(2024)

Abstract: Taxonomy completion, enriching existing taxonomies by inserting new concepts as parents or attaching them as children, has gained significant interest. Previous approaches embed concepts as vectors in Euclidean space, which makes it difficult to model asymmetric relations in taxonomy. In addition, they introduce pseudo-leaves to convert attachment cases into insertion cases, leading to an incorrect bias in network learning dominated by numerous pseudo-leaves. Addressing these, our framework, TAXBOX, leverages box containment and center closeness to design two specialized geometric scorers within the box embedding space. These scorers are tailored for insertion and attachment operations and can effectively capture intrinsic relationships between concepts by optimizing on a granular box constraint loss. We employ a dynamic ranking loss mechanism to balance the scores from these scorers, allowing adaptive adjustments of insertion and attachment scores. Experiments on four real-world datasets show that TAXBOX significantly outperforms previous methods, yielding substantial improvements over prior methods in real-world datasets, with average performance boosts of 6.7%, 34.9%, and 51.4% in MRR, Hit@1, and Prec@1, respectively.

heading: 1 Introduction
text: Taxonomy, a critical knowledge graph with an "isa" relationship, plays a vital role in information retrieval, recommendation systems, and question answering (Chatterjee and Das, 2022; Chuang and Chien, 2003; Kejriwal et al., 2022; Kerschberg et al., 2001; Suchanek et al., 2007; Huang et al., 2019; Yang et al., 2017; Yu et al., 2021). However, manual taxonomy enrichment is inefficient and costly due to the constant emergence of new concepts. To address the challenge of incorporating new concepts, taxonomy completion has been
† Corresponding author.
introduced, with new concepts either inserted as both parents and children or attached only as children (Jiang et al., 2022; Zhang et al., 2021; Wang et al., 2022; Zeng et al., 2021). This task goes beyond taxonomy expansion, which primarily treats new concepts as leaf nodes and tends to have limitations in downstream applications (Shen et al., 2020; Liu et al., 2021; Yu et al., 2020; Manzoor et al., 2020; Phukon et al., 2022; Jiang et al., 2023).
Taxonomy completion entails a more comprehensive incorporation of new concepts with two operations: insertion and attachment. For instance, in Figure 1, new query concepts such as cat and insect are added to the existing animal taxonomy. The process requires enumerating all possible candidate positions within the original taxonomy, including existing edges like <Animal, Vertebrate> and implicit edges from each node to its descendants such as <Animal, Tiger>. Each candidate position is then paired with the query concept, and a confidence score is calculated. Finally, insect is
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attached as a child of animal and cat is inserted as a parent of Siamese cat and children of Domestic Animal and Vertebrate according to their confidences.
Recent research on taxonomy enrichment has examined various practical methods (Jiang et al., 2022; Zhang et al., 2021; Wang et al., 2022; Zeng et al., 2021). Nevertheless, all of these approaches embed concepts as vectors in Euclidean space, which makes them less capable of modeling the asymmetric relationship ("is-a") in taxonomy. BoxTAXO (Jiang et al., 2023) tried to employ box embedding, a representation method that can capture more prosperous and asymmetric relationships like inclusion, disjoint, and proximity among concepts through its geometric properties. However, this method is limited in real-world applications for its reliance only on the volume property, rendering it suitable only for the taxonomy expansion and even incapable of discerning optimal ancestor concepts and handling multiple parents during inference. Moreover, methods for taxonomy completion (Zhang et al., 2021; Wang et al., 2022) suffer from using a "pseudo-leaf" as a child node in attachment cases, leading to confusion in the matching. It is attributed that attachment cases often predominate due to leaf nodes’ prevalence in real taxonomies. Therefore, learning too much about the pseudo-leaf in the attachment cases may reduce the network’s perceptual ability for child nodes in the insertion cases.
To overcome these limitations, we present a novel framework for taxonomy completion called TAXBOX, which is the first to apply box embedding to taxonomy completion. This approach adopts a structurally enhanced box decoder, representing concepts as box embeddings (Vilnis et al., 2018) encompassing the information of children, furnishing richer semantics. Most importantly, TAXBOX combines two probabilistic scorers to unify the process of insertion and attachment in the box embedding space and incorporates both the volume and center closeness properties of box embedding. Such a design effectively exploits the fine-grained geometric attributes of box embeddings, circumventing the need for a pseudo-leaf and yielding optimal, feasible results during the ranking process. Additionally, we propose two novel training objectives, optimizing both box volume and position, and rectifying scorer numerical imbalances.
The specific contributions of this paper are outlined as follows:
• We introduce TAXBOX, the first framework using box embedding for taxonomy completion with a structurally enhanced box decoder.
• We establish insertion and attachment scorers, obviating the need for pseudo-leaves and ensuring the determination of optimal results.
• We design box constraint loss, focusing on both volume and center closeness, and dynamic ranking loss, rectifying scorer numerical imbalance.
• Experimental outcomes from four datasets demonstrate our model’s efficacy, achieving 6.7% MRR, 34.9% Hit@1, and 51.4% Prec@1 improvements over the previous methods.

