separate types of entities. We design the loss function according to Lemma to achieve its equivalent effect.
Lemma 1. Given one positive sample p* for q, and N negative samples {p7}* (node set: {g,p+) } ), employing the following loss function is equivalent to conducting spectral clustering on similarity graph v with the temperature hyper-parameter X:
exp(A sim(g.p*)
infoNCE(g.F Ap }N)
-log exp(A (amn(((pt)) Σj exp(A sim(g.
(5)
The equivalence has been discussed in previous studies problem, the positive samples are the corresponding pairs whereas the negative samples are those sampled unaligned pairs. The equivalence derived as follows:
32 Regarding our proposed
exp(Asim(q.P") + Ex cxp()sim(q.P7)
infoNCE(q.P) p" M) = log
exp(A sim (g.
N exp(A sim P3 ))
= logg[1
+
exp(1 sim(g, p+)
= logg
1+
exp(A (im(q.p)) - sim(g.p*))
3
The spectral contrastive learning loss could be obtained by replacing the exponent term with Eq. 4
Remark. In the alignment loss function, we observe that dangling entities are only the negative samples, and the entities in the positive samples are all matchable. Such stark asymmetry provides an advantage in distinguishing between dangling and matchable entities.
We also prove that Linto (Eq, 3 can promote model learning by Lemma (see appendix
B for proof).
Lemma 2. The loss Linta can mine high-quality negative samples, which we show has an equivalent effect truncaled uniform negative sampling (TUNS) in (35).
Minimizing the spectral contrastive loss of Eq. maps matchable and dangling entities into a unified but distinguishable feature space for improved entity alignment while facilitating dangling detection. In practice, we adopt the loss normalization trick (8] on (-) to speed up training.
4 Iterative Positive-Unlabeled Learning for Dangling Detection
We expect to avoid any additional computational overhead for EA if few entities are matchable for the source and target KG. Thus, address more challenging problem in EA: given partial pre-alia pre-aligned
5

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matchable entities as positive samples (i.e.. Xp). how to jointly predict the proportion of matchable entities in the unlabeled nodes (i.e., T and identify those entities? If Tp could be predicted, it could serve as an indicator whether we should proceed to EA. We propose to address the problem by PU learning.
Unbiased Risk Estimator. First, we propose a new unbiased estimation for PU learning without any constraint (i.e., (x = p(z) in [33 30 on unlabeled samples distribution Pa(2)) concerning the overall distribution p(x). Assuming that Tp and Tp are known (estimation strategy would be given
later), we have:
Theorem 1. Suppose that g E G: Rd IR is a binary classifier, and é IR [+1] + R is the loss function by which E(t, y) means the loss incurred by predicting an output t when the ground truth is Rou (g) is the unbiased risk estimator of R(g):
Rpu(g) = Tp (g)
Fn
(9) (
(6)
T
where Tn = --y---1 and म. Pol = -) are estimable class priors given Tp and Tp. 12 (g) Ex (=) +1)] and R, (g) = E Ex-p,(=) (xx=))))))((((((X),-)))))))))))) -1)] (see appendix for proof).
= By our proof, Rm (g) is an unbiased risk bound for the PU learning. More importantly, the bound provided by Thm. 2 is tighter uniform deviation bound than the classic Non-negative Risk Estimator 13:
Theorem 2. Let Var(R) denote the uniform deviation bound of risk estimator R. and Non-negative
Risk Estimator be I2' pa (g), then: Var (g)) < Var(It pu (s)) appendix proof).
Positive Unlabeled Loss Function. Since it is evident that all negative samples exist in unlabeled data, we have Iy < and thus we apply hyper-parameter a to scale Tp R2 (g) equivalently
"h
x
me
and max(-) to restrict the estimated R (g) 2 O. The PU learning loss function is formulated as:
pu = amp (g) + max R, (g) Ro (g)).
We specify the corresponding risk function using cross-entropy losses below:
P (0)
log P,
1 [
Σ aEE.
log logú)(-1), R
log ūs( (-1)
eEX,,
GEXx,
where the output logit for 6 E E2 U Fr, being labeled as state E (+1,-1) is D: (u) so (MLnaz based on KEESA output hy Hence each term in the final loss can be calculated or estimated without the negative labels.
Iterative PU Learning with Prior Estimator How could we estimate prior Tp and n!!? Inspired by [II], we introduce hidden variable in the model as well as iterative approach. We adopt variational approximation strategy and a warm-up phase to tackle the cold start problem, as shown in Alg. [ First, we estimate and fix the class prior Tp and 7p by the ratio of the anchor points in the trainnn set. Liato is optimized together with Lpu for a discriminative embedding space in the warm-up phase. Finally, we minimize Cpu update the class prior and the model parameters alternately till convergence.
The convergence guarantee is provided in Thm. which mostly follows the convergence of EM algorithm. We collect the proof in appendix E
Theorem 3. Given the assumptions of marginalization Eq. 16 and Eq. 17 the objective function of cuu is the same as the expectation function Q of Eq. 13 where the loss function is the cross en-
tropy function CE Ui) = T. (+11 log Ts(+1) - D.( Log (-(-1)
1) on the preference condition:
Ejeu M log (((+))
(-1- ΣiEp P log (x(+1)
(-13
The iterative process of our method is a special case of the EM algorithm. We hold the same assumptions as the EM algorithm and we further assume the training of f is able to find the globally optimal 0. Although the assumptions seem to be too strict, the algorithm typically converges in practice we verified in the experimental section.

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Algorithm iPULE (iterative PÚ Learning with Prior Estimator)
Require: G2 and G. are treated as one input graph G (V,E), positive-node set P = Xp. unlabeled-node set U Xy,, classifier ʃ with initial parameters 6o, KEESA Enc(G, (G, 1p) with initial parameters do and warm-up epoch N. L represents training loss.
Ensure: Model parameters 0, 1b and estimated prior up and Tp
I 00; "p f- Tp- + |
IP
i 3= Bo;
//Initial value //Loss of warm-up
2: L C+B. Linto + (1- B) Ppii
3: repeat
4: X + Enc(G,
//Entity embedding matrix X //Optimize Enc(-) and jointly
5: 0, +- arg ming.y [(0) P,U,)
6: 1 C,0;X,y P,U);
7: until N epochs is over
//Warm-up phase to solve cold start
8: L Lpui
9: repeat
10: X + Enc(G, v) Os + f(X, 1; for all EEV;
|P|+|U|-
11: Ap + - ieu [+ū; > 0.5], Mp +
I/E step
Pl+
12:
l 1 C(0; X, Y.y,))
13:
0,00 f- arg max [(0 y.
//M step
14:
until I1' Il converge OR Ap converge
15:
return
5
Experiments
Our method is evaluated on datasets GA16K, DBP2.0, and GA-DBPI5K. DBP2.0 and GA-DBPI5K are used for the verification of iPULE. To address incomparability caused by inconsistent metrics, we adopt the GA16K dataset enable compromised comparison of the Dangling- Dangling-Enttiies--nn EA method. We further compare our method with dangling aware baselines on DBP2.0. Statistics of experimental dataset in appendix F and additional experiment in appendix G
Datasets. The training/test sets for each dataset are generated using a fixed random seed. For entity alignment, 30% of matchable entity pairs constitute the training set, while the remaining form the test set. For dangling entity detection, we did not utilize any labeled dangling entity data, in contrast to prior work which labels extra 30% of the dangling entities for training [33 19]
Baselines. Since our work does not take advantage of any side information, we emphasize its comparison with the previous methods purely depending on graph structures. These works majorly incorporate two types:
Dangling Bangling-Entities-- Align [1). RSNs (9) MuGNN 4). KECG [16). Methods with bootstrapping to generate semi- supervised structure data are also adopted: BootEA (35), TransEdge [36], MRAEA (26,, AliNet (37] and Dual-AMN
We include advanced entity alignment methods in recent years: GCN-
25
Dangling- Enrities--warre To the best of our knowledge, the method of 33 is the most fairly comparable baseline which is based on MTransE [5] and AliNet [37]. Because MHP [19 over- emphasized more use of labeled dangling data like high-order similarity information which is
also based on the above two methods, while SoTeaa [22] and UED 24 utilize additional side-
information. SoTTaad 22 and UED 24 can only execute the degraded version on DBP2.0 cause no inde-information is available on that. We exclude them from baselines for our methods. (33 introduces three techniques to address the dangling entity issue: nearest neighbor (NN) classification, marginal ranking (MR), and background ranking (BR).
Implementation Detail. We use the Keras framework for developing our approach. Our experiments conducted on a workstation with NVIDIA Tesla A A100 GPU, and 80GB memory.
By default, the embedding dimension is set to 128 with the depth GNN set to and a dropout rate of 0.3. A total of 64 proxy vectors are used and margin = 1. R.SSrrp optimizer is adopted with a learning rate 0.005 and batch size of 5120. X set to 30. ,3 set as le-3 for all datasets. CSL.S [4] is adopted as the distance metric for alignment. As we found, the tanh function changes rapidly
7

