NeuSymEA: Neuro-symbolic Entity Alignment via Variational Inference
Authors: Shengyuan Chen
Zheng Yuan
Qinggang Zhang
Wen Hua
Jiannong Cao
Xiao Huang
Venue: NeurIPS 2025 poster ()
Keywords: entity alignment
neuro-symbolic reasoning
knowledge graphs
Primary area: probabilistic_methods
TL;DR: NeuSymEA combines symbolic and neural models in a neuro-symbolic framework to robustly align entities across knowledge graphs, improving effectiveness, robustness, and interpretability.

Abstract: Entity alignment (EA) aims to merge two knowledge graphs (KGs) by identifying equivalent entity pairs. Existing methods can be categorized into symbolic and neural models. Symbolic models, while precise, struggle with substructure heterogeneity and sparsity, whereas neural models, although effective, generally lack interpretability and cannot handle uncertainty. We propose NeuSymEA, a unified neuro-symbolic reasoning framework that combines the strengths of both methods to fully exploit the cross-KG structural pattern for robust entity alignment. NeuSymEA models the joint probability of all possible pairs' truth scores in a Markov random field, regulated by a set of rules, and optimizes it with the variational EM algorithm. In the E-step, a neural model parameterizes the truth score distributions and infers missing alignments. In the M-step, the rule weights are updated based on the observed and inferred alignments, handling uncertainty. We introduce an efficient symbolic inference engine driven by logic deduction, enabling reasoning with extended rule lengths. NeuSymEA achieves a significant 7.6\% hit@1 improvement on $DBP15K_{ZH-EN}$ compared with strong baselines and demonstrates robustness in low-resource settings, achieving 73.7\% hit@1 accuracy on $DBP15K_{FR-EN}$ with only 1\% pairs as seed alignments. Codes are released at https://github.com/chensyCN/NeuSymEA-NeurIPS25.

NeuSymEA: Neuro-symbolic Entity Alignment via Variational Inference
Shengyuan Chen
Zheng Yuan
Department of Computing
Department of Computing
The Hong Kong Polytechnic University Hung Hom, Hong Kong SAR sheng-yuan yuan. chen@poly yu. edu.hk
The Hong Kong Polytechnic University Hung Hom, Hong Kong SAR yzheng. yuan@connect, polyu.in hk
Qinggang Zhang*
Wen Hua
Department of Computing
Department of Computing
The Hong Kong Polytechnic University Hung Hom, Hong Kong SAR ginggangg zhang9connect polyu. hk
The Hong Kong Polytechnic University Hung Hom, Hong Kong SAR wency hua@polyu. edu. hk
Jiannong Cao
Xiao Huang
Department of Computing
Department of Computing
The Hong Kong Polytechnic University Hung Hom, Hong Kong SAR cs caj8commppp polyu. edu hk
The Hong Kong Polytechnic University Hung Hom, Hong Kong SAR xiachuang@comp polyu. edu. hk
Abstract
Entity alignment (EA) aims to merge two knowledge graphs (KGs) by identify- ing equivalent entity pairs. Existing methods can be categorized into symbolic and neural models. Symbolic models, while precise, struggle with substructure heterogeneity and sparsity, whereas neural models, although effective, generally lack interpretability and cannot handle uncertainty. We propose NeuSymEA, unified neuro-symboll reasoning framework that combines the strengths of both methods to fully exploit the cross-KG structural pattern for robust entity align- ment. NeuSymEA models the joint probability all possible pairs' truth scores in Markov random field, regulated by set of rules, and optimizes it with the variational EM algorithm. In the E-step, a neural model parameterizes the truth score distributions and infers missing alignments. In the M-step, the rule weights are updated based on the observed and inferred alignments, handling uncertainty. We introduce efficient symbolic inference engine driven by logic deduction, enabling reasoning with extended rule lengths. NeuSymEA achieves a significant 7.6% hir@@ 1 improvement DBP15KZL EN compared with strong baselines and demonstrates robustness in low-resource settings, achieving 73.7% hit@ accuracy DBP15KFREN with only 1 pairs as seed alignments. Codes are released http:///ttoo..ooocc///c/////////
IPS25
Introduction
Knowledge graphs (KGs) are crucial for organizing structured knowledge about entities and their relationships, enhancing search capabilities across various applications. They are widely used in
"Conrepsponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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question- aantionnnnswrring systems [ 2) recommendation systems B3 4 SQL generation [5] 61, nat- ural language processing 79. etc.. Despite their utility, real-world KGs often face issues like incompleteness, domain specificity, and language constraints, which hinder their effectiveness in cross-disciplinary or multilingual contexts (10 To address these issues, entity alignment (EA) aims to merge disparate KGs into unified, comprehensive knowledge base by identifying and linking equivalent entities across different KGs. For example, aligning entities between a biomedical KG and pharmaceutical KG allows for mining cross pross-discipline relationships through the aligned entities, such as identifying the same drugs and their effects on different diseases to enhance drug repurposing efforts. This alignment enables more nuanced exploration and interrogation of interconnected data, providing richer insights into how entities function across multiple domains.
Entity alignment models aim to determine the equivalence of two entities by assessing their alignment probability. Existing methods can be broadly categorized into symbolic models and neural models. Symbolic models [12]44] provide interpretable and precise inference by mining ground rules, but they struggle with aligning low-degree entities, especially those without aligned neighbors. In such cases, the lack of supporting rules leads to low recall. Conversely, neural models, such as translation models [15] [6] and graph convolutional networks (GCNs) (7-233, excel in recalling similar entities by embedding them in a continuous space, yet they often fail to distinguish entities with similar representations, causing a drop in precision as the entity pool grows. deuro-symbolic models aim to combine the strengths of both approaches, offering robust reasoning ability for entity alignment in challenging scenarios.
However, neuro-symbolic reasoning in entity alignment (EA) faces several challenges. First, combin- ing symbolic and neural models into a unified framework suboptimal due to the difficulty in aligning their objectives. Current approaches either use neural models auxiliary modules for symbolic models to measure entity similarity []4] or employ symbolic models to refine pseudo-labels [24] 25 Second, in EA task, the search space for rules is large. Deriving ground rules from both int and inter-KG structural patterns results an exponentially growing search space as rule length increases, complicating efficient rule weight estimation and inference. Finally, generating interpretations for EA remains underexplored. Effective interpretations should not only generate supporting rules but also quantify their confidence through rule weights.
To overcome these challenges, we propose NeuSymEA, neuro-symbolic framework that combines the strengths of both symbolic and neural models. NeuSymEA models the joint probability of truth score assignment for all possible entity pairs using Markov random field, regulated by set of weighted rules. This joint probability is optimized via a variational EM algorithm. During the E-step, a neural model parameterizes the truth scores and infers the missing alignments. In the M-step, the rule weights are updated based on both observed and inferred alignments. To leverage long rules without suffering from the exponential search space, we employ logic deduction to decompose rules of any length into a set of unit- sub-rules. This allows for efficient inference and weight updates for long rules. After training. the missing alignments are jointly inferred by both components. Additionally, we introduce rule decomposition process, we extract long rules as explicit supporting evidence for alignments and recover rule weights as quantified confidence scores. Our contributions are summarized below:
explainer to enhance interpretability. By reversing the
While variational EM has been utilized in KG completion tasks (66 27,. adapting it directly to the EA task is nontrivial because they only consider single-KG KG structures. We bridge this gap by formulating truth scores and weighted cross-KG rules, and modeling the joint probability of the truth scores in a Markov random field regulated by the weighted cross-KG rules.
A principled neuro-symbolic reasoning framework via variational EM:
Efficient optimization via logical decomposition: We introduce a logic deduction mechanism that decomposes long rules into shorter ones, significantly reducing the complexity of rule inference and enabling efficient reasoning over large knowledge graphs.
Interpretable inference: The explainer utilizes leameed rules generate support paths for inter- preting both aligned and misaligned pairs. It offers two modes: (1) Hard-anchor mode-generates supporting paths from prealigned anchor pairs; and (2) Soft-anchor inde--incorporates inferred anchor pairs for more informative interpretation.
Empirical validation and superior results: NeuSymEA demonstrates state-of-the-art perfor- mance on benchmark datasets, delivering robust alignment accuracy and rule-based interpretations.