heading: 2 Related work
text: Taxonomy Expansion and Completion. Taxonomy expansion, the process of attaching novel concepts into an existing taxonomy, has evolved over time with various approaches (Shen et al., 2018, 2020; Yu et al., 2020; Manzoor et al., 2020; Liu et al., 2021; Ma et al., 2021; Phukon et al., 2022; Jiang et al., 2023). Although effective, these methods have limitations in addressing real-world applications. Thus, Zhang et al. (2021) introduced taxonomy completion, a generalization that allows for the insertion of a concept as a parent to existing nodes, generating wider-reaching solutions. Subsequent research (Wang et al., 2022; Jiang et al., 2022; Zeng et al., 2021) sought to tackle this more challenging version of taxonomy expansion. Jiang et al. (2022) incorporated contextual embeddings into input embeddings, leveraging dual LSTMs to encode ancestor and descendant information (Staudemeyer and Morris, 2019). Meanwhile, Zeng et al. (2021) devised a generative strategy that concurrently generates concept names and classifies valid candidate positions. Wang et al. (2022) introduced the Quadruple Evaluation Network (QEN), which utilized pretrained language models (PLM) (Devlin et al., 2018; Sanh et al., 2019) to augment initial embeddings with semantically rich term representations. Arous et al. (2023) learns a positionenhanced node representation through anchor sets to better find the candidate. Box Embedding. Box embedding represents a mapping technique that embeds concepts or objects
within hyperplane boxes. Initially proposed by Vilnis et al. (2018), this approach employs probabilistic box lattices to encapsulate entities in knowledge graphs as n-dimensional rectangles. Subsequently, various studies have applied box embedding across diverse domains. For instance, Rau et al. (2020) predicted visual overlap in images, while Onoe et al. (2021) and Patel et al. (2021) focused on entity typing and multi-label classification, respectively. Moreover, Dasgupta et al. (2022) mapped words to capture set-theoretic semantics, and Hwang et al. (2022) and Messner et al. (2022) explored relation extraction and knowledge graph completion. These works highlight box embedding’s suitability for nuanced semantic relationship modeling.