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in the region close to 0 but stays stable in the region beyond [-3,3]. Hence we initialize the rey to 1 to prevent gradients oscillation or near-zero gradients.
5.1 Experiments of iPULE Convergence and Class Prior Estimation
DBP2.0 Fp between 20%-50% contains more entities to be aligned, trained by pre-aligned 19% nodes then judged by iPULE as aligned KGs. GA-DBPI5K 15K Tp between 10%-25% are treated as unaligned KGs ignoring the pre-aligned part, trained by all pre-aligned 10%-25% nodes. We get accurate estimation and convergence results shown in Fig. As iPULE progresses, the estimated class prior gradually approaches the true value as the first row for GA-DBPI5K and the second for DBP2.0. T 0 for GA-DBPI5K while DBP2.0's for GA-DBPISK is stably consistent as pre-aligned proportion due to accurate estimation of its Tp' iPULE treats more nodes as positive when Tp R 50% in FR-EN.
given by red dotted line respectively. The "p
As common in PU learning
43
GAEN
GA
GA-JA
GA-ZH
Pro olggeo 10%6
Pro aligaidddd
2u-2
Pro aligned 20%
Pro alignnl 25%
1.2
1.2e:3
3
66-3
8e5
46-5
76-4
3e-3-
4e-3
12
18
Epoch
mg
true %
true x5
aligned
allgned
JA aligned
allgned
0.5
0.20
0.3
Pior
0.15
Preor
0.2
0.3
Priot
0.4
0.10
0.05
0.1
0.1
0
8
12
Epoch
Epoch
Epoch
Epoch
Figure 3: Prior estimation GA-DBPI5K and DBP2.0. (loss convergence in appendix P
5.2 Experiments Unaware of Dangling Entities
We show the experiments on baselines without considering dangling entities in this section.
Dangling-Entitiee Entities----aaaae Baselines Comparison. The direct comparison between our method and the adopt the GA16K dataset as compromise and do not remove any detected dangling entities for entity alignment. Thus the ranking list S in Hits K only contains (matchable) entities in the source graph since GA16K only contains dangling entities in the target KG. Tab. 2 Dual-AMN demonstrates competitive performance but is inferior to ours at Hits I. MRAEA pertorms similarly to Dual-AMN since the latter is built on the former. TransEdge performs poorly since the method adopts superviee bootstrapping to mine anchor entities iteratively. The presence of dangling entities could lead false anchors and spread of error. Meanwhile, it is also relation- approach that suffers from insufficient relation information GA16K. Other baselines exhibit up-to-par performance but our method delivers consistently superior or state-of-the-art Hits @ Ks.
baselines is unavailable due to inconsistent metrics used. Hence,
5.3 Experiments Aware of Dangling Entities
We provide a comparison of dangling detection and entity alignment with baselines aware of dangling.
Dangling Entities Detection Performance. We test our method's dangling detection performance compared with baselines aware of dangling entities. The results on DBP2.0 the consolidated setting are reported in Tab. [2] Note that the comparison is unfair as we don't 't use 30% of the labeled

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GA16K
Method
H@1
H@10 10
H@50
BootEA
13.95
37.25
49.08
TransEdge
0.03
0.12
0.14
MRAEA
63.97
76.64
81.06
GCN-Align
29.48
45.64
57.15
RSNs
9.40
42.70
46.70
MuGNN
62.17
76.25
80.87
KECG
44.18
57.73
63.41
AliNet
48.53
67.72
74.50
Dual-AMN
64.49
80.55
84.67
Ours
67.59
80.33
84.35
Figure 4: Visualization of entity representations Table 2: Performance comparison with dangling- learned by our method on GA16K dataset. entities-unaware baselines on GA16K.
Methods
ZH-EN
EN-ZH
JA-EN
EN-JA
FR-EN
EN-PR
Prec.
Rec.
F1
Prec.
Rec.
F1
Prec.
Rec.
Fl
Prec.
Rec.
F1
Prec.
Rec.
Prec.
Rec.
Fl
MitannE: AhNet
NNC
.666
.419
-517
.738
.558
634
.597
.82
.534
.761
.120
.207
466
365
.009
.545
.162
.250
MR
.752
.538
.627
828
.505
627
.779
.580
.665
884
.543
.664
.552
.570
.686
.549
.609
BR
.662
.356
6643
829
.515
.35
.783
.591
.673
846
.546
.663
.547
.556
.552
.674
.556
.609
NNC
.604
.485
-388
.719
.11
598
.622
.491
.549
686
.506
.833
445
.47
.433
.557
.543
.500
MR
.781
.702
740
866
.675
.759
.799
708
.751
864
.653
.744
.482
.575
.524
.639
.613
.625
BR
811
.288
767
700
785
816
733
772
.888
.311
801
.539
686
604
.692
.35
.13
Ours
.95
836
844
.909
875
807
836
.211
880
.809
.615
.772
.685
.732
.749
.740
Table 3: Dangling detection results on DBP2.0 in the consolidated setting.
ZH-EN
EN-ZH
JA-EN
EN-JA
FR-EN
EN-FR
Methods
Prec.
Rec.
Prec.
Prec.
Prec.
Rec.
Proc.
Rec.
Prec.
Rec.
FI
MThaaa AhiNet
NNC
.121
.193
.14
.085
.138
.155
.11
.146
.127
.067
.208
.101
.126
.148
.136
.086
.161
.112
MR
207
.299
.245
.159
.320
213
.231
.321
269
.178
.244
.195
.193
.160
.178
BR
.003
.286
.238
.155
.308
.007
.233
.306
.258
.170
.311
.222
.183
.181
.182
.164
.200
.180
NNC
.164
.215
.86
.118
.207
.150
180
.238
.205
.101
.167
.125
.185
.189
.187
.135
.140
.138
.344
231
288
.33
338
.227
.280
.60
238
213
.188
BR
312
.362
.35
241
.366
294
314
363
336
.251
338
.255
.265
208
233
.211
211
.222
Ours
.279
.447
.344
.219
.49
_303
.324
.409
.362
.234
.460
.310
.244
.320
.271
.192
.363
.251
Table 4: Entity aligmment results on DBP2.0 in the consolidated setting.
dangling entities as the baselines. Nevertheless, our approach maintains SOTA performance across all six datasets, excelling in almost every metric except for slightly inferior precision.
Dangling-Entionn Entities-tiirrrr Baselines Comparison. Tab, reports the entity alignment performance comparison in the consolidated setting DBP2.0. The precision, recall, and Fl scores are computed according to Eq. (19), (20). (2)) in Metric part of appendix respectively. We test the entity alignment performance of our method in comparison with baselines that are aware of dangling entities. Our method still maintains almost state-of-the-an performance, but there is still a slightly inferior precision problem. It makes wonder about the reasons behind it.
How does our method work? To understand why our method works and its precision slightly suffers, we visualized all entity embeddings of GA16K in Fig. [ As shown above, matchable entities are denoted as red and green in source and target KG respectively, while dangling as blue. The distribution of three types of entity in Fig. a() suggests our method maps all nodes into unified embedding space where matchable entities exhibit considerable overlap and are appropriately aligned (shown in Fig. b)). Fig. 4 (d) depicts that part of the dangling entities is intertwined with the matchable ones, suggesting that this part resides the decision boundary and easily leads to false positives which explain the lesser precision of our method.

---

5.4 Ablation Studies and Varying Anchor Nodes
We conduct ablation studies show each module's impact, and similarly pre- vre-aligned entities' impact.
Ablation Studies The impact of adaptive dangling indicator and relation projection attention in our method are investigated. We denote the counterpart removing fe, w/o Tee and replacing h with the original hr, as w/o h Fig. 5 gives the ablation study results on DBP2.0, where 'Ours' represents an all-inclusive model. We observe that the t has a more substantial impact than Te, consider to the alignment performance. As to why the Te, has minor impact on the alignment, may be attributed to the lower degrees dangling entities on DBP2.0. The degrees of dangling entities generally lower than that of matchable ones, indicating that the dangling is more isolated in the graph and thus has less impact on matchable nodes in the neighborhood aggregation.
665
0.5
wio hrg.
0.55
Preshiie
Ours
0.15
N2 N2112
NPR
0.
N2
Figure 5: The ablation study of entity alignment performance in the consolidated setting on DBP2.0.
0.7
0.25
0.55
0,
21EIN
0.6
0.3
0.47
0.35
Prestien
Recall
JA-EN
0.5
0.25
02.00
0.1
FNJA
04
9.2
0.31
0.25
83
0.15
0.33
02
0.15
0.15
Figure 6: The entity alignment performance on varying pre-aligned anchor nodes ratios on DBP2.0.
Varying Anchor Nodes. Pre-aligned entities may be far scarce in reality. The sensitivity of our method the proportion variation of anchor nodes is investigated. As the proportion increases, the alignment performance enhances as provided in Fig. 6
Notably, even with anchor ratio low 5%, alignment accuracy still well exceeds 30% on most datasets except for FR-EN and EN-FR. Cause they contain twice as many entities and triples ZH-EN and E,,,, which introduces intricate dependencies among entities and thus greater challenges in alignment. Moreover, a larger graph may require a higher dimension of representations learn, but the embedding dimension is restricted to merely 96 due to the out-of-memory problem.
6 Conclusion
We found that previous EA methods suffer from great performance decline if dangling entities are considered. Our goal is address the EA problem with unlabeled dangling entities. A novel framework Lambda for detecting dangling entities and then pairing alignment is proposed. The core idea to perform selective aggregation with spectral contrastive learning and to adopt theoretically guaranteed PU learning to relieve the dependence on the labeled dangling entities. Experimental results on multiple representative datasets demonstrate the effectiveness of our proposed approach. This work also has important implications for real-world applications, such as EA of different scales, KG plagiarism detection, etc.
10