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Efficient reasoning with decomposee rule
Variational EM
Rule mining
Long rule
Unit-length rules
Joint probabiity (())
Aligned
Rule decompase
720 14
Source KG
Rule weight "ppp"
Woury
Target KG
Eniliy embedlding 9
Parameterizatin
Matching scores
E-step: fix update @
Target KG
e
M-step: fix update W
Neural modeling
Figure 1: Framework illustration of NeuSymEA. The yellow solid line represents pre-aligned entity pair. The symbolic model computes the path-level matching probability of entity pairs by mining rules with learned weights. The neural model learns embeddings and calculates entity-level matching scores. NeuSymEA models their agreement using joint probability distribution over observed pairs and parameterized truth scores for hidden pairs, optimizing through a variational EM algorithm.
2 Preliminaries
2.1 Problem statement
knowledge graph G comprises triples 'I where each triple (e4, Tk, ey) E T represents a directional relationship between its head entity and tail entity, Given two KGs
of entities ε, set of relations R. and of relation
{E, (., G' {E', R', T'). and set of observed aligned entity pairs O = {(e,, e;)le; E E, e, E the goal of entity alignment is to infer the missing alignments by reasoning with the existing alignments. This problem can be formulated in probabilistic way: each pair (e,e'),
E E' is associated with binary indicator variable (e,ee) V(e,e') means (c, el) ) is an aligned pair, and (eeee") = 0 otherwise. Given some observed aim to predict the labels of the remaining hidden entity pairs H & 8" O, i.e., UH = (((e,e") (e.e)))
alignments to (V(e.e)) (),ee,))
2.2 Symbolic reasoning for entity alignment
Given an aligned pair (ej, (ej,ef), a new aligned pair (ei, e() can be inferred with confidence score wp.p they are each connected to the existing pair via a relational path p and p' respectively, formally:
Up.p' (eg = c,)) A p(e,, e;) A pp' (' e()) - (a== c(),
(1)
where p= [ R are pair of paths each consisting of L connected relations, and uppp' measures the rule quality that considers the mG structure and inter-KG structure, such as the indicative of each path, and the similarity between two paths. By instantiating such rule with the constants (real entities and relations) in the KG pair, symbolic model predicts the label distribution of entity pair (c, e') by:
Pe ( (G. for (e,e') E (OUH).
Using logic rules to infer the alignment probability can leverage the high-order structural information for effective alignment as well as provide interpretability. However, exact inference is intractable due to the massive amount of possible instantiated rules (exponential to I)), limiting its applicability to real-world KGs.
3 Neuro-symbolic reasoning framework for entity alignment
3.1 Variational EM
Given a set of observed labels vo, our goal to maximize the logelikeldoood of these labels, i.e., log Pw (vo). Directly optimizing this objective is intractable because it requires computing
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integral over all the hidden variables. Instead, we optimize the evidence lower bound (ELBO) of the log-likelinn log-likelihood as follows:
(3) here, ((vH) is variational distribution of the hidden variables UH. This inequality holds for all because Pa (vo) ELBO(g, to; wo;w) + DR (pWH)))))(()) vo)), where D ( vo)) 2 0 is the KL-divergence between g(vH) and Pur vo). Under this framework, the log-likelihood (vo) can be optimized using an EM algorithm, efficient method to find the maximum likelihood where the model depends on unobserved hidden variables: during the E-step, we fix and update the variational distribution q; during the M-step, we update to maximize the log-likelihood of all the entity pairs, i.e., Eg(wx) [log Pa voo UH )] as illustrated in Figure !
(w( 2 Eg(wy) Po 2H) - log g ) = ELBBO vo;
Explicitly representing the variational distribution q is parameter intensive, which requires & |E|[E") variables because the observed pairs are very sparse. To this end, we parameterize q with neural model as qa. with being the parameters of the neural model.
3.2 E-step: inference
In this step, we fix w and update qo to minimize the KI. divergence DEL. Directly minimizing the KL divergence is intractable, as it involves computing the entropy of qo- Therefore, we follow (26 and optimize the reverse KL divergence of and Pw, leading to the following objective:
von.0
Σ (e,E")EH
(vz).
(4)
To optimize this objective, we first use the symbolic model with weighted rules to predict ((( vo) for each (e, e') E H. If Pw (V(e,e)) voo) > 6, where ô a threshold, we treat this entity pair as positive label; otherwise,
regard the pair as negative pair that can be selected during negative sampling process of the neural model.
The observed labels can also be used as training data for supervised optimization. The objective is:
@uo
Σ
log go (V(e,ee)) 1).
(e,6")EO
The final objective for qo is obtained by combining these two objectives: Φo = von + Peo.0-
3.3 M-step: rule weight update
In this step, we fix Oo and update the rule weight w to maximize ELBO(q: vo: w). Since the right term of the ELBO in Equation constant when go fixed, the objective is equivalent maximizing the left term (q)(v)) [log Pw Pw (Vo, UH)], which is the log-likelihood function.
Specifically, start by predicting the labels of hidden variables using the current neural model. For each cc) E H. we predict the labels "(e,,') (0) and obtain the prediction set ( = ( ( this way, maximizing the likelihood practically becomes maximizing the following objective:
ow = Po Pw (wo. H (f)).
(6)
To obtain the pseudo-label (eeee)) using qo, we employ the trained neural model to compute the matching score of any entity pair e') E H. However, this strategy can easily introduce false positives into the pseudo-label set especially when the number of entities is large. To mitigate this, we consider one-to-one matching to sift only the most confident pairs. Practically, we first sort all pairs by their confidence score, then we annotate the pairs as positive following the order of the confidence. If pair contains an entity observed in the annotated pairs, then this pair skipped. This simple greedy strategy significantly reduces the amount of false positives.
Optimization and inference
4.1 Efficient optimization via logical deduction
Inference and learning with logic rules of length I. can be intractable, as the search space for paths grows exponentially with increasing L. To enhance reasoning efficiency, we decompose rule in

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Equation Jsiig logic deduction, inspired by 28 in KG completion:
(^ (elea-11ex) a (
Wp. (e, =())
= (e; = e.)
(7)
Here A Nh ( ex) represents the path formed by r1, r2,...., FL connecting e, to e, with eo ea
and 6 Eg. This long rule can be reorganized as the combination of a series of unit-length logic reasoning:
( (A [ratec-1, A A=1
Wpp.'
(ej =
r(((6--1,))) Arla))rr((eeee
= (e: e.).
(8)
In this way, each logic rule of length L can be viewed as a deductive combination of short rules of length 1. At each step, following [2). we perform one-step inference update Pw (((eee)) for each (e, e') E H by aggregating the alignment probability from neighbors:
II (e.T.o))TT (E)''
(1 -1- -r-(()oooost C (') x (1 (rrr))sus((" C (r) (9)
where 7(r) is a relation pattern r measuring the uniqueness of e through relation r given a specified re)))T)
tail entity quantified by n(r) = ((c Pous(r S r') denotes the probability that relation is subrelation of r'. This technique enables inference with confidence by explicitly quantifying confidence w during each inference step by introducing 7) and Psus C rr'.. Moreover, in this way. the update of the weight simplifies updating Psub(r r') during the M-step (Equation 6), as m)(r for each relation is constant. In practice, the update of Paus S r') can be computed by:
Σ (ehr eDEE (1 -1-00 (en ((eree))))))))) Plene))
(10)
Σ (1-11o,, (1- Veen,ef e ((ene())
where ef) and (f).ee)) are labels (or pseudo-labels) from wo U ( (0).
After optimization, rule weights can be computed by taking the product of the importance scores n of relations and the sub-relation probabilities of the corresponding relation pair:
Wp.p' II m((x) -(rk) (ruul(k S rr)) + Peut (T'k C rk)
(11)
2
In Appendix A.2 we provided detailed complexity analysis, demonstrating that parameter complex- ity scales linearly with dataset size, while computational complexity quadratic. Our implementation enables efficient execution through parallel computing and batch processing.
4.2 Inference with interpretability
To predict new alignments, there are two approaches: using the symbolic model the neural model. The symbolic model infers alignment probabilities with the optimized weights w. Due to scalability concern, symbolic methods generally adopt a lazy inference strategy that only preserves the confidently inferred pairs during inference. On the other hand, the neural model computes similarity scores for all entity pairs ee, using the learned parameters 0, generating a ranked candidate list for each entity.
The evaluation of these models thus differs. Symbolic models are generally evaluated by precision, recall, and F1-score for their binary outputs, while neural models are assessed using hit@k and mean reciprocal ranks (MRR) for their ranked candidate lists. Following the practices in [14] and 24), unify the evaluation metrics by treating the recall metric of symbolic models as equivalent to hit@1. 1, facilitating comparison with neural models.
To enhance the interpretability of predictions, we adapt the optimized symbolic model into an explainer. For any given entity pair, the explainer generates set of supporting rule path pairs that
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justify their alignment, each associated with confidence score indicating its significance. The explainer operates in two modes: 1 hard-anchor mode, which generates supporting paths only from prealigned pairs, and @ soft-anchor mode, which includes paths from both prealigned and inferred pairs, providing more informative interpretations.
By integrating a breadth- search algorithm (detailed in Appendix A.3), the explainer efficiently generates high-quality interpretations. For truly aligned pairs, typically produces high-confidence interpretations, while for non-aligned pairs, the interpretations may result in an empty set (indicating no supporting evidence) or have low confidence scores. See Figure 4 for a visualized comparison.
5 Experiments
5.1 Experimental settings
Main experiments use the DBPI5K dataset, comprising three cross-lingual KG pairs: JA-EN, FR-EN, and ZH-EN. The original full version [2] of DBP15K resembles real-world KGs, posing challenges for GCN-based models due to sparsity and scale. Recent GCN-based models [19 16 17
Datasets,
24 remove low-degree entities to get smaller version with higher average degree. For thorough evaluation, we utilize both full and condensed versions. Dataset statistics are provided in Appendix B.1. For additional experiments on large KGs, we employ OpenEA
30 and DBPIM.
Two different dataset split strategies are used the EA literature: a 3:7 train/test split, and @ a 5-fold cross- validation scheme with 2:1:7 ratio for training, validation, and test sets, used in OpenEA [30]. We adopt the latter for all algorithms to ensure fair comparison.
Baselines and metrics. Baseline models include seven neural models GCNAlign 19] AlignE, BootEA [6), RREA [7], Dual-AMN [8] LightEA 200, PEEA (31, one symbolic models
We use Hit@ 1, Hit@ 10, and MRR as the evaluation metrics. For PARIS and PRASE that have binary outputs, we report their recall as @it@ 1, following [24). For RREA, Dual-AMM, and LightEA, which offer both basic and iterative versions, we adopt the iterative ones due to their generally superior performance.
PARIS 12, and two seuro-symbolic models PRASE 14
EMEA
24
Hyperparameters. NeuSymEA involves two main hyperparameters: the number of EM iterations and the symbolic model's threshold 8 for selecting positive pairs. We tune them on the validation set, searching in 0.6, 0.7, 0.8, 0.9, 0.95, 0.98, 0.99 and the number of iterations from I to 9. As shown in Section 5.3 NeuSymEA is robust to ō and converges quickly.
5.2 Results
5.2.1
Comparison with baselines
Table [ compares NeuSymEA with baseline models on two versions of the DBP15K dataset: the full and the condensed version. Results for PRASE and EMEA on the condensed DBP15K are sourced from the original EMEA paper. The results yield three key observations:
First, NeuSymEA surpasses both symbolic and neural models. By integrating symbolic reasoning with neural representations, it: 1) captures multi-hop relational structures across KGs using rules; and 2) learns effective entity representations compute pairwise similarities. Second, NeuSymEAA outperforms other neuro-symbolic models. This improvement can be largely attributed to the model objective design in our framework. While PRASE and EMEA treat the symbolic and neural models as separate components, NeuSymFA unifies them under a joint probability objective. This enables joint optimization via Variational EM. yielding more coherent and convergent solution with superior performance. Finally, NeuSymEA demonstrates robustness across both full and condensed datasets. Comparisons between two groups of results offer an interesting insight: neural models experience significant performance degradation in the full version of DBP15K (e.g., MRR of Dual-AMN decreases from 0.815 to 0.717 on JA-EN), while symbolic models, in contrast, show improvements. We attribute this to their different matching mechanisms: (1) neural models rely on entity-level matching, which sensitive to dataset size. The full DBP15K includes more low-degree entities, increasing similar embeddings and reducing precision. (2) Symbolic models use level matching, which is less affected by dataset size but vulnerable to substructure heterogeneity. The full DBP15K's additional connections via long-tail tail entities enhance rule-minie mulemmining, boosting symbolic

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Table 1: Entity alignment results DBPI5K dataset. The suffixes "-D" and "-L" indicate the of Dual-AMN and LightEA as the neural models. The results of RREA and EMEA are omitted on the full dataset due to OOM (Out of Memory) emror.
JA-EN Hit@1
FR-EN
ZH-EN Hit@1 Hit@ MRR
Category
Model
@
MRR
H
Hit@ 10
MRR
Results the full DBPI5K dataset
GCNAlign
0,221
0.461
0.302
0.205
0.475
0.295
0.189
0,438
0.271
BootEA
0,454
0.782
0.564
0.443
0.799
0.564
0.486
0.814
0.600
Neural
AlignE:
0.356
0.715
0.476
0.346
0.731
0.475
0.333
0.690
0,455
Dual-AMN
0.627
0.883
0.717
0.652
0.908
0.744
0.650
0.884
0.732
LightEA
0.736
0.894
0.793
0.782
0.919
0.832
0.725
0.874
0.779
Symbolic
PARIS
0.589
0.618
0.603
Neuro-symbolic
PRASE
0.611
0.647
0.652
NeuSymEA-D
0.806
0.942
0.855
0.827
0.952
0.871
0.801
0.925
0.843
Ours
NeuSymEA-L,
0.781
0.907
0.826
0.834
0.937
0.871
0.785
0.894
0.825
Results the condensed
DBP15K dataset
GCNAlign
0.331
0.662
0.443
0.325
0.688
0.446
0.335
0,653
0..44
BootEA
0.530
0.829
0.631
0.579
0.872
0.961
0.575
0.847
0.668
AlignE
0.433
0.783
0.552
0.457
0.821
0.580
0.474
0.806
0.587
Neural
RREA
0,749
0.935
0.818
0.797
0.958
0.859
0.762
0.938
0.827
Dual-AMN
0,750
0.927
0.815
0.793
0.954
0.854
0.756
0.919
0.816
LightEA
0.778
0.911
0.828
0.827
0.943
0.830
0.770
0.894
0.816
PEEA
0.703
0.912
0.777
0.748
0.937
0.815
0.726
0.905
0.790
Symbolic
PARIS
0.565
0,584
0.543
PRASE
0.580
0.622
0.593
Neuro-symbolic
EMEA
0.736
0.807
0.773
0.841
0.748
0.815
NeuSymEAAD
0.805
0.930
0.849
0.835
0.953
0.879
0.815
0.926
0.855
Ours
NcuSymEA-L
0.811
0.928
0.854
0.858
0,954
0.894
0.804
0,904
0.840
model performance. By integrating symbolic reasoning with KG embeddings, NeuSymEA overcomes these limitations, ensuring robustness to variations dataset scale and structure.
5.2.2 Evolution of rules and embeddings
6000
0.85
8000
0.75
infered
fres infered demminiered tra al-es inkreed tue
5000
ja-n infeaed all 2 inford
0.70
Eret
Beration
Beration
Figure 2: (Left) Evolution of rule inferred pairs, with solid lines representing total inferred pairs and dashed lines representing true inferred. The shaded areas indicate the number of false pairs. Precision values annotated each data point. (Right) Convergence of MRR of the neural model.
We study how rules and embeddings evolve and interact with each other during the EM steps, with results shown in Figure 2 Results the left subplot indicate that in each EM iteration, the number of rule-inferred pairs grows consistently with high precision, implying that the embedding model continuously improves the inference performance of rules. These precise pairs, in turn, enhance the performance of the neural model. As shown in right subfigure, MRR of the neural model converges within few iterations.
5.2.3 Scalability on large datasets
Figure 3 illlsstrate the scalability of NeuSymEA. We evaluated its hit@1 accuracy and runtime efficiency across datasets of varying entity sizes, namely DBP15K, OpenEA 100K, and DBPIM.
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10
Exntrise Proconted pir Secoedd
2,554
0.874
3,529 0.845
Metbod LighttA NeaSymiAA
Syszbble
10000
Netral
11211
0.6
$030
5030
3509
0,142
4030
1968
Oped go.
Opent PreaEAAHiKKg.... Hitkkge H
Die (M
DBPIM
DUP15K
Openl At 100t Pt.
DBP 1M
Figure 3: Scalability analysis on large scale KGs. (Left) Hit@1 alignment performance on large KGs.
(Right) Per-second processed entities of neural and symbolic components on different scales of KGs.
Performance scalability. In the left subfigure, we compare NeuSymEA with LightEA the only strong baseline capable of processing large-scale knowledge graphs with million-scale entities. The results demonstrate NeuSymEA's superior scalability in terms of performance.
Runtime Scalability. We separately evaluate the runtime performance of NeuSymEA's neural and symbolic reasoning components. For the neural component, efficiency (entities processed per second) increases with dataset size. We attribute this to higher GPU utilization larger datasets, which enhances computational efficiency. Conversely, the symbolic component's efficiency initially increases but then slightly decreases. Upon investigation, we found that this phenomenon is related to the multiprocessing and batch-processing implementation. For smaller datasets (e.g.. DBP15K), the overhead from process initialization and termination significant. As dataset size grows to OpenEA 100K, this overhead becomes negligible relative to inference runtime, leading to an efficiency improvement. When moving from OpenEAI00K to DBPIM, the per-second processed entities slightly decline as the quadratic complexity of inference computation dominates.
5.2.4 Interpretations by the explainer
Rule Confidence Distribution
Soft Anehor Mode
Hard Anchor Mode
Poskive Pain
Probability Density
4
Nagzove Par
0.2
24:7
27:3
24:
25.1
12.4
33.1
3
Thresheld
0.4
5.5
9:4
16.2
10.2
102
4.1
5.7
5X
5.5
53
20
2
0.6
1.6
Nuoke
1.3
0
0.0
0.2
0.4
0.6
0.8
1.0
2
5
Rule Confidence
Max Rule Length
Max Rule Length
Figure 4: (Left) Probability density of the top supporting rule's confidence; (Middle) Number of supporting rules relative to the maximum rule lengths under the soft anchor mode; (Right) Number of supporting rules relative to the maximum rule lengths under the hard anchor mode.
We investigate the interpretations generated by the explainer on the FR-EN dataset in Figure The left subfigure displays the probability density of confidence scores for supporting rules associated with entity pairs. Positive pairs come from the test set, while negative pairs are generated by randomly replacing one entity in each positive pair. The distinct confidence distributions show that positive pairs generally have stronger alignment evidence, as expected. Upon further examination, we found that many test pairs are isolated, i.c., they lack directly aligned neighbors. Despite this, NeuSymEA successfully generates supporting rules for isolated pairs by exploiting multi-hop dependencies.
The middle and right subfigures examine the impact of rule length on the explainer's s effectiveness, showing the number of supporting rules for positive pairs as the maximum rule length increases. In soft anchor mode, the explainer produces more high-quality supporting rules than in hard anchor mode by using inferred pairs as complementary anchors, mitigating substructure sparsity. While longer maximum rule lengths yield more high-quality supporting rules, however, the long rules have a lower confidence distribution compared to short rules. This can be attributed to our method for calculating confidence: the logical deduction- based approach computes a rule's 's confidence as the