---

Acknowledgments and Disclosure of Funding
The research was supported in part by NSF China (No. 61960206002, 62272306, 62032020, 62136006).
The authors would like to thank the reviewers for their constructive comments and appreciate the Student Innovation Center of SJTU for providing GPUs. Hang Yin personally thanks Chenyu Liu, Yuting Feng, Jingyuan Zhou, and Qingyang Liu for feedbacks on early versions of this paper. Hang Yin would also like to thank Professor Yuan I Luo for his inspiration in the information theory course (CS7317) of Shanghai Jiao Tong University.
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14

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Appendix
A Notation
A.1 Definitions
Definition (Knowledge graph). Knowledge graph (KG) is directed graph G = (E, R) comprising three distinct sets: entities E, relations R, and triples T S R E. KG stored in the form of triples <entity, relation, entity>, with entities denoted by nodes and the relation between entities defined by edges.
Definition 2 (Entity alignment). Given source KG and target KG, corresponding to G. R,,TT) and G2 = (E,, Ti) respectively, and A = {(u, e) u E Eg, E,, = o} a set of pre-aligned anchor node pairs, where indicates equivalence, the goal of entity alignment is to identify additional pairs of potentially equivalent entities using information from Gg. Gt, and A. This task typically assumes one-to-one correspondence between F., and Eg.
Definition (Entity alignment with dangling cases). Let entities in the source and target graphs be composed of two types of nodes: E, D, U M,, E, De M, where D., Dr denote dangling sets that contain entities that have no counterparts, and M., Mt are matchable sets. A set of pre- aligned anchor node pairs are S {(u, v)lu E M,, Mc, v}. The task seeks to discover the remaining aligned entities given G., G2, and S.
A.2 Transductive Learning:
Transductive learning models are trained from observed, specific (training) cases to specific (test) cases, employing both training and test information except for test labels. In contrast, the inductive learning model is reasoning from observed training the test cases. Let's get down to the EA task. As KG structure accessible through given triples, which can accurately describe the connections between entities. When we attempt to figure entity alignment tasks, the KG structure information of the whole source and target KGs is what we could exploit, covering potentially (test) equivalent entities' relative positions. That's why confine the problem field to transductive learning.
general rules, which then applied to
A.3 Graph Convolutional Networks
For graph convolutional networks (GCN) [22. the embedding h of node at the ( + 1-th layer is updated iteratively by aggregating node features of the neighboring nodes Ne, from the prior layer:
h 1+1
Σ
Ori
U(es)
where each embedding h² represents the d-dimensional embedding vector of e;, 0rjj denotes the weight coefficient between e; and ej. Wl+1 being the transformation matrix of the (1 + 1)-th GNN layer, and being the activation function.
B Proof for Lemma 2
Proof. The Linto also has good effect on mining high-quality negative samples, which we show has an equivalent effect to truncated uniform negative sampling (TUNS) in (35]. TUNS points out that negative samples obtained by uniform random sampling are highly redundant since only high-quality negative samples improve the model. Thus TUNS chooses the K-nearest neighbors of the as the negative samples, which are most challenging to distinguish. In the special case of = 1, the loss can be written as LTUNS Ee,x,, max, (H (e,, e'_, )))). If we approximate the max function by the Sog the contrastive loss function turns
CrUNS ~ Σ log eiEN,
Σ exp(2 H(e,, ((en),,))))
(8)
15

---

minimizing which is equivalent to minimizing Eq. (3). Hence our contrastive learning loss is actually special form of TUNS. For randomly sampled negative samples, iinto can play role preferentially optimizing high-quality negative samples.
C Proof for Theorem 1
Proof. The risk of g is R(g) Bex.y) (((((X))))) Tpp Tn (g) in posi- tive negative learning problems. In positive-unlabeled learning where Xn is unavailable, we can only upproximate R(g) by positive samples and unlabeled samples. We represent the un- labeled distribution as Pu(r) (T)u(x) + (x))((), so that the negative distribution can be
written as (2) mo [pu(x) (x)p Provided Rp (g) (x~ [((g(X), -1)] and
Ri (g) = (c) - 1)], we obtain that
Tn Tu (g)
(R) Tp R (g)
(9)
T
and
Rg) 7p R
(1) -5) (0)
(10)
Specifically, =1 1 Tp. Ti = 1 Th could be derived as class-prior probubility given up and up In particular, the ratio of labeled positive samples could be precisely figured out as x in transductive learning, given which T. 1-
m Th mp could be derived as class-prior probability.
D Proof for Theorem 2
Proof. Rngg is defined as the Rademacher complexity of the class of classifiers G for the sampling of size from distribution (2x). From (30) we have that with probability at least 5/2, the uniform deviation bounds below hold separately:
ln(4/)
sup (R(()) 14 (g)l S 2L, (9) + gEG
2n
A M > 0,
In(4/0)
sup Rg) Ru,-(g) S R (9)
gEG
2n
& M_ 0,
where Le the Lipschitz constant loss in its first parameter and n. is the number of positive samples while n,, is that of unlabeled. Following the symmetric condition assumption in (30), it is obviously holds that:
Var(Ht pu po()) M + M_, while Var(Rpu(o))
Tn "p M
(Tp
M_
TH
holds in our setting. Then it is evident that 7p
i.e.,
Tp and
<1
Consequently, comparing the coefficients of M. and M_ leads to the conclusion that R (g) could
possess tighter uniform deviation bound than that of Non-negative Risk Estimator [3]
16

---

E Convergence Proof
(EM) algorithm is an iterative approach to maximize the likelihood p(y X; 0) of target variables over input variables X and parameters 0. The EM algorithm works iteratively, and each iteration consists of an expectation (E) step and maximization (M) step. The step computes the expectation of the log-likeliine lieelikiidoodd concerning the conditional distribution of the latent variable z given the current parameters ((9) at the 1-th iteration:
The expectation-maxiiizant
Q( (()) = Bxxpp(xxX.,xx.cooee oe (oog z X, 8)].
(11)
Then, the M step finds set of parameters that maximizes the computed expectation:
0(++1) arg max Q(0 (l+))
(12)
Due to the maximization step, we get the following inequality naturally:
Q(a(++1)
((()) Q(((+) ((()) O.
(13)
Lemma (Convergence of EM algorithm (29]), ). It is guaranteed that the EM algorithm always im-
proves log ply X, 0) by increasing the value of Q( ((t)). Since log logply (x,0) 0) is monotonically bounded, the EM must converge.
Proof. The following equation holds for any z:
logp(y X, 0) logp(y, z X, 0) - logp(z x,y, 0).
We take the expectation over p(z X, y, y, 0(1) for both sides as follows:
Ep [ly X,
EpleXXyy,ooo ply z
X, 0)] Eot X,y,
= Q(0 (00) + Hp((z X,y,0) p(z x, ((t)()
A Q(0 ((0) 11 (po Poco)
where stands for the entropy. we substitute g(t) for 0, we get the following:
logply X, 0()) = Q(0()) (00) - (Wgro) Pace:).
(14)
Gibbs' inequality states that ((q | p) H(p p) 2 0 always holds for any distribution p and
Hence we have:
logply X, x( - ply X, 0(0)
Q(0((+2) (10)) + ((port))) PP(c)) Q(0(() ((t)) H (pots) Paac)) )
Q(((++11) ((())) Q(0(!)) ((()) + (P Poct)) H (p P
2 Q(0(++1) ((()) - Q(0(t)) |0(0) = ((0(++1) ((0)) - Q(((()) 2 0.
Now we provide the proof for Theorem
Proof. According to Lemma we have that the EM algorithm converges. Next, we model our problem and give the approximate equivalence between our algorithm and the EM algorithm to prove our algorithm converges.
The latent variables z represent the true label distribution of unlabeled samples, where (u( = f(X, i; ()) is the probability of node being labeled as {+1,-11 by the current classifier Given Th the label distribution of all unlabeled samples is:
if
2 +1,
p(2:)
(15)
1
Au
P
if
24
-
17