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product of the confidences of its decomposed unit-length sub rules (as in Equation I1 For instance, a rule comprising two unit-length sub-rules, each with a confidence of 0.8, has a combined confidence of 0.8 x 0.8 0.64. Thus, confidence scores typically decrease as rule length increases.
Table 2: Examples of supporting rules for query pairs in FR-EN. Anchor pairs are shown in bold.
Query Pur
Supporting Rule
Confiennc
Maison, de, Sevoic
Hosse_o(_Snn Savuy
Hunben JI Lioe a d'halie, dyzasti,, Maison de Suoiies (Huneer RLooe of e conjoinn Marie Jusé de. Belique)
(Unbeeto Jfly, house, House Se00y, (Unbeto_ oL July, spouse Maric Just of Begium)
0.30
L.tion, copegrale
Sparisl__egion
(Lighasspennie commadeeffoirrigg Franzisen, Foacol, (Franciaoo Franso, conjan Carmen, Polo) (ppuish Legion, Commlneo Fruncsco Franco, FrunescceFratton spons. Carmon, Laiyy of Meiris)
0.90
Premict minierc da. Duscnask ThimeMMniioe of Deamark
(Penic miniiii _u, Dancuul, lilhircAsttel, Larx, Lakke Remmssn)) Miiiter of Demmer incumbent Lars Lakke Ban.Lakke_Rommmsen
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5.2.5 Robustness in low resource scenario
Table 3: Comprehensive results with different ratios of training data, with best shown in bold.
Dataset
Model
1%
5%
10%
20%
H@l
H@ 10
MRR
H@1
H@ 10
MRR
H@I
H@ 10
MRR
H@1
H@ 10
MRR
AlignE
0.007
0.034
0.016
0.080
0.268
0,143
0.244
0.588
0.356
0.433
0.783
0,552
BootEA
0.010
0,040
0.021
0,379
0.683
0,481
0.468
0.779
0.573
0.530
0.829
0.631
GCNAlign
0.029
0.128
0.063
0.127
0.368
0.206
0.209
0.515
0.310
0.331
0.662
0.443
PARIS
0.145
0,340
0.450
0.565
PRASE
0.163
0.432
0.508
0.580
JA-EN
Dual-AMN
0,239
0.519
0.334
0.509
0.795
0.611
0.652
0.887
0.738
0.750
0.927
0.815
RREA
0,253
0.486
0.332
0.558
0,830
0.653
0.672
0,903
0.756
0.789
0,956
0.853
LightEA
0.291
0,514
0,363
0.627
0,806
0.689
0.714
0.874
0.778
0.911
0.828
EMEA
0.411
0.488
0.630
0,710
0.688
0.764
0,736
0.807
PEEA
0,519
0.490
0.785
0.589
0.612
0.834
0.679
0.912
0,777
NeuSymFA-D
0.481
0.684
0.550
0.692
0.855
0,749
0.742
0.895
0.796
0.835
0.953
0.879
NeuSymEA-L
0.632
0.779
0.683
0.733
0.870
0.781
0.773
0.900
0.818
0.858
0.954
0.894
AlignE
0.008
0.040
0.019
0.408
0.347
0.733
0.475
0.457
0.821
0.580
BootEA
0.009
0.041
0.020
0.418
0.746
0,529
0.490
0.809
0.598
0,579
0.872
0.681
GCNAliga
0.119
0.388
0.215
0.539
0.325
0.688
0.446
PARIS
0.195
0.401
0.479
0,584
PRASE
0.227
0,514
0.575
0.633
FR-EN
Dual-AMN
0,293
0.631
0.407
0,598
0,886
0,703
0.717
0.928
0.797
0.793
0.954
0.854
RREA
0,289
0,583
0.389
0.628
0.895
0.725
0.717
0.932
0.796
0.789
0,956
0,853
LightEA
0.430
0.663
0,509
0.723
0,885
0,781
0.779
0.914
0.828
0.827
0,943
0,870
EMEA
0.480
0.565
0.677
0.757
0.727
0.802
0.773
0.841
PEEA
0.285
0,588
0.385
0.552
0.812
0.642
0.665
0.875
0.738
0.748
0.937
0.815
NeuSymEA-D
0.642
0,833
0.709
0.768
0.916
0.820
0.811
0,939
0.856
0.835
0.953
0.879
NeuSymEAAL
0.737
0.874
0.785
0.806
0.921
0.848
0.827
0.937
0.867
0.858
0.954
0.894
AlignE
0.006
0.033
0.016
0.127
0.368
0,206
0.296
0.635
0.407
0.474
0.806
0.587
BooEAA
0.006
0.029
0.014
0.396
0.689
0.495
0.498
0.782
0.594
0.575
0.847
0.668
GCNATign
0.041
0.155
0.080
0,147
0.396
0,229
0.225
0.519
0.323
0.335
0.653
0.442
PARIS
0.059
0.333
0.429
0.543
PRASE
0.241
0.461
0.522
0.593
ZH EN
AMN
0.375
0.666
0.480
0.582
0,830
0.672
0.676
0.892
0.755
0,756
0.918
0.816
0.316
0.564
0.605
0,858
0.696
0.686
0.901
0.765
0.760
0.934
0.823
LightEA
0.507
0.673
0.565
0.670
0.819
0.723
0.727
0.860
0.775
0,770
0.894
0.816
EMEA
0.517
0.591
0.665
0.738
0.706
0.777
0.748
0.815
PEEA
0,288
0.586
0.388
0,532
0.801
0.622
0.649
0.871
0.710
0.725
0.905
0,790
NeuSymEA-D
0.589
0,750
0.645
0.704
0.856
0.757
0.763
0.897
0.809
0.815
0.926
0.855
NeuSymEAA-L
0.676
0.799
0.720
0.735
0.858
0.779
0.773
0.882
0.811
0.804
0.934
0.841
Table 3 demonstrates the model performance under low-resource settings. As the percentage of training data decreases, all models experience noticeable drops in Hit@| performance. Despite this, NeuSymEAA exhibits remarkable robustness across all datasets, consistently outperforming other models. Notably, with only 1% of pairs used as training data, NeuSymEA achieves Hit@ I score exceeding 0.7 on FR-EN, rivaling or even surpassing the performance of some state-of-the-ar models trained on 20% of the data.
5.3 Parameter analysis
We present the hit@1 performance of NeuSymEA across three datasets, varying hyperparameters, illustrated by three-dimensional graph. The threshold hyperparameter d is explored within the set (0.6, 0.7, 0.8, 0.9, 0.95, 0.98, 0.99), while the number of EM iterations ranges from I to 9. Performance levels are indicated using colormap. Performance sensitivity analysis in Figure

---

reveals that for all datasets, performance generally improves as the iteration increases. On the other hand, the performance is less sensitive to the threshold 8.
1ig@ for jaren
High for fr-en
Hingl for en
008000
9.810
0.800 0.85
0.500
0.850
0.750
90.95
0. 8 855
3.3 e.s. 6 7788 Bersiion
0.545
0,790
The /
3 5 Nersiion
Threuber 039
0.95
Wernson
Figure 5: Performance sensitivity to hyperparameters iteration and threshold 6.
6 Related work
Neuro-symbolic reasoning on knowledge graphs. Neuro-symbolic methods aim to combine symbolic reasoning with neural representation learning, leveraging the precision and interpretability of symbolic approaches alongside the scalability and high recall neural methods. In KG completion
task, [32] and [33] employ horn rules to regularize the learning of KG embeddings; [34] and [28) model the rule-based predictions as distributions conditioned on the input relational sequences, and parameterize these distributions using recurrent neural network; (26,, 27 and [35] models the joint probability of the neural model and the symbolic model with a Markov random field, and employ gradient descent for weight updates. Despite extensive advancements of neuro-symbolic reasoning in KG completion, these studies only consider single-KGG KG structures, thus cannot be directly adopted to entity alignment which requires consideration of iGter-KG structures.
Entity alignment. Recent models have sought to combine symbolic and neural approaches for entity alignment. For instance, [44] enhances probabilistic reasoning with KG embeddings to measure entity-level and relation-level similarities. [24] implements self- sef-bbootsrapppng with lseudo-labeling
in neural framework. using rules to choose confident pseudo-labels. However, it relies solely on unit-length rules, which restricts its effectiveness for long-tai entities. In contrast, we employ logic deduction to scale symbolic reasoning with long rules of any length. Unlike existing work that separately handles two reasoning components, our work models the unified joint probability of symbolic and neural inference within markov random field. The symbolic component captures intra-KG and inter-KG structures using weighted logic rules, while the neural model learn expressive patterns in the embedding space.
7 Limitations
NeuSymEA is currently designed for EA between two KGs. Extending align multiple KGs simultancously may require iterative pairwise alignments, which could be inefficient. A more sophisticated optimization paradigm is needed to adapt NeuSymEA for scalable multi-KG alignment.
8 Conclusions
We presented NeuSymEA, unified and extensible neuro-symbolic framework for entity aligmment. By unifying neural and symbolic reasoning, NeuSymEA addresses the challenges of substructure het- erogeneity, sparsity, and uncertainty real-world KGs. Empirical results demonstrate NeuSymBA's clear improvements over baselines and robustness under limited resources. By delivering interpretable alignment predictions with uncertainty scores, NeuSymEA advances knowledge fusion, enabling effective and trustworthy entity alignment.
10