---

First, the conditional distribution p(z x, of latent variables given the current parameters g(t) is approximated by:
p(z X, 0
I
(16)
VEU
Second, the joint distribution ply, z X. 8) of labeled and unlabeled nodes with new parameters 0 is approximated by the classifier f. which is also considered as marginalization function that gives the label distribution of each node based on all given information:
ply 2 X, 0)
II iEP
0i(+1)
πI (5(g)) jEU
(17)
We propose to use the average log-likelihood differences measure the classification preference of the model. It can be transformed into the following form by Au and Ap, representing that on domain U/ and P:
Au
log ū5 (+1) - log ( (-1)
[
jEU
Ap
log D: (+1)
- log (i(-
P
REP
the mean value is positive, it means that the logarithmic probability of positive classes is higher than that of negative classes in the given domain, and vice versa. When we ignore the preference of the classification model, this actually describes the category feature bias in certain domains.
Let's rethink the meaning of preference condition. If the model has similar classification preference in domain 2/ and P, we can express as Au ze Åp. This condition can be arranged as the mathematical expression of the condition given by Theorem through the properties of the log function.
(+11) log
Au
log
jEU
We derive Lpu from Eq.() since the goal of training is to minimize the objective function:
E ee1 (o) X,
N Σ p(z log X)
Σe p(z (())
log (e(((()))))
Σ JEU
log Log (z()))
log (+1))
Σ
p(2z) log (24)
N
REP
JEM #
IP|
[
log (g +
@ log új (+1) (1 =/) log ú, (-1))
N
N
iEP
jel
P
M
# [
log (ogū((+)++ +
log (
Au
N
N
N
PEP
yEU
We replaced Au with Ap, and this process is completed by the above preference condition.
IP]
[
2 A
1
log (o
log 0y (-1)+
log 0i(+1) log (-1
N
P
N
[
N
P
iEP
jeH
IP|
+ U
log 9. (+1)
P R4
Σ
log 03 (-1)
- A
log D. (x(-1)
Pl
+ U
P
M
U
Ap
P
NEP
[
JEU
iEP
18

---

Datasets
# Entities
# Rel.
# Triples
# Dang
# Align
Chinese
84,996
3,706
286,067
51,813
DBP2.0zH-2
33,183
English
118.996
3,402
586,868
85,813
Japanese
100.860
3,243
347.204
61,090
DBP2.0JA-EN
39,770
English
139.304
3,396
668.341
99,534
French
221.327
2,841
802,678
97.375
DBP2.0FR- FN
123,952
English
278.411
4.598
1,287,231
154,459
Chinese
19,388
1,701
70,414
4,388
DBP15K2H Eie--EEEnggg
15,000
English
19,572
1.323
95,142
4,572
Japanese
19,814
1,299
77,214
4,814
DBP15K -AAAAAJ--NEEEEE
15,000
English
19,780
1,153
93,484
4,780
French
19,661
903
105,998
4,661
DBP15KFR-
Engglissh
19,993
1,208
115,722
4,993
15,000
None
6,208
8
68,534
0
GA16K
6,208
None
16,363
12
151,662
10,155
Table 5: Statistics DBP2.0, DBP15K and GA16K.
Datasets
# Entities
Rel.
Triples
# Dang
# Align
Chinese
69,386
3,455
241,588
36,302
DBP2.0 Plus
33,084
English
94,026
3,131
470.284
60,942
Japanese
82,192
3,011
291.406
42,588
DBP2.0JA-EN Plus
39,604
English
110,362
3,054
532,988
70,758
Chinese
72,252
3,351
200,400
54,594
DBP2.0ZHEEN.MinH
17,658
English
107,853
3,140
421,597
90,195
Japanese
86,241
3,014
236.546
64,841
DBP2.0AA-E Minus
Minuus
21,400
English
126,558
3,166
485,133
105,158
Table 6: Statistics of DBP2.0-Plus and DBP2.0-Minus
Denote Tp- =
P1+ |P|+|U
W and T
W
The above formula is equivalent to:
IP+R
arg max
E
Bampiz y,,,))
( [log pyy, (,, z x, 0)]
N
arg max Tp R
R (g) कp R, (g)
T
(18)
Ryu(s)
arg min Rg) arg max-
Cpu
F Statistics of Experimental Dataset and Baselines
Datasets. The training/test sets for each dataset are generated using a fixed random seed. For entity alignment, 30% of matchable entity pairs constitute the training while the remaining form the test set. For dangling entity detection, we did not utilize any labeled dangling entity data, in contrast to prior work which labels 30% of the dangling entities and matchable pairs respectively for training B33. Hence our method imposes minimal restrictions on annotated data. All datasets are briefly introduced in the following and some statisties provided in Tab.
In addition to the existing datasets, we also constructed DBP2.0-minus & -plus as supplementary to DBP2.0, GA16K enabling comparison between Dangling Mangglin-EEttttess- Entities-Unaware baselines, and GA- DBP15K for evaluation of iPULE.
DBP15K 34
DBP15K consists of three cross-lingual subsets constructed from DBpedia: English- Crench(DBP... EN). English-Chinese (DBPzH
EN). Each subset
hhtps//ppperssiitcoode..ooo http://wwwwwwweocoocc//////////a
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GA-DBPISK
Entities
Dang
Align
GA EN
GA
16,363 19,388 + Align
16,363 Align
16,363#c%
EN-share
19.30 19,388 16,363 Align
GA ZH
GA
16,363 19,572 + Align 16,363 19,814 + Align 19,388 19,661 + Align
16,363+c%
ZH-share
19,572 16,363 Align
GA - JA
GA
16,363
JA-share
19,81
GA
16,363 Align
GA FRR
16,363
FR-share
19,661
Table 7: Statistics of GA-DBPISK. c= 125%,20%,,159,,000
contains 15,000 pre-aligned entity pairs. This dataset includes a small proportion of dangling entity samples which is yet mostly ignored in previous entity alignment tasks.
DBP2.0 is an entity alignment dataset with a considerable proportion of dangling entities, constructed from the multilingual Infobox Data of DBpedia (2). The dataset contains three pairs of crosslingual KGs, ZH-EN (Chinese to English), JA-EN (Japanese to English), and FR-EN (French to English). Since there are dangling nodes both the source and target graphs, we separately test source-to-targee and target-to-source alignment, consistent with the established approach. A representative feature of the dataset that the matchable and dangling entities exhibit similar degree distributions which are hard to distinguish, displaying a real-world challenge in aligning knowledge graphs. Based on DBP2.0, we extend the following -minus & -plus datasets for verification of iPULE on different positive proportions between 20%-50%
D 33
DBP2.0-plus: In the construction of the plus dataset, our goal is construct the dataset that has higher Tp. and we realize this by reducing a few existing dangling entities on ZH-EN and JA-EN. We randomly delete dangling entities from both source and target KG equally and remove triples containing them. The constructed DBP2.0-plu are reindexed and thus obtain higher TP value than the original dataset.
DBP2.0-minus: In contrast, to lower the Tp value. Given the constraint of preventing new dangling entities that could introduce false information the KG, we can only reduce the number of matchable entities. Given source and target KG, removing one entity from a pair makes the remaining entity dangling. We randomly delete matchable entities from one side of the pair on both source and target KG uniformly. The constructed DBP2.0-minus are reindexed and thus obtain a lower Tp value than the original dataset.
GA16K: This dataset constructed by us exclusively contains dangling nodes in the target graph, facilitating a comparison between work and baselines that neglect dangling entities. GA16K is extracted from G 77. a Geoscience Academic Knowledge Graph. We first order each type of entity in GAKG according to their degrees and select the entities with large degree into the entity set. A total of ((363((6)) separate entities and their relations were extracted to compose the target graph. Then extract 6,208 entities from the target graph comprise the source graph. Hence there are 6,208 ground-truth matchable pairs between the source and the target. The remaining 10,155 entities in the target graph are regarded as dangling entities.
The GA-DBP15K dataset is derived from a subset of entities within GA16K, along with their associated triples, which are then concatenated with the DBPI5K dataset, such as EN, resulting in a new dataset pair that shares a proportion of common entities. To achieve the goal, we first extract a certain proportion of triples from GA16K. We then reindex all the entities from the extricated 161 and DBP 15K datasets. Finally, we update the entity and relation indices in the triples, replacing them with the newly assigned indices.
GA-DBP15K:
Baselines. Since our work does not take advantage of any side information, we emphasize its comparison with the previous methods purely depending on graph structures. These works majorly incorporate two types:
Bongling-EEnities--UU Unaware. We include advanced entity alignment methods in recent years: GCN- Align 41). RSNs 9 MuGNN 4 KECG (16). Methods with bootstrapping to generate semi-
20

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supervised structure data are also adopted: BootEA [31, TransEdge [36]. MRAEA 26 AliNet 37 and Dual-AMN (25]
Entities--warr. To the best of our knowledge, the method of 33 is the most fairly comparable baseline which is based on MTransE [5] and AliNet [37]. Because MHP [19 over- emphasized more use of labeled dangling data like high-order similarity information which is
also based on the above two methods, while SoTead [22] and UED [4] utilize additional side-
information. SoTea [2)] and UED 24 can only execute the degraded version on DBP2.0 cause no side-information is available on that. We exclude them from baselines for our methods. [33]
introduces three techniques to address the dangling entity issue: nearest neighbor (NN) classification, marginal ranking (MR). and background ranking (BR).
Metrics are set for the dangling entity detection task and the entity alignment task separately. For evaluate the detection performance by the standard precision, recall, and F1
the entity detection, score. To align the previous dangling detection baselines, we detect dangling entities as 'positive' samples and align matchable entities for entity alignment.
For the entity alignment, the metrics slightly differ in the dangling-entitiien unaware and dangling- entities-aware settings. We evaluate the baselines unaware of the dangling entities by following their assumptions and using their metric Hits @ K (K= 1, 10, 50, H@K for short) on the ranking list S. This setting is referred to relaxed setting when S is composed of all ground-trume -outh entities without the dangling ones:
ss| (rank; S k))
Hits@K
In contrast, refer to a consolidated setting for baselines aware of dangling entities. In this setting, the ranking list S also contains all dangling entities. We use H@K K in the consolidated setting to evaluate the performance of baselines aware of but not removing dangling entities in the alignment. For baselines where dangling entities are detected and removed before alignment, the direct use of H@K K to evaluate entity alignment may not be precise, since errors are introduced in the detection phase. Thus we follow the convention of (33) to apply a set of metrics for evaluating the accuracy of entity alignment the consolidated setting. Each of them is derived and introduced as follows.
The standard precision and recall is given as
TP
TP TP FN
precision
TP + FP recall
for dangling entity detection. We denote the dangling entities as O and matchable ones as 1. The subseript tly suggests that an entity with ground truth y classified as matchable. Likewise, ty1 represents matchable entity that is classified as by the detection classifier. If source entity dangling but not identified by the detection module, its alignment result is always considered incorrect, i.e., H&K,,, = 0. Hence we have the precision for entity alignment as
H@l,, = precision H@1,, + (1- precision) H@1
(19)
precision H @@L,,,
Similarly, if a matchable entity is falsely excluded by the dangling detection module, this test case also regarded as incorrect H H 0 since the alignment model has no chance to search for alignment. Hence we have the recall for entity aligmment as
H@l,, = recall - H@I,, + (1 recall) H H@l
(20)
recall H@1,,,
For the methods that are aware of dangling entitics, we use H@L, and H H@L,, to denote the precision and recall of the entity alignment task. Similarly, we define the Fl score of the entity alignment as the harmonic average precision and recall:
2.H@1,, H01
(21)
H@I, + H H@1,,
Later, H @ and H ② Le, are referred to Prec. and Rec. in reporting alignment performance.
21

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G
Additional Experiment
RQ1: How do current network alignment methods perform in unlabeled dangling cases? (see appendix G
RQ2: Loss convergence GA-DBPI5K and DBP2.0. (see appendix G.2
RQ3: How do we select the embedding dimensions? (see appendix G.3
RQ4: What is the actual efficiency of our approach? (see appendix G.4
RQ5: Baseline comparison under different pre-aligned seeds? (see appendix G.5
RQ6: Additional experiments involved LightEA strong baseline? (see appendix (.6)
G.1
The Non-Negligibility of dangling Problem (RQ1).
We investigated the performance degradation of various existing EA methods in the face of the dangling problem, which shows that this problem is worth considering.
DEPISKatt-P
DEPI3KIA-
DBP15KFn-EN
Method
1GR10
HP:500
H
HGR10
H8:50
H
HR 10
H
BootEA
31.30] 20..96
59.70] 16.18
71.51
12.01
33.77|
16.27
(2.66) 11.64
73.09]
10:29
23.111 26.72
58.39L 18.77
71.54,
Transkidge
49,91]
15.21
76.62]
9.79
$3.44]
7.16
54.071
13.42
78.011
8.25
$4.00)
6.21
48.23|
17.34
79,321 9.70
86.69L 6.24
MRAEA
59.45| 5.62
K3.04]
88.68
61.60]
83.48| 2:21
1.50
61.55]
85.85
2.61
1.00
GCN-Align
31.99] 10.70
62:211 6.45
71.931 4.31
32.08)
10.08
61.04) 5.86
70.34] 3.52
30.711
10.30
61.64, 7:07
72.45L 5.55
RSN:
43.001 8.50
62,90] 8.00
69.701
7.00
20.60]
31.60
44.60) 26,00
53.20) 23.00
36,30]
15,30
63.30) 10.10
71.70, 7.80
MuGNN
34.66]
66.68] 8:60
78.631
$1.67,..50 5:32
KBCG
35.921
12.87
63:70] 10.35
76.44] 8.06
32.311
15.48
63.19) 11.96
74.42) 9:29
32.841
15.47
64.78) 11.58
76.70) 8.35
AllNer
53.84| 0.66
73.73]
3.16
80,30]
1.50
52.691
1.30
74.01 2.60
80.911 1.90
54.011
0.58
76 191
2.74
83.25, 1.40
Dual-AMN
60.72] 12:20
83.93)
5.22
89.45|
3.54
62.291
10.62
83.38) 5:35
$8.80] 3.21
65.331 10.48
87,76] 4.17
92,47,
Table 8: Network alignment performance DBPI5K in the consolidated setting. The blue numbers suggest the drop from the relaxed setting (as with their original implementation).
We reproduce the baselines unaware of dangling entities on DBP15K in the relaxed setting. On the same dataset, we rerun their methods but in a consolidated setting that takes the dangling entities into account. Even though DBP15K only comprises small percentage of dangling entities, the drop in the consolidated setting is significant, as shown Tab.
The reason behind such a performance drop is mainly because most previous works remove dangling entities from the ground truth in measuring their alignment performance. In particular, Dual-AMN takes advantage of the bootstrapping module by incorporating labeled pairs in training. In the relaxed setting, such labeled pairs are ground-truth aligned pairs, but in the consolidated setting, the dangling entities could bring in erroneous alignment which contaminates the alignment of other pairs.
G.2 Class Prior Estimation Supplementary Experiment (RQ2).
We hope further to verify the estimation and convergence results of iPULE of loss convergence. We list the corresponding loss convergence results Fig. The losses under different pre-aligned proportions (0.25, 0.2, 0.15, 0.1) on the GA-DBP15K 15K constitute a group of statistical data, and the corresponding loss mean and standard deviation of this set of statistical data are displayed.
On the other hand, the loss difference is direct indication of convergence in iPULE's implementation. Thus, we plot the histogram figure of the DBP2.0 (w/ -minus & -plus) of the corresponding loss difference for more comprehensive. With the progress of the algorithm, and the statistical number of the difference of the smaller loss function occupied the maximum. This shows the convergence of iPULE in this data set from another aspect.
is worth noticing that, there are performance fluctuations during the constitution of an ideal embedding space during the cold start stage. The figure plotted covers only the cold start subsequent procedure.
G.3 Embedding Dimension Selection (RQ3).
Although a higher embedding dimension may encode richer information, an overly high dimension leads to performance decline. We select the GNN dimension according to the principle of [22). Let
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F9:2N
J4.EN
23:E2
33
E705
5/88
830
732
600
01x
Spach
Figure 7: Visualization of loss convergence on DBP2.0 and GA-DBPI5K.
the dimension of embedding be d and the number of entities is N. According to the feature entropy in [22], it holds that d 8.33 log N by the is at (log N) order. In most of our settings, N is approximately 10 and thus d is set to 128.
lemma [15] that the vector dimension
ZH-EN
EN-ZH
JA-EN
EN-JA
FR-EN
EN-FR
Dimension
Prec.
Rec.
F1
Prec.
Rec.
F1
Prec.
Rec.
F1
Prec.
Rec.
F1
Prec.
Rec.
F1
Prec.
Rec.
F1
Ours
64
.278
.46
.342
224
501
.100
.25
.410
362
239
.470
.377
.144
224
.189
.135
.257
.178
.281
.451
.36
226
.505
312
329
.415
367
2241
474
320
.122
235
.19
.143
.271
.187
128
.280
448
.44
225
.502
331
.330
.416
.68
.243
.477
.322
.177
222
.204
.151
.287
.198
Table 9: The entity alignment performance over different embedding dimensions on DBP2.0.
As shown in Tab. due to the varying number of entities in datasets, the embedding dimension at the optimal performance varies. For example, the top performance is achieved on ZH-EN and EN-ZH when the embedding dimension is 96 but is obtained on JA-EN, EN-JA, FR-EN, and EN-FR with an embedding dimension of 128. As compromise, we simplify the embedding representation of edges to enable FR-EN and EN-FR to run with an embedding dimension of 128 with limited memory (For more details, please check our open source code). A higher alignment performance can be achieved if no compromise is made. As we observe, the optimal performance is typically achieved at the theoretically chosen d. This also indicates our approach has a memory cost at the order of Olog N).
G.4 Efficiency (RQ4)
The previous works concerning the dangling problem have not analyzed its efficiency in their experiments. Thus we only report the efficiency of our methods without baseline comparison. We evaluate the efficiency our work on Tab, 10 including alignment search time as 'Inference Time', KEESA average training time as 'Average Training Time', and GPU memory cost on three different datasets of DBP2.0. O. Data obtained from these three datasets with the top three node numbers is a robust indicator of the efficiency of our method.
We gathered the mean value of 5 inference time costs for each dataset with the corresponding CPU and GPU memory consumption. Mcanwhile, the average training time for each period from early to late calculated. We enumerate the average training time of epochs 1-20, 21-25, 26-30, 31-35, 36-40, and 41-45.
Cause GPU is employed for not only model training but also inference, as shown Tab. I0 the inference speed is still very impressive. Specifically, we split the large similarity matrix into multiple independent row blocks to perform the nearest searches within each block, which are well suited for GPU parallel processing.
It's noteworthy that the average training time correspondingly increases training progresses from carly to late stages. More quasi-supervised information incorporated by us accounts for that. To be specific, the training deepens, we repeatedly conduct preliminary alignment tests while we gather more and more entity pairs mutually closest under a given metric. The entity pairs serve as the pre-aligned anchor nodes, i.e., the qupsi-supervisory information mentioned above.
Besides, we list the CPU and GPU memory consumption required for our work. Memory consumption is influenced by various factors such as complex allocation algorithms, model parameter scales, and hyperparrmeters. In this problem, we put more attention on triples which characterize one KG, revealing approximate proportionality between the number of triples and memory consumption.
23