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Acknowledgement
The work described this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 25208322).
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A Notations and algorithms
A.1 Notations
Table 4: Notations
Notation
Description
9,9
The source and target knowledge graphs, respectively
E
The sets of entities G and G", respectively
R. R'
The sets of relations in G and g', respectively
T.T
The sets of relational triplets in 9 and g', respectively
O
The set of observed aligned entity pairs between two knowledge graphs 9 and g' Set of unobserved entity pairs, i.e. & E
H
((a,e)
Binary indicator variable for entity pair c') where V(c,e)) indicates alignment
Uppp'
Confidence score of rule-inferred alignment based paths p and p' Probability distribution the alignment indicator (eeee)) given knowledge graphs G and Parameters of neural model
Pio (eeee))))
Threshold to select positive pair from the symbolic model Relation pattern measuring the uniqueness of an entity through relation
n(r)
A.2
Complexity analysis of the symbolic reasoning
In the following, we present the analysis of runtime complexity and parameter complexity one by one.
A.2.1 Runtime complexity
In variational inference, the process of learning and inferring long rules (Equation 7 is simplified by decomposing them into unit-length rules (Fquation [8. Consequently, rule weight learning (Equation 10) only conducted for unit-length rules. The inference process for an L-length rule is then estimated by iteratively applying inference steps with unit-length rules (Equation 9 for L iterations. This strategy effectively avoids the exponential search space associated with longer rules, making the computational complexity of the inference linear with respect to the rule length L.
Each iteration of reasoning with unit-length rules comprises inference step (Equation 9) and rule-weight learning step (Equation 10). These steps require computing the matching probability for all possible entity pairs and relation pairs, respectively. As result, the computational complexity of the inference step and the weight updating step are ( and O(R[[('))) I), respectively.
Thus, the total computational complexity for reasoning with an L-length rule is O(L ((E11"") [R][ Given that entity sizes are typically much larger than relation sizes, this complexity can be approximated as O(L (8)
Notably, the computations involved in Equation 9 and Equation 10 can be accelerated through parallel processing, which we have implemented. This optimization reduces the runtime complexity to O (I. E(|E1) where n represents the number of CPU cores available for parallelization.
A.2.2 Parameter complexity
The total number alignment probabilities for all entity pairs is |E||E", which is large when the entity sizes increase. We adopt lazy inference strategy to enhance parameter efficiency. This strategy involves only saving the aligmment probabilities of the most probable alignments:
(el E, E E', Pw max
max Pue (V(),)))) max Pe ((((ee))))) (12) e'Eg!
EEE
Probabilities of other entity pairs can inferred from these saved alignment probabilities using Equation In this way, parameter complexity is reduced to O ([E) + (g")).
14

---

A.3 Pseudo-code of Explainer
Below is the pseudo-code of how the explainer generates supporting rules as interpretations for the query pair. It consists of two stages: searching reachable anchor pairs, and parsing rule paths as well as calculating rule confidences.
Algorithm I Generating Interpretations for the Queried Entity Pair with Weighted Rules
Inputs: Subrelation probabilities Prub(r C r'), Paub(r' S r) for r, E R; Knowledge Graph pair (9, g"): Maximum rule length L; Anchor pairs A with source-to-target mapping S2T and target-to-source mapping T28: Query entity pair (eq.
Outputs: Ranked rules based on confidence
Search Reachable Anchor Pairs within Max Depth
RN (e: G, C)
/+ Search reachable neighbors of eg using breadth-fire first search, max depth L +/
RN + BFS(C, , G¹, L)
/* Scarch reachable neighbors of e, using breadth- first scarch, max depth C t/
RN. + RN U (2S(RN":
/a Find souree nodes of reachable anchor pairs using hash mapping */ /* Identify reachable anchor pairs */
RA {(e, (e; A)) )
Parse and Rank Rules Based on Confidence
for e(e,e') ) E RA do Extract paths; ple, eq) =11 (c, =n A
if (p(e,e,) pe eg) then
path lengths don't match, rule confidence /
else
"p(e.e)). < I M(rr) - v(r.) Prnsf
products of subrelation probabilities and relation functionalities */ /* Compute rule confidence by
end end for Sort the rules (p.p') o by Wpyy in descending order Return the ranked rules
B Experimental details
B.1 Dataset statistics
The DBP15K dataset, designed cross-lingual knowledge graph alignment, has two versions: full and condensed. The original full version resembles real-world knowledge graphs, including comprehensive data across three language pairs with many sparsely connected low-degrec entities. The condensed version, derived by JAPE and adopted by later methods (GCNAlign, RREA, Dual- AMN, LightEA), removes these low-degree entities and their connected triples to create smaller (and higher average degree) dataset suitable for SCNN-ased methods. Detailed information about the two dataset versions can be found the "dataset" section of readme JAPE's s official implementatior
The dataset statistics of them are shown in Table Table In this paper, we adopt both versions of DBPI5K for comprehensive evaluation. Specifically, the full dataset sparser and larger in scale in scale due to the inclusion of low-degree entities, thus suitable for evaluating the models' robustness to sparsity and large scale.
Table 5: Data statistics of the full DBP15K dataset.
Datasets
KG
Entities
Relations
Rel. Triplets
Aligned Entity Pairs
Chinese (rh)
66,469
2,830
153,929
ZH-EN
English (en)
98,125
2,317
237,674
15,000
Japanese English (en)
65,744
2,043
164,373
JA-EN
15,000
95,680
2,096
233,319
French (fr)
66.858
1,379
192, 192,191 191
FR-EN
English (en)
105,889
2,209
278,590
15,000
15

---

Table 6: Data statistics of the condensed DBPI5K dataset.
Datasets
KG
Entities
Relations
Rel. Triplets
Aligned Entity Pairs
Chinese (2h)
19,388
1,701
70,414
ZH EN
15,000
English (en)
19,572
95,142
Japanese (ja) English (en)
19,814
1,299
77.214
JA-EN
15,000
19,780
1,153
93,484
French (fr) English (en)
19,661
903
105,998
FR-EN
15,000
19,993
1,208
115.722
B.2 Efficiency analysis
To provide a comprchensive understanding of the computational efficiency of our model, we report the runtime and memory usage during the experiments. The results, as summarized in Table Table 7 demonstrate that our model achieves efficient performance with a runtime consumption of 868 MB, and GPU memory usage of 4.33 GB. These metries highlight the practi- cality of our approach in terms resource utilization. Hardware configurations of the experiments presented in Table g
minutes, a memory
Table 7: Runtime and Memory Usage
Runtime
Memory
GPU Memory
15 minutes
868 MB
4.33 GB
Table 8: Machine configuration.
Component
Specification
GPU
NVIDIA GeForce RTX 3090
CPU
Intel(R) Xeon(R) Silver 4214R CPU @ 2.40GHz
16