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Datasets
Triples
Inference Time
Average Training Time (from 1 to training epochs)
CPU Memory
GPU Memory
1-20
21-25
26-30
31-35
36-40
41-45
DBP2 -2
872,935
48.78s
11.2
21.16wAt
25.67a/i
28.175/i
29.21s/it
50. 14./it
10.8GB
32,5GB
DBP2 DBP2.0xX-IX -IN
1,015,545
120.76s
28.1 18.14x/t
53.99</t
63.43c/it
68.27s/it
70.61s/tt
72.80
1199B
3266BB
D1112.0 2,089,909
382.48s
90.18
188,18.51
190.65wft
27.7GB
60.2GB
Table 10: Efficiency performance of our work on DBP2.0. The measurement of average training time is 's/it', which indicates seconds per iteration. One iteration here represents one training epoch. "-' indicates the absence of data due to training termination.
G.5 Baseline Comparison Under Different Ratios of Pre-aligned Seeds (RQ5).
Comparing the proposed method with strong baseline models under different ratios of pre-aligned seeds would better demonstrate Lambda's superiority. The experimental baseline includes MtransE w/ BR the SOTA method in previous works, which is also the only open-source method. The results are shown the Table.
Methods
7HLEN
EN ZH
JA-EN
EN-JA
FR-EN
EN-FR
Ratios
Prec.
Roc.
Proc.
Rec.
Prec.
Rec.
Prec.
Roc.
Prec.
Rec.
F1
Prec.
Roc.
F1
10%
161
.141
.148
.105
.127
.114
.102
102
.188
.102
.128
.113
.133
.085
.122
099
.022
.083
MiransE: BR
20%
264
2677
-265
186
.251
.213
.179
180
-251
.180
.251
.210
.215
.151
-179
-67
.138
.150
30%
.312
362
335
.241
.376
.294
314
363
336
.358
.295
.265
208
233
.231
.222
10%
.236
.346
.280
.197
.385
.261
.262
.315
.286
.266
.360
.262
.179
.230
.201
.153
.260
.193
Lambda
20%
2262
.399
.316
.215
446
.290
.000
368
.330
226
.417
.293
.217
.286
.247
.182
.324
.233
30%
.279
.447
.344
.219
.489
.303
.344
.409
.362
.244
.460
.310
.234
.300
.271
.122
.363
.251
Table 11: Performance of Lambda and MtransE w/ BR under different ratios of pre-aligned of 10%, 20%, and 30%. Bold indicates optimal performance.
G.6 LightEA as Strong Baseline for Comparison (RQ6).
LightEA (88 is recommended as strong baseline for Lambda. We fixed LightEA's code to include dangling entities into the alignment candidates and evaluated its performance on DBP2.0. Hits I and Hits @ 10 are evaluated in similar way the dangling-unaware methods in our paper, listed below. In comparison, Lambda still outperforms LightEA.
Methods
ZH-EN
JA-EN
FR-EN
H@1
H@ 10
H@1
H@1
H@
LightEA Lambda
60.5%
82.9%
61.4%
84.1%
62.6% %
84.7% %
62.1 1%
84.0%
44.1 % 69.3%
Table 12: Comparison of Lambda and LightEA under relaxed setting. indicates the absence of data due to out of time.
H Discussion
H.1 Alignment Direction
As we found, the alignment problem with dangling cases has a deeper issue concerning the classifica- tion of imbalanced datasets. It originated from the observation that the alignment performance from the source to the target is different from the other direction. The work of (33) has observed that on DBP2.0, choosing the alignment direction from a less populated KG (c.g., ZH, JA, FR) to a more populated KG (e.g.. EN) enjoys higher alignment accuracy but the other way around would lead to a noticeable performance drop. Meanwhile, the dangling entity detection on EN-XX has a higher F1 score than XX-EN, shown in Tab. B
By analysis, we think it may be attributed to an improper indication the dangling entity detection power on imbalanced datasets. This error in removing the predicted dangling entity would accumulate hurting the alignment task. To verify the point, we introduce trivial classifier that makes a simple choice to classify all entities dangling (positive) ones, and the detection results are reported in
24