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NeurIPS Paper Checklist
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Reviews:
[{"review_id": "XjLkGCR9gX", "reviewer": "Reviewer_J4kQ", "summary": "This research introduces NeuSymEA, a system for entity alignment between two knowledge graphs. The system combines symbolic reasoning (which is accurate and interpretable) with neural models (which handle large-scale noisy data well).\nPrevious work focused on single knowledge graphs, but NeuSymEA specifically handles alignment across different knowledge graphs (currently no more than two at a time).\nThe system treats alignment as a joint probability problem using a Markov random field with logical rule guidance. It uses a variational EM algorithm for training: the neural model predicts alignment probabilities (E-step), then the symbolic model updates rule weights (M-step). To speed up inference, long logic rules are broken into smaller steps.\nThe system also includes an explainer module that shows which logic paths support each predicted alignment. NeuSymEA achieves strong performance on benchmark datasets, especially with limited labeled data, demonstrating both effectiveness and interpretability.", "questions": "1) In M-step description the paper states that the model uses a greedy one-to-one matching strategy to reduce false positives during pseudo-label generation. However, this step appears to be a heuristic and is not rigorously analyzed. Could you elaborate on how NeuSymEA's performance is affected by this heuristic and which alternative approaches have you considered?\n\n2) I understand that the authors mentioned this following as limitation of their current work. Still, how do you envision NeuSymEA extending to multi-KG scenario? Could you briefly describe how such an extension could be designed and what would be the pain points?", "limitations": "The main limitation of this work (as mentioned by the authors) the fact that NeuSymEA is designed for pairwise alignment only and multi-KG cases are currently not supported.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n\nQuality\n\nThe proposed method is technically sound and well-motivated. The use of a variational EM algorithm to jointly optimize symbolic rule weights and neural parameters is principled and well-grounded in probabilistic modeling.\nThe paper provides a clever decomposition strategy for long logic rules, which significantly improves inference scalability without compromising symbolic reasoning depth.\nEmpirical evaluation is comprehensive and includes standard EA benchmarks (e.g., DBP15K) with both high-resource and low-resource settings, showcasing consistent gains over prior baselines.\n\nClarity\nThe overall structure is logical, with motivation, methodology, and experiments clearly stated.\nKey components such as the symbolic explainer, logical decomposition, and variational EM training are explained with sufficient technical detail.\nDespite the technical complexity, the authors make a commendable effort to keep the exposition readable, especially around the symbolic-neural interaction.\n\nSignificance\nThis paper tackles a key challenge: matching information across different knowledge sources, not just fill in missing facts within one.\nThe proposed method improves both accuracy and explainability—two things that usually don’t go well together in ML systems.\nIt also includes a feature (explainer component) that explains why the system made a match, which is especially useful in fields like medicine or language translation, where understanding the reasoning behind results is very important.\n\nOriginality\n\nThis work introduces a new way to combine logic rules with a probabilistic learning method to match entities across different knowledge graphs.\nIt also includes a smart trick: breaking down long and complex logic rules into smaller, simple steps to keep the system fast and manageable.\nFinally, it adds an explanation tool that works in two modes—one using only known matches, and another using both known and predicted matches—making the system’s decisions easier to understand and trust.\n\n\nWeaknesses\nQuality\n\nThe paper should better explain how inference quality is affected by pseudo-label noise in the early stages.\nThe one-to-one matching heuristic used to reduce false positives is not rigorously analyzed or justified — a more formal treatment or ablation would strengthen the argument.\n\nClarity\n\nCertain parts of the paper, especially those introducing dense notation (e.g., η, p_sub, rule decomposition), could be more clearly explained or accompanied by examples. In particular, equations sometimes appear without adequate narrative grounding.\nThe treatment of evaluation metrics across symbolic and neural models is briefly reconciled, but the equivalence assumption between recall and hit@1 could be more carefully justified.\n\nSignificance\n\nRight now, the method can only match two knowledge graphs at a time, and the authors don’t yet explain how it could be expanded to handle many at once.\nIt’s also not clear how well the system works when the graphs are very different in structure or topic, which would be useful to test.\nWhile the paper presents the idea of combining logic and learning as something new, some parts seem borrowed from earlier work. It would help if the authors made it clearer what parts are truly original.", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the detailed comments from reviewer, and address the two questions below:\n\n##  **[Q1]**\n\n```\nQuestion: In M-step description the paper states that the model uses a greedy one-to-one matching strategy to reduce false positives during pseudo-label generation. However, this step appears to be a heuristic and is not rigorously analyzed. Could you elaborate on how NeuSymEA's performance is affected by this heuristic and which alternative approaches have you considered?\n```\n\n\nThank you for your question, we would like to take this oppotunity to present the details here. Below, we first list two alternative approaches we have considered, then we show how our heuristic reduce false positives by showing the truth positive rate, finally show how the NeuSymEA's alignment performance is affected by this heuristic.\n\n\n\n### **1.1 Alternative approaches**\n\nWe include two algorithms: (1) Threshold: select entity pairs by using a preset threshold to select pairs with high matching scores; (2) Top1: each candidate entity along with its top1 matched entity is added to the pseudo-label set.\n\nThe shortage of these alternative solutions are: \n\n- The thresholding method is sensitive to the preset threshold, introducing burden to the hyperparameter tuning.\n- Both these algorithms can potentially generate conflict pairs (identical head or tail entities in different entity pairs).\n\nThe greedy one-on-one matching is essentially a top1 strategy that naturally avoids conflit pairs.\n\n\n\n### **1.2 The true positve rate (TPR) of pseudo-labels is increased by the one-on-one matching**\n\nTaking the `fr-en` dataset as an example, the true postive rate in the 5 steps are below.\n\n| Step                | 1     | 2     | 3     | 4     | 5     |\n| ------------------- | ----- | ----- | ----- | ----- | ----- |\n| Thresholding        | 0.452 | 0.604 | 0.639 | 0.655 | 0.654 |\n| Top1                | 0.731 | 0.826 | 0.839 | 0.847 | 0.848 |\n| One-on-One matching | 0.748 | 0.839 | 0.859 | 0.860 | 0.860 |\n\nWe can observe a significant improvement of truth postive rate by the one-on-one matching algorithm. And we also observe that the TPR of pseudo labels grows as the EM-steps process.\n\n\n\n### **1.3 Impact of the strategies on NeuSymEA's final Perofrmance** \n\n|                     | Hit@1     | MRR       |\n| ------------------- | --------- | --------- |\n| Thresholding        | 0.836     | 0.875     |\n| Top1                | 0.848     | 0.884     |\n| One-on-One matching | **0.858** | **0.894** |\n\nResults reveal that: One-on-one matching enhance the final performance with its precise pseudo-labels.\n\n---\n\n##  **[Q2]**\n```\nQuestion: I understand that the authors mentioned this following as limitation of their current work. Still, how do you envision NeuSymEA extending to multi-KG scenario? Could you briefly describe how such an extension could be designed and what would be the pain points?\n```\n\nWe envision extending NeuSymEA to multi-KG scenario with two directions of design: (1) recursive pair-wise KG alignment; and (2) simultaneous multi-KG alignment in a unified space. Below we detail the two designs and their painpoints.\n\n### **2.1 Recursive pair-wise alignment**\n\nThe first design is naive. Simply apply the current NeuSymEA framework to perform entity alignment between every KG pair, finally recursively merge these multiple KGs. \n\n**The painpoints in the first design** are: (1) The number of KG pairs grows quadratically to the number of KGs, presenting efficiency and effectiveness challenges in scheduling the recursion. (2) Conflicts among alignments. For instance, the model infers three alignment pairs $(e^1_i=e^2_i), (e^2_i= e^3_i), (e^3_i= e^1_j)\\text{ for three KGs } G^1, G^2, G^3$, there is an conflict if $i\\neq j$.\n\n\n\n### **2.2 Simultaneous multi-KG alignment** \n\nThe second direction is more ambitious and theoretically appealing to jointly align all KGs in a unified probabilistic framework. This requires several key modifications:\n\n**Modification 1.** Generalized alignment variables\n\nInstead of binary variables $v\\left(e, e^{\\prime}\\right) \\in\\{0,1\\}$, we would define alignment clusters: each entity belongs to one equivalence class (i.e., same real-world entity). This leads to a clustering-based formulation rather than pairwise binary classification.\n\n**Modification 2.** Multi-source symbolic rule inference:\n\nCurrent rules involve relational paths in two KGs (e.g., $p, p^{\\prime}$ ). In multi-KG settings, we must reason over triplets of paths (or more), i.e., $p_{i j}, p_{j k}, p_{i k}$ across $G_i, G_j, G_k$ and respect transitive consistency: i.e., if $e_i \\equiv e_j$ and $e_j \\equiv e_k$, then $e_i \\equiv e_k$.\n\n**Modification 3.** Joint probabilistic model across all KGs:\n\n- Extend the Markov random field to include higher-order potentials that model transitive consistency among multiple KGs. \n- The neural model would also need to be extended to learn shared embedding space across all KGs.\n\n**The painpoint in the second direction** is that: the increased heterogeneity in multiple KGs, challenging the robustness of the extended framework. \n\n\n\nAbove is how we envision extending NeuSymEA to multi-KG scenario, in terms of both the potential design and painpoints. We hope this vision can inspire future research on scalable and robust multi-KG alignment."}, {"review_id": "rsv5QJFDL4", "reviewer": "Reviewer_E7Dm", "summary": "The paper proposes a variational EM method to align two sets of entities from two knowledge bases/graphs. The idea is to have two entity-matching score models: NN model parameterizing a score function, and symbolic model relying on rules. The two models are trained together using a variational EM method. The paper shows that the proposed approach outperforms several baselines (NN and symbolic) on several benchmarks.", "questions": "Please see above.", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\nThe paper shows strong results on several benchmarks. \n\n## Weaknesses\n\n* The paper is quite difficult to read, and especially to check the correctness of the EM. For instance, why optimizing (4) & (5) will have the same effect of minimizing the KL? Could the authors explain the intuition behind (9, 10, 11)? why do they make sense?