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Dangling Detection
Entity Alignment
Datasets
Our Work
Trivial
Our Work
Prec.
Rec.
F1
Prec.
Rec.
F1
Prec.
Rec.
F1
ZH-EN
.763
.955
.836
.583
.736
.279
.447
.344
EN-ZH
.844
.909
.875
.609
.756
.219
.489
.303
JA-EN
807
.836
821
.800
.734
.324
.409
.362
EN-JA
.880
.809
.843
.605
.753
.234
.320
.271
FR-EN
.615
.772
.685
.439
.610
.234
.320
.271
EN-FR
.732
.749
.740
.554
.715
.192
.363
.251
Table 13: Dangling entities detection by our classifier v.s. a trivial one on DBP2.0.
Tab. 13
As all unlabeled entities are trivially classified as dangling ones, the detection metries of the trivial classifier are all falsely high. The more populated source KG usually has more dangling entities (positives) and thus yields higher precision detection. Meanwhile, since the detection classifier actually is not working, more dangling entities participate in the alignment phase, resulting in poor alignment performance. This has explained why EN-XX has a higher dangling detection performance but a lower alignment accuracy compared to the other direction.
The root of this issue is that matchable and dangling entities comprise imbalanced categories in the classification task, but the corresponding metric is inappropriate. Hence boosting the detection performance does not necessarily improve the alignment performance. We believe more practical indicators of imbalanced datasets should be introduced to the alignment problem.
H.2 The Similarity between Dual-AMN and Lambda
The differences between the proposed GNN and Dual-AMN include:
Aggregation:
1. The adaptive dangling indicator Tey is included in Lambda for eliminating dangling pollution.
2. The indicator rey is concatenated part of the entity feature.
Attention:
The attention scaled by Tey to filter dangling information.
2. Relation r 's embedding hr, is linked to the adaptive dangling indicator of the associated entity
Tey and thus the attention in Eq. models the relationship between the relation and the entity.
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NeurIPS Paper Checklist
1. Claims
Question:
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: see Abstract and Introduction.]
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
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The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: We explained the trade-off of our method in How does our method work? 5.3 for slightly inferior precision reported in Tab. 12 and Tab. 4
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the paper. The authors should use their best
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
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Justification: According to the order of appearance, we sort out and give the specific proof in the appendix.
The problem setting about PU learning as sec.
Proof for Lemma 3
Proof for Theorem
Proof for Theorem
Proof for Theorem
Proof for Lemma
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
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4. Experimental Result Reproducibility
Does the paper fully disclose all the information needed to reproduce the main perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
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Reviews:
[{"review_id": "rUvs9ajcZv", "reviewer": "Reviewer_RNuV", "summary": "This paper introduces a novel entity alignment framework called Lambda for aligning entities with dangling cases. It includes a GNN encoder, KEESA, to aggregate information within and across KGs, and an iterative positive-unlabeled learning algorithm, iPULE, to detect dangling entities. The authors provide both theoretical proof and empirical evidence to demonstrate the superiority of the proposed method.", "questions": "Why do the authors call L_info ``spectral contrastive learning''?", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "The idea of using positive pairs to support unlabeled dangling detection is interesting and effective.\n\nResults on dangling entity detection are promising.\n\nTheoretical proofs are provided to further support the proposed method.", "weaknesses": "The writing may be improved. Figures need more detailed captions. The methodology section introduces too many new terms. For instance, in Lemma 1, perhaps using e_i, e_+ and e_j is better than q, p^+, p^-. The overall subscripts and superscripts are also inconsistent.\n\nThe motivation of this paper is not clear enough. There are already many methods leveraging inter-graph and cross-graph GNNs for entity alignment. In this sense, the novelty of KEESA is limited. The so-called spectral contrastive learning seems no different from the existing ones.\n\nThen, the core contribution of this paper is iPULE. This paper could be improved if the authors delve deeper into the discussion of iPULE instead of KEESA, especially regarding the application of this module on different EA methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's valuable comments and address the reviewer's concerns as follows. The writing issues will be fixed in the revision.\n\n```\nThe motivation of this paper is not clear enough. There are already many methods leveraging inter-graph and cross-graph GNNs for entity alignment.\n```\n\nHere we restate the motivation: our focus is a new setting where dangling entities exist in the entity alignment problem. Since those entities do not have a match and remain unknown in prior, the alignment problem cannot be addressed by inter-graph and cross-graph GNNs alone.\n\nAlthough KEESA leverages both the intra-graph and inter-graph information, it emphasizes on learning the unified matchable embedding space $\\textbf{in the presence of dangling entities}$. The idea is to avoid the $\\textbf{pollution}$ of dangling entities to the embeddings of matchable ones in neighborhood aggregation. Conventional approaches not considering dangling nodes would assign a match to the dangling nodes, which leads to error spreading to the matchable nodes.  \n\n```\nThe so-called spectral contrastive learning seems no different from the existing ones.\n```\n\nThe form of the loss function in spectral contrastive learning may already exist, but our purpose is to illustrate its role in serving both tasks, i.e., entity alignment and dangling detection, by mining high-quality negative samples. The loss  $L_\\text{info}$ is critical to our problem as PU learning is based on the assumption of the existence of discriminative classes in the feature space, while this could be offered by the equivalent spectral clustering effect of $L_\\text{info}$ (L127-138).    \n\n```\nThen, the core contribution of this paper is iPULE. This paper could be improved if the authors delve deeper into the discussion of iPULE instead of KEESA, especially regarding the application of this module on different EA methods.\n```\n\nWe agree with the reviewer. However, we would like to point out that KEESA is important to iPULE, as the effectiveness of iPULE depends on the powerful and discriminative embedding produced by KEESA. The discriminative features are the prerequisite that PU learning works. Hence KEESA can be considered as an essential part which complements iPULE in our framework. According to our investigation, the encoder module by other EA methods lacks consideration of dangling nodes, and thus does not work with iPULE as well as KEESA does.\n\n```\nWhy do the authors call L_info ``spectral contrastive learning''?\n```\n\nThe word 'spectral' comes from spectral clustering as the loss is equivalent in performing spectral clustering in the embedding space (telling dangling entities apart from matchable ones). The word 'contrastive' indicates that the loss also performs contrastive learning over positive and negative sample pairs (for entity alignment).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JSq97qM1Cq", "reviewer": "Reviewer_HTtm", "summary": "This paper elaborates the unique challenges of unlabeled dangling entities in EA task. To address the challenge, it proposes the framework, namely Lambda, for dangling detection and then achieves entity alignment. The main idea is to perform selective aggregation with spectral contrastive learning and to adopt theoretically guaranteed PU learning to relieve the dependence on the labeled dangling entities. Extensive experiments demonstrate the effectiveness of each module and its ability to handle Unlabeled Dangling Cases.", "questions": "1. The paper mention that the initialization of r_{e_j} is critical (L94-95). What are the potential consequences of poor initialization? How is a good initialization chosen? Is it based on experience or theoretical support?\n2. How is the equivalent form of infoNCE (L137-137) derived?\n3. What point of aligned entity sparsity does the model stop all subsequent processes and determine that the two KGs cannot be aligned? Is there experimental evidence supporting this?", "limitations": "NA", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.  The paper is well-motivated.\n2.  The paper propose innovative methods to addresses the challenge of unlabeled dangling entities in entity alignment tasks\n3.  Extensive experiments demonstrate the model's effectiveness, and the experiments unaware(aware) of dangling entities is particularly insightful. \n4. The paper is well written and organized.", "weaknesses": "1. The section on KG Entity Encoder with Selective Aggregation section adds an adaptive dangling indicator based on the Dual-AMN[1] model. The paper would be better to provide a detailed explanation of why the adaptive dangling indicator is effective.\n2.  In some evaluation metrics of datasets, Lamada's results still need improvement. For instance, the results for H@10 and H@50 in Table 2 are lower than those of Dual-AMN[1], but H@1 is more higher. And the precision in Tables 3 and 4 is slightly inferior. \n3. Comparing the proposed method with strong baseline models under different ratios of pre-aligned seeds would better demonstrate the method's superiority.\n\n[1] Xin Mao, Wenting Wang, et al. Dual attention matching network with normalized hard sample mining. WWW 2021", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We extend our gratitude to the reviewer's invaluable feedback and address the reviewer's concerns as follows.\n\n```\nThe paper would be better to provide a detailed explanation of why the adaptive dangling indicator is effective.\n```\n\nSorry for not explaining it clearly. We introduce the adaptive dangling indicator as a global weight, instead of a local weight, in that any dangling entities with a similar structure can be assgined a high weight in the propagation. Our experiments confirm the point that, as shown in Figure 5, Sec. 5.4, the results without the adaptive dangling indicator are inferior indicating the power of the design.  \n\n```\nComparing the proposed method with strong baseline models under different ratios of pre-aligned seeds would better demonstrate the method's superiority.\n```\n\nThanks for the advice and we have included the experimental results on different ratios of pre-aligned seeds in the global rebuttal PDF. The experimental baseline includes MtransE w/ BR the SOTA method in previous works, which is also the only open-source method.\n\n```\nThe paper mentions that the initialization of r\\_{e\\_j} is critical (L94-95). What are the potential consequences of poor initialization? How is a good initialization chosen? Is it based on experience or theoretical support?\n```\n\nAs stated in $\\textbf{Implementation Detail}$ (L213-215), the $\\tanh$ function changes rapidly in the region close to $0$ but stays stable in the region beyond $[-3, 3]$. Since all nodes are considered equally important at the start of training, we thus initialize all the $r_{e_{j}}$ to $1$ to prevent gradients oscillation or near-zero gradients. \n\nOur past experiments have shown that if we ignore the above setup and choose to initialize $r_{e_{j}}$  to $[0, 0.5] \\cup [1.5, +\\infty)$, for example \\{$0, 0.5, 1.5, 2, 3$\\}, the alignment performance would be poor. \n\n```\nHow is the equivalent form of infoNCE (L137-137) derived?\n```\n\nDue to the page limits, we omitted the following derivation details: \n\n$$\\text{infoNCE}(q, p^{+}, \\\\{p^{-}\\\\}^{N}) = -\\log \\frac{\\exp(\\lambda~ \\textrm{sim}(q, p^{+}))}{ \\sum_{j}^{N} \\exp(\\lambda~ \\textrm{sim}(q, p^{-}_j)) + \\exp(\\lambda~ \\textrm{sim}(q, p^{+}))}$$\n\n$$=\\log\\frac{\\exp(\\lambda\\text{sim}(q, p^{+}))+\\sum_{j}^N\\exp(\\lambda\\text{sim}(q, p^{-}_j))}{\\exp(\\lambda~ \\textrm{sim}(q, p^{+}))}$$\n\n$$=\\log [1+ \\frac{ \\sum_{j}^N \\exp(\\lambda~ \\textrm{sim}(q, p^{-}_j))}{\\exp(\\lambda~ \\textrm{sim}(q, p^{+}))}]$$\n\n$$=\\log\\left[1 + \\sum^N_{j} \\exp(\\lambda~ \\textrm{sim}(q, p^{-}_j)-\\lambda~ \\textrm{sim}(q, p^{+}))\\right]$$\n\nIf we substitute the exponent with $H(\\cdot)$ defined in Eq.(4), the loss turns into $L_\\text{info}$ defined in Eq. (3). We will include the proof to avoid confusion in the revision.