\n\n* Following the \"heuristic\" of selecting \"positive\" examples, and the trick for equation (4), in the E and M steps, I'm not pretty sure whether this is EM (could the authors prove that the log-likelihood of the observed data increases?). Instead, I think this is more like a co-training approach because:\n   * in step E: the symbolic model annotate data for the NN model\n   * in step M: the NN model annotate data for the symbolic \n\n* In section 4.1, it's unclear to me how a new rule is created. Do we have to consider all possible combinations of unit rules? \n\n* Line 217 \"NeuSymEA unifies them under a joint probability objective.\" -- which \"joint probability\" is that? \n\n* One baseline is PRASE. However, this baseline, from the original paper, is unsupervised learning. Then is it fair for comparison? Also, from the PRASE paper, different benchmarks are used. How well does the proposed approach perform on those benchmarks?", "was_revised": true, "final_justification": "The authors' response addresses my concerns, especially two of them:\n1. the authors showed that their approach is EM (rather than simply co-training), though employing some heuristics to make the approach work. \n2. the authors showed that the comparison with PRASE if fair.", "rebuttal": "We sincerely appreciate the time and effort the reviewer has invested in reviewing our paper and providing detailed comments. We address the concerns in the following.\n\n---\n##  **[Q1]**\n```\nThe paper is quite difficult to read, and especially to check the correctness of the EM. For instance, why optimizing (4) & (5) will have the same effect of minimizing the KL? Could the authors explain the intuition behind (9, 10, 11)? why do they make sense?\n```\n\nWe answer these two questions one by one.\n\n### **1.1 Eq. (4) & (5)**\n\nOptimizing Eq. (4) &. (5) approximates minimizing the KL divergence between the variational distribution $q_\\theta$ and the true posterior $p_w\\left(v_H \\mid v_O\\right)$. This is because:\n\n- Equation (3) shows the evidence lower bound (ELBO) formulation:\n\n$$\n\\log p_w\\left(v_O\\right) \\geq E_{q\\left(v_H\\right)}\\left[\\log p_w\\left(v_O, v_H\\right)\\right]-E_{q\\left(v_H\\right)}\\left[\\log q\\left(v_H\\right)\\right]\n$$\n\n- Optimizing ELBO is equivalent to maximizing the log-likelihood (lower bounded by ELBO) while minimizing the KL divergence.\n- However, we points out that directly minimizing the KL is intractable, so we instead optimize a *reverse* KL via a sampling-based surrogate (Eq. (4)):\n\n$$\n\\sum_{\\left(e, e^{\\prime}\\right)} E_{p_w\\left(v\\left(e, e^{\\prime}\\right) \\mid v_O\\right)} q_\\theta\\left(v_H\\right)\n$$\n\n- **Intuition**: To optimize the above objective, the symbolic model generates high-confidence pairs $\\rightarrow$ these are treated as positives to supervise the neural model $q_\\theta$. The neural model thus approximates the true posterior $p_w\\left(v_H \\mid v_O\\right)$, closing the KL gap indirectly. Equation (5) simply adds supervision from observed data to stabilize training.\n\n### **1.2 Eq. (9) & (10) & (11)**\n\nAllow us to clarify any confusion. These three equations describe the symbolic update process using decomposed rules:\n\n **Eq. (9): Inference via neighbor aggregation**\n\n- This computes the alignment probability $p_w\\left(v\\left(e, e^{\\prime}\\right)\\right)$ by aggregating support from neighboring aligned pairs $\\left(e_t, e_t^{\\prime}\\right)$ via one-step (unit-length) rules.\n- Each neighbor contributes a confidence score based on:\n\n  - $\\eta(r)$ : the relation pattern of $r$, quantifying its uniqueness (e.g., 1-to-1 vs many-to-many).\n\n  - $p_{\\text {sub }}\\left(r \\subseteq r^{\\prime}\\right)$ : the subrelation probability, i.e., how likely $r$ is structurally or semantically a subrelation of $r^{\\prime}$.\n\n\n- The product form ensures that support from multiple neighbors is combined in a noisy-OR fashion: as more confident neighbors exist, the overall alignment probability increases.\n\nThis equation reflects a forward chaining deduction, where alignments propagate from neighbors under weighted logical rules.\n\n**Eq. (10): Updating subrelation probabilities**\n\n- This equation updates the confidence that $r \\subseteq r^{\\prime}$ holds across the two KGs.\n- It does so by measuring how often aligned entity pairs co-occur under matching relation pairs.\n\nThe result is a data-driven estimation of relation-level correspondence across KGs.\n\n**Eq. (11): Rule confidence via composition**\n\n- This computes the final confidence score of a rule $p, p^{\\prime}$ of length $L$ by multiplying the confidences of each unit step.\n\n\n---\n##  **[Q2]**\n\n```\nFollowing the \"heuristic\" of selecting \"positive\" examples, and the trick for equation (4), in the E and M steps, I'm not pretty sure whether this is EM (could the authors prove that the log-likelihood of the observed data increases?). Instead, I think this is more like a co-training approach...\n```\n\nThank you for your insightful comments, and allow us to clarify any confusion. While our E/M steps may resemble co-training heuristics in form, our method is theoretically grounded in variational EM.\n\nSpecifically, we formalize a latent-variable model where the truth scores $v_H$ are hidden variables, and model the joint probability $p_w(v_O, v_H)$ over observed and hidden variables(truth scores). Our E-step updates the variational distribution $q_\\theta$ to approximate the true posterior, and the M-step maximizes the expected complete log-likelihood with respect to $w$, given $q_\\theta$. This alternating optimization provably increases the ELBO (a lower bound of the true likelihood), which satisfies the formal criteria of EM.\n\nI acknowledge that the use of heuristics such as greedy one-to-one filtering in the E-step may blur the boundary. But these heuristics are used only for practical implementation of pseudo-labeling and do not violate the variational EM principle, since the neural model is still trained to minimize divergence from symbolic inference. Fig2 shows proxy metrics (neural MRR, symbolic precision) improve over iterations.\n\n\n---\n##  **[Q3]**\n```\nIn section 4.1, it's unclear to me how a new rule is created. Do we have to consider all possible combinations of unit rules?\n```\n\nThe rule creation process is clarified in Equations (7)-(8):\n\n- A long rule of length $L$ is decomposed into a conjunction of $L$ unit-length subrules.\n- During optimization, only unit-length rules are explicitly learned, and longer rules are constructed by deductive composition.\n\nThe symbolic model doesn't consider all combinations exhaustively. Instead, we exploit the local sparsity of KGs to mine rules efficiently:\n\n- For each candidate alignment ( $e, e^{\\prime}$ ), it uses breadth-first search to find matching paths of length $\\leq$ $L$.\n- Each matched pair of paths forms a potential rule $p, p^{\\prime}$, and its confidence is computed using Equation (11).\n\nThus, the search is constrained to reachable anchor pairs and their paths, avoiding exhaustive enumeration. As a result, the number of rules mined is a small number (as shown in middle & right in Fig. 4).\n\n---\n##  **[Q4]**\n```\nLine 217 \"NeuSymEA unifies them under a joint probability objective.\" -- which \"joint probability\" is that?\n```\n\nThis joint probability refers to:\n\n$\np_w\\left(v_O, v_H\\right)\n$\n\nmodeled via a Markov Random Field as described in Section 3:\n\n- The Markov Random Field captures interactions among all entity pairs via symbolic rules.\n- This joint distribution is optimized by variational EM (Equation 3), making it the central objective function unifying symbolic and neural inference.\n\nThis \"joint probability\" represents the coherent agreement between symbolic rule-based reasoning and neural embeddings, governed by shared latent variables $v_H$.\n\n\n---\n##  **[Q5]**\n```\nOne baseline is PRASE. However, this baseline, from the original paper, is unsupervised learning. Then is it fair for comparison? Also, from the PRASE paper, different benchmarks are used. How well does the proposed approach perform on those benchmarks?\n```\n\nWe respectfully clarify the **fairness** of the experimental setting by comparing our methods under both their \"unsupervised\" setting and our supervised setting.\n\n### **5.1 Unsupervised setting with text attributes for alignment.**\n\nDifferent from our setting that only structure information is used, the \"unsupervised\" setting in PRASE uses text attributes as **extra information**. Specifically, they treat attribute values as entities and exploit the same attribute values as seed alignments to guide the entity alignment. \n\nNeuSymEA can also be run by constructing attributed KGs, and enable unsupervised learning. We attach the hit@1 performance under this setting below, on the benchmark datasets they adopted (D-W-15K, D-W-100K, D-Y-100K, EN-FR-100K, EN-DE-100K), for comprehensive comparison.\n\n| Hit@1    | D-W-15K | D-W-100K | D-Y-100K | EN-FR-100K | EN-DE-100K |\n| -------- | ------- | -------- | -------- | ---------- | ---------- |\n| PRASE    | 0.875   | 0.804    | 0.993    | 0.930      | 0.955      |\n| NeuSymEA | 0.920   | 0.845    | 0.994    | 0.942      | 0.956      |\n\n\n\n### **5.2 Our experimental setting enable fair comparison by supervised learning of PRASE with same input.**\n\nOur setting is a more general setting where structure information is the main information and no attributes are available. In this case, PRASE utilize the **same input** (two KGs, and a set of seed alignments)  as NeuSymEA for supervised learning, rather than unsupervised, ensuring fair comparison.\n\n| Hit@1    | JA-EN | FR-EN | ZH-EN |\n| -------- | ----- | ----- | ----- |\n| PRASE    | 0.611 | 0.647 | 0.652 |\n| NeuSymEA | 0.811 | 0.858 | 0.804 |\n\n\n\nThank you for taking the time to review our response. We hope it has clarified any confusion and addressed your concerns."}, {"review_id": "JYBlS0c4Tr", "reviewer": "Reviewer_xG55", "summary": "This paper introduces NeuSymEA, a neuro-symbolic framework for entity alignment in knowledge graphs. The authors propose a unified model that leverages a variational EM algorithm to combine the strengths of neural and symbolic reasoning. The key ideas include modeling the joint probability of alignments in a Markov random field and using a novel logical deduction method to efficiently handle long-range rules. The experimental results on several benchmark datasets demonstrate the effectiveness and interpretability of the proposed approach.", "questions": "See the Weakness.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n1.  The unification of neural and symbolic methods via a variational EM algorithm is elegant and effective. By optimizing a single joint probability distribution, the framework allows the neural (E-step) and symbolic (M-step) components to mutually enhance one another. This principled approach creates a more coherent and powerful model compared to methods that treat the components separately.\n\n2.  The use of logical deduction to improve efficiency is a significant contribution. By decomposing long rules into unit-length sub-rules, the method avoids the exponential search space typically associated with symbolic reasoning. This makes the inference complexity linear with respect to rule length, allowing the model to leverage complex, multi-hop patterns in a scalable manner.\n\n3.  The framework demonstrates excellent performance, achieving state-of-the-art results on benchmark datasets and showing impressive robustness in low-resource scenarios. A key strength is its built-in explainer, which provides rule-based justifications and confidence scores for its predictions. This combination of high accuracy and strong interpretability is a major advantage.\n\n**Weaknesses:**\n\n1.  The comparison could be strengthened by including more recent baselines. The latest model in the comparison is from 2023.