\n\n```\nWhat point of aligned entity sparsity does the model stop all subsequent processes and determine that the two KGs cannot be aligned? Is there experimental evidence supporting this?\n```\n\nSuch a termination point should be decided upon the requirements of downstream tasks --- whether the downstream task considers the alignment of two KGs is worthy. For example, for $1$\\% entities remaining to be aligned, it may take a disproportionate amount of computing resources to get these negligible amount of entities aligned.\n\nThrough experiments, we consider $1 - 5$\\% sparsity threshold is sufficient for most applications.\n\nFurthermore, we have also considered rigorous estimation of the corresponding sparsity threshold. One way is to draw the ROC curve given varying sparsity thresholds. Such statistical results are based on a large corpus of paired and unpaired KGs which requires heavy human annotations.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VSg6GT3AdV", "reviewer": "Reviewer_cR2t", "summary": "The paper tries to tackle the challenge of entity alignment (EA) with unlabeled dangling cases in knowledge graphs (KGs), where some entities lack counterparts in another KG. \n\nIt presents a framework to detect dangling entities and align matchable entities using a GNN-based encoder and a positive-unlabeled learning algorithm, respectively. \n\nIt also provides theoretical guarantees for the proposed methods, including unbiasedness, uniform deviation bounds, and convergence. \n\nExperimental results demonstrate the promising performance of the framework over baselines on real-world datasets, even without labeled dangling entities for training.", "questions": "- According to Table 2, the proposed GNN greatly outperforms MTransE and AliNet. Why does it not show such a huge advantage in Table 4?\n\n- (Open question) Is the proposed method applicable to dangling entity detection with labeled data? What about its performance in this case?", "limitations": "NA.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper addresses the challenge of entity alignment in a dangling-aware context, even when labeled data for dangling entities is unavailable. This practical scenario is explored, and can inspire future work on this task.\n\n- In my opinion, the proposed method, selective neighborhood aggregation combined with positive-unlabeled learning, holds promise for tackling the problem under investigation.\n\n- Experimental results across multiple datasets demonstrate that the proposed method can outperform baselines for dangling entity detection and entity alignment in most metrics.", "weaknesses": "- The discussion on related work appears insufficient. I recommend enhancing the appendix by providing a detailed discussion and comparison. For instance, although the proposed GNN looks similar to Dual-AMN, no explicit discussions regarding this similarity are currently included.\n\n - In Table 2, the proposed GNN exhibits significant superiority over MTransE and AliNet. However, in Table 4, the advantage is reduced. Although this could be attributed to various factors, such as dataset characteristics, evaluation metrics, or specific scenarios, it is essential to carefully analyze the experimental setup and consider potential confounding variables to fully understand this discrepancy.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's valuable comments and address the reviewer's concerns as follows.\n\n```\nThe discussion on related work appears insufficient. For instance, although the proposed GNN looks similar to Dual-AMN, no explicit discussions regarding this similarity are currently included.\n```\n\nThanks for the advice and we will provide a detailed discussion in the revision.\n\nThe differences between the proposed GNN and Dual-AMN include: \n\n$\\textbf{Aggregation}$: \n\n1.  We proposed adaptive dangling indicator  $r_{e_j}$ into GNN for eliminating dangling pollution.\n2.  The indicator $r_{e_j}$ is concatenated as a part of the entity feature.\n\n$\\textbf{Attention}$: \n\n1. We scale the attention by $r_{e_j}$ to filter dangling information.\n\n2. We link relation $r_k$'s embedding $h_{r_k}$ to the adaptive dangling indicator $r_{e_j}$ of the associated entity $e_j$, and thus the attention in Eq. (2) models the relationship between the relation and the entity.\n\n```\nAccording to Table 2, the proposed GNN greatly outperforms MTransE and AliNet. Why does it not show such a huge advantage in Table 4?\n```\n\nIn Table 2, MTransE and AliNet are both dangling-unaware methods. They are trained with 30\\% aligned entities under the same setting as our method. While in Table 4, MTransE and AliNet are dangling-aware for being $\\textbf{extended}$ by three techniques (NNC, MR, and BR), and have an $\\textbf{additional}$ 30\\% labeled dangling data for training. In contrast, our method does not leverage any labeled dangling data for training but has a superior performance, hence showing the power of our approach.\n\n```\n(Open question) Is the proposed method applicable to dangling entity detection with labeled data? What about its performance in this case?\n```\n\nThanks for proposing the interesting question. Our PU learning is an unbiased estimation of the loss on the labeled data (only positive) and unlabeled data (both positive and negative). In the case of dangling entity detection with labeled data, both positive and negative labels are known. Hence the approach is adapted as follows.\n\nThe new loss function is  $L=\\lambda L_N + (1-\\lambda) L_{PU}$ where the first term is the loss for the labeled negative data and the second is a PU-learning loss. The only extension is an additional estimation $\\lambda$ making the new loss an unbiased one. The performance depends on how we estimate $\\lambda$. This is an interesting field to explore in future work.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1fd1Pbls1t", "reviewer": "Reviewer_jgr6", "summary": "The paper proposes to match entities in dangling settings, where the entities may not have a link to any other entity. New dataset GA16K is proposed.", "questions": "See above", "limitations": "No limitations section is provided.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The task is important in the knowledge graph field. \n\nThe method achieves better F1 in the experiments.\n\nThere are proofs for the correctness of the algorithm.", "weaknesses": "Writting should be improved. For example, there is no definition of 'GA16K' in the whole paper. The first clear definition of 'GA16K' is in the appendix.\n\nBaselines are old. Many related works are missing (https://arxiv.org/abs/2210.10436, https://aclanthology.org/2022.acl-long.405/, https://dl.acm.org/doi/abs/10.1145/3404835.3462870, https://aclanthology.org/2021.emnlp-main.226/)\n\nThe GA16K dataset seems to be multi-modal, but there are well-established datasets for multi-modal entity matching (https://dl.acm.org/doi/10.1145/3534678.3539244, https://arxiv.org/abs/2212.14454) that are not mentioned or used in the experiments.\n\n No large scale experiments are conducted. The paper should conduct experiments on large-scale datasets (https://arxiv.org/abs/2108.05211) to show the scalability of the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's comments and the corresponding discussion will be added in the revision. However, we still would like to clarify some misunderstanding of our work:\n\n1. In terms of baseline comparison, we think our comparison is fair enough since the problem we focus on is the alignment issue with unlabeled dangling nodes.\n\n2. Multi-modal and literal-based tasks are not concerned with ours and thus there are some misunderstandings towards our experimental setup.\n\n3. For the entity alignment on large-scale graphs, we have presented experimental results to verify the scalability of our method on large-scale entity alignment with dangling entities.\n\nWe explain them one by one in the following.\n\n```\nBaselines are old.\n```\n\nIn selecting the baseline for our experiments, we carefully considered several factors. Although the GNN-based method of Sun et al. [1] is the closest related one to fairly compare our work with, a direct comparison is unfair as we do not apply any $\\textbf{dangling label}$ as they do. In our setting, we do not use any $\\textbf{side information}$ such as literal information to avoid name bias [6,7]. Thus our goal is to compare with baselines purely relying on $\\textbf{graph structures}$.\n\nAccording to the above principles, we consider none of the works mentioned by the reviewer could serve as a fair baseline. Specifically, EASY [2] and SEU [3] are literal-based methods, where additional side information for alignment is used. The LightEA [4] is a non-neural method based on label propagation algorithm with no consideration on the dangling entities. DATTI [5] is a plug-in that could be added onto Dual-AMN [8], RSNs [9], and TransEdge [10], of which the prototypes have been used as baselines in our paper.\n\n$\\textbf{Reference:}$\n\n[1] Sun Z, Chen M, Hu W. Knowing the No-match: Entity Alignment with Dangling Cases[C]//Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). 2021: 3582-3593.\n\n[2] https://dl.acm.org/doi/abs/10.1145/3404835.3462870\n\n[3] https://aclanthology.org/2021.emnlp-main.226/\n\n[4] https://arxiv.org/abs/2210.10436\n\n[5] https://aclanthology.org/2022.acl-long.405/\n\n[6] Liu X, Wu J, Li T, et al. Unsupervised entity alignment for temporal knowledge graphs[C]//Proceedings of the ACM Web Conference 2023. 2023: 2528-2538. \n\n[7] Zhang Z, Liu H, Chen J, et al. An Industry Evaluation of Embedding-based Entity Alignment[C]//Proceedings of the 28th International Conference on Computational Linguistics: Industry Track. 2020: 179-189.\n\n[8] Mao X, Wang W, Wu Y, et al. Boosting the speed of entity alignment 10×: Dual attention matching network with normalized hard sample mining[C]//Proceedings of the Web Conference 2021. 2021: 821-832.\n\n[9] Guo L, Sun Z, Hu W. Learning to exploit long-term relational dependencies in knowledge graphs[C]//International conference on machine learning. PMLR, 2019: 2505-2514. \n\n[10] Sun Z, Huang J, Hu W, et al. Transedge: Translating relation-contextualized embeddings for knowledge graphs[C]//The Semantic Web–ISWC 2019: 18th International Semantic Web Conference, Auckland, New Zealand, October 26–30, 2019, Proceedings, Part I 18. Springer International Publishing, 2019: 612-629.\n\n```\nThe GA16K dataset seems to be multi-modal.\n```\n\nThere are some misunderstandings to clarify. GA16K is $\\textbf{not}$ muti-modal. Although the original GAKG is a multi-modal Knowledge Graph, the extracted GA16K consists of pure graph structures in the form of URL links and their triples. \n\nOur problem has nothing to do with multi-modality, and there is $\\textbf{no dangling}$ entity in the multi-modal dataset mentioned by the reviewer [1] [2]. We can study the multi-modal problem as an independent one.\n\n$\\textbf{Reference:}$\n\n[1] https://dl.acm.org/doi/10.1145/3534678.3539244\n\n[2] https://arxiv.org/abs/2212.14454\n\n```\nNo large scale experiments are conducted.\n```\n\nLargeEA [1] mentioned by the reviewer is excluded from the baseline for its literal-based property and irrelevance to our dangling problem. As to large-scale datasets, DBP2.0 is one with dangling entities used in our evaluation: it has a total number of 943,894 entities, more than 20,000 relations, and 3,000,000 triples. It is approximately at the same order of scale as those in [1]. \n\nThe scalability test has been evaluated on DBP2.0. More experimental results could be found in Appendix $\\textbf{H.4 Efficiency}$ concerning the training time, inference time, GPU, and CPU memory consumption. We hope our response could address the concern of the reviewer. \n\n$\\textbf{Reference:}$\n\n[1] https://arxiv.org/abs/2108.05211", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
For reviewer HTtm.

```
Comparing the proposed method with strong baseline models under different ratios of pre-aligned seeds would better demonstrate the method's superiority.
```

Table.1 contains experimental results on different ratios of pre-aligned seeds. The experimental baseline includes MtransE w/ BR the SOTA method in previous works, which is also the only open-source method.

Meta-review: The paper deals with entity alignment and the problem of entities without a counterpart in the other graph.
It proposes a GNN-based encoder and an iterative learning algorithm supported by theoretical and empirical arguments.
The strengths of the paper are (1) addressing an important practical problem (`RNuV`, `HTtm`, `jgr6`), (2) using an innovative approach (positive-unlabeled learning) (`HTtm`, `cR2t`), (3) extensive experiments (`RNuV`, `HTtm`, `cR2t`), (4) theoretical analysis (`RNuV`, `jgr6`).
It could be further improved in terms of writing (`RNuV`, `jgr6`), additional experiments (`HTtm`, `cR2t`, `jgr6`), and inclusion of the outline of differences to Dual-AMN in the paper (which was already given in the discussion).
Most reviewers actively participate in the discussion, sometimes over several rounds.
Decision: Accept (poster)