\n\n2.  The paper lacks a direct, head-to-head efficiency comparison with its baselines. While an analysis of NeuSymEA's own scalability and resource usage is provided, there is no data comparing its runtime, memory, or GPU consumption against key competitors. This information is essential for fully evaluating the practical performance trade-offs of the framework.", "was_revised": true, "final_justification": "The paper is solid. I decide to maintain my positive score.", "rebuttal": "---\n##  **[Q1]**\n```\nThe comparison could be strengthened by including more recent baselines. The latest model in the comparison is from 2023.\n```\n\nThank you for the constructive suggestion, we include two baselines, ASGEA[1] and TFP[2] in the last two years to strengthen the comparison. Results are below. \n\nNote that these two algorithms adopt the 3:7 (train: test) split of datasets, different from our experiment setting in the manuscript. Therefore, we align our experimental setting with theirs in the comparison below.\n\n|           | ZH-EN |       | JA-EN |       | FR-EN |       |\n| --------- | ----- | ----- | ----- | ----- | ----- | ----- |\n|           | Hit@1 | MRR   | Hit@1 | MRR   | Hit@1 | MRR   |\n| ASGEA [1] | 0.560 | 0.660 | 0.595 | 0.690 | 0.653 | 0.745 |\n| TFP [2]   | 0.816 | 0.868 | 0.812 | 0.868 | 0.853 | 0.899 |\n| NeuSymEA  | 0.835 | 0.870 | 0.829 | 0.868 | 0.872 | 0.905 |\n\n\n\n- [1] ASGEA: Exploiting Logic Rules from Align-Subgraphs for Entity Alignment, CoRR, 2024\n- [2] Rethinking Smoothness for Fast and Adaptable Entity Alignment Decoding, NAACL 2025\n\n\n\n##  **[Q2]**\n```\nThe paper lacks a direct, head-to-head efficiency comparison with its baselines. While an analysis of NeuSymEA's own scalability and resource usage is provided, there is no data comparing its runtime, memory, or GPU consumption against key competitors. This information is essential for fully evaluating the practical performance trade-offs of the framework.\n```\n\n\nBelow, we compare the efficiency with strong baselines on the `fr-en` dataset. Here, we adopt the lightEA as our base neural model.\n\n|            | GPU Mem(/MB) | Mem(/MB) | Run Time(s) | Hit@1 accuracy |\n| ---------- | ------------ | -------- | ----------- | -------------- |\n| RREA       | 8903         | 2330     | 2605        | 0.797          |\n| Dual-AMN   | 9002         | 4009     | 96          | 0.793          |\n| LightEA    | 1328         | 3801     | 8           | 0.827          |\n| NeuSymEA-L | 1328         | 4153     | 101         | 0.858          |\n\nResults show:\n\n- **GPU Memory-consumption**: NeuSymEA's GPU memory usage is comparable to that of the base neural model it employs, enabling it to be as efficient as highly optimized models like LightEA.\n- **RAM memory consumption**: While memory usage depends on the neural model, the loaded dataset, and the symbolic model, NeuSymEA maintains acceptable overhead by avoiding the creation of large intermediate variables during processing.\n\n- Its **runtime** is slower than Dual-AMN and LightEA, but faster than RREA. The major reason is that neuro-symbolic reasoning generally has a symbolic reasoning process executed on CPUs, which is slower than the neural model that is executed on GPUs. We have enabled parallel computation and batch processing in our implementation of the symbolic model (Appendix A.2), so theoretically, it can be further accelerated by using a larger batch size and more parallel processes.\n\nThank you again for your detailed review. We sincerely appreciate your constructive comments that help us improve this manuscript, and we will include these results in the final version."}, {"review_id": "lmzQOB8Ncw", "reviewer": "Reviewer_KxkU", "summary": "This paper proposes a principled neuro-symbolic reasoning framework via variational EM. It provides efficient optimization through logical decomposition, allowing effective reasoning over large knowledge graphs. The paper also proposes interpretable inference with Hard-anchor and Soft-anchor modes and provides empirical validation demonstrating NeuSymEA’s high alignment accuracy and informative rule-based interpretations on benchmark datasets.", "questions": "1. LLMs contain large amount of world knowledge and thus may be easily applied to the EA task. Why didn't you consider this as a baseline or additional source of information in the framework?\n2. eq. 1 and 2: It's unclear how the confidence and probability is inferred in the symbolic setting.\n3. line 194: What exactly does the split mean? is it a split over entities or relations? how would it work if you apply something learned from one dataset to another?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Quality: The proposed neuro-symbolic reasoning framework is derived in a principled way, although some assumptions and simplifications are applied. The experimental results seem to show the effectiveness of the proposed approach.\n\nClarity: Mostly clear but can benefit from additional information.\n\nSignificance: KG is no longer something critical in the LLM era. For this reason, the work is less important than before. In addition, If LLM is used, the performance of entity alignment may be further improved.\n\nOriginality: The proposed neuro-symbolic reasoning framework, although still falls into a standard framework, is sufficiently novel in the way the loss is selected and the optimization process is conducted.", "was_revised": true, "final_justification": "The authors answered my questions and justified my rating.", "rebuttal": "Thank you for your detailed review and the constructive questions. Below are our responses. \n\n##  **[Q1]**\n```\nLLMs contain large amount of world knowledge and thus may be easily applied to the EA task. Why didn't you consider this as a baseline or additional source of information in the framework?\n```\n\nWe first expain why no extra source of information is considered in this framework, then discuss the LLM-based method that exploits extra information and LLMs' knowledge, finally compare our work with LLM-based methods.\n\n### **1.1 Why not consider additional information in the framework**\n\nIn this work, we focus solely on structural information within our framework, as it represents a more general and widely applicable setting for the entity alignment task. In the EA field, structural signals are considered the **foundational resource** for alignment, enabling applicability even in low-resource or text-scarce scenarios.\n\n\n\n### **1.2 Discussions of LLM-based entity alignment**\n\nWhile LLMs do offer significant potential for EA, especially in datasets rich with textual attributes. Recent studies have revealed the resource-intensive challenges in this research line. Specifically, (1) The **$O(|\\mathcal{E}|^2)$** annotation space makes **LLM-API-based methods** costly; (2) The large parameter size of LLMs poses scalability limitations for **locally-deployed LLM-based methods**.\n\n\n\n### **1.3 Comparison between NeuSymEA and LLM-based entity alignment**\n\nIt is worth noting that our approach is **orthogonal** to LLM-based methods and can in fact **complement their limitations**. As shown in Table 2 of the manuscript, NeuSymEA is particularly effective in low-resource settings, making it well-suited to leverage LLM-generated pseudo-labels to improve alignment performance.\n\n\n\nTo empirically support this, we compare NeuSymEA with a recently published LLM-based method [1] by\n\n(1) adopting the annotated labels they generated; and \n\n(2) the text attributes for lexical-based simlarity intialization in the symbolic component\n\n|                     | D-W-15K |       | D-Y-15K |       | EN-FR-15K |       | EN-DE-15K |       |\n| ------------------- | ------- | ----- | ------- | ----- | --------- | ----- | --------- | ----- |\n|                     | Hit@1   | MRR   | Hit@1   | MRR   | Hit@1     | MRR   | Hit@1     | MRR   |\n| LLM4EA [1]          | 0.742   | 0.810 | 0.891   | 0.926 | 0.875     | 0.909 | 0.977     | 0.983 |\n| NeuSymEA+labels     | 0.887   | 0.903 | 0.924   | 0.933 | 0.892     | 0.914 | 0.976     | 0.981 |\n| NeuSymEA+attributes | 0.920   | 0.943 | 0.985   | 0.989 | 0.981     | 0.987 | 0.970     | 0.980 |\n\n- [1] LLM4EA: Entity alignment with noisy annotations from large language models. NeurIPS, 2024.\n\n\n\n---\n\n##  **[Q2]**\n```\neq. 1 and 2: It's unclear how the confidence and probability is inferred in the symbolic setting.\n```\n\n\nEquations (1) and (2) introduce the formulation of symbolic inference rules and alignment probabilities, and the subsequent sections (particularly Equations 9–11) explain how they are instantiated.\n\n#### **Eq. (1)** introduces rule templates:\n\n$$\nw_{p, p^{\\prime}}:\\left(e_j \\equiv e_j^{\\prime}\\right) \\wedge p\\left(e_i, e_j\\right) \\wedge p^{\\prime}\\left(e_i^{\\prime}, e_j^{\\prime}\\right) \\Rightarrow\\left(e_i \\equiv e_i^{\\prime}\\right)\n$$\n\n- Here, $p$ and $p^{\\prime}$ are relational paths in source and target KGs .\n- $w_{p, p^{\\prime}}$ is the confidence of the rule, computed compositionally in Eq. (11).\n\nEssentially, this template formulates how a new alignment is inferred in a symbolic model: they have a pair of aligned neighbors, and they have a pair of similar relational paths to these neighbors. The confidence of this rule support is related to the similarity of this path pair.\n\n\n\n**Eq. (2) defines alignment probability**:\n\nEq. (2) is a global modeling of the alignment probability, by aggregating all the rules in the format of Eq. (1):\n$$\np_w\\left(v\\left(e, e^{\\prime}\\right) \\mid G, G^{\\prime}\\right)=\\text { aggregated support from symbolic rules }\n$$\n\nPractically, this is computed using Equation (9), which aggregates the confidence scores of all symbolic rules supporting a candidate pair $(e, e')$ via aligned neighbors.\n\n\n---\n##  **[Q3]**\n```\nline 194: What exactly does the split mean? is it a split over entities or relations? how would it work if you apply something learned from one dataset to another?\n```\n\n\nWe first clarify the meaning of the split in the experimental setting, and discuss how framework generalize from one dataset to another.\n\n### **3.1 Clarification of the split**\n\nThe split refers to a split over aligned entity pairs, i.e., ground truth labels.\n\n- It is not a split over entities or relations individually, but over the alignment labels between entities from the two KGs.\n- For example, in DBP15K there are 15,000 ground-truth aligned entity pairs. A 3:7 split means 4,500 are used for training and 10,500 for testing. \n\n\n\n### **3.2 Transferability across datasets**\n\n- The rules and relation patterns (e.g., $\\eta(r), p s u b\\left(r \\subseteq r^{\\prime}\\right)$ ) are KG-specific, as they depend on the structure and semantics of relations in each dataset.\n- Therefore, the symbolic rule weights are not directly transferable between datasets with different schemas or relation vocabularies.\n- However, the general mechanism (logic deduction, confidence composition, rule mining process) is dataset-agnostic and can be applied across datasets. What's learned is specific to each dataset, but the framework generalizes.\n\n\nWe appreciate you time and effort in reviewing our feedback. And we hope it addresses your concerns."}]

Author rebuttal:

Meta-review: NeuSymEA presents a unified neuro-symbolic framework for pairwise entity alignment (EA) between knowledge graphs.  It models the joint probability of all candidate alignments via a Markov random field whose potentials are defined by first-order logical rules. The paper unifies historically disjoint neural and symbolic EA communities in a single, theoretically grounded framework, achieving accuracy, robustness, and interpretability simultaneously—an important milestone for trustworthy knowledge-graph integration.
Decision: Accept (poster)