heading: 3 Preliminary
text: Box embedding (Vilnis et al., 2018; Chheda et al., 2021) refers to a mapping that represents a concept or object as a hyperplane box. A box x = [xm, xM ] is a hyperrectangle such that xm ∈ Rd and xM ∈ Rd where xm and xM represent the minimum and maximum endpoints of the box respectively along the d axis and xm,i ≤ xM,i holds for each axis i ∈ {1, 2, ..., d}. The center of box embedding is formulated as:
Cen(x) = xM + xm
2 (1)
There are two important operations: Intersection and Volume which are required for the calculation of the conditional probability of boxes’ containment. Given two box embedding x = [xm, xM ], y = [ym, yM ], the Intersection of them is defined
as follows:
Inter(x, y) = [max(xm, ym),min(xM , yM )] (2)
where min(·, ·) and max(·, ·) in Equation 2 perform element-wise operations. Specifically, min(a, b) = [min(a1, b1), ...,min(ad, bd)], and similarly for max(·, ·). The Volume is defined as:
Vol(x) = d∏
i=1
τ∗softplus(xMi − xmi τ )
softplus(a) = log(1 + exp a)
(3)
where τ is a hyperparameter to adjust the smoothness. The probability of box x containing box y or the conditional probability of x given y is:
Pr(x|y) = Vol(Inter(x, y)) Vol(y)
(4)

heading: 4 The TAXBOX Framework
text: In this section, we elaborate on the proposed TAXBOX framework, as shown in Figure 2. We begin by defining the problem in Section 4.1. Then, in Section 4.2, we introduce the structurally enhanced box decoder, which maps concepts into box embeddings with hierarchical information enhanced. Section 4.3 focuses on the discussion of two probabilistic scorers that evaluate the query and candidate boxes, providing attachment and insertion scores. Finally, in Section 4.4, we elucidate the learning objectives that contribute to improved optimization of box decoding and scorer balancing.

heading: 4.1 Problem Definition
text: A taxonomy is a directed acyclic graph and is defined as T 0 = (N 0, E0) where each node n ∈ N 0 represents a concept and each edge ⟨p, c⟩ ∈ E0 represents the "is-a" relationship edge between concepts. Given a seed taxonomy T 0 and a set of new concepts C, the definition of taxonomy completion is to construct a new taxonomy T = (N , E) where N = N 0 ∪ C and E is updated by adding new edges among C and N 0. To fulfill the task, all the candidate positions P = {⟨p, c⟩|∀p ∈ N 0, ∀c ∈ descendants(p)} have to be evaluated given a novel concept n ∈ C. The whole training paradigm follows self-supervised learning. For each node in the seed taxonomy, we pretend it to be a query and optimize it with a reconstructed taxonomy without the node.

heading: 4.2 Structurally Enhanced Box Decoder
text: The structurally enhanced box decoder includes a graph aggregation module to aggregate the hierarchical features from the ego subtree, as well as two box projectors map aggregated features and query embedding to box embedding space, respectively. An ego subtree of node n is defined as a tree only containing n and its one-hop children, denoted by T(n).
For a query q and a possible candidate ⟨p, c⟩ ∈ P , we first obtain the embedding of each concept in the candidate along with their hierarchical information. As illustrated in Figure 3, we design a graph aggregation module to achieve this. The formulation is given by Equation 5:
Fk =Lin(R(GAT(T(k)) + T(k))), k ∈ {p, c} (5)
where Fk is the aggregated feature and R(·) is a readout method, which implies that we only read out the root embedding of an ego subtree. Lin denotes a linear layer with activation. To effectively fuse more information from relevant child
nodes, we opt for GAT (Graph Attention Network) (Veličković et al., 2018) to aggregate these trees in our implementation.
Next, two box projectors with identical Highway network(Srivastava et al., 2015) structure project aggregated features and query embedding to box embeddings, respectively, as formulated in Equation 6. To avoid potential conflicts arising from different latent spaces, we do not use a shared weight module for the aggregated parent/child features and query embedding.
Bq = QProjector(Fq) Bk = CProjector(Fk), k ∈ {p, c} (6)
where Fq denotes query embedding and Bq, Bp, Bc represent the box embedding of query, candidate parent, and candidate child, respectively. QProjector is the query box projector, and CProjector is the candidate box projector.

heading: 4.3 Insertion and Attachment Scorer
text: To make the best use of the geometric properties of box embedding like volume and center closeness, we design insertion scorer and attachment scorer to separately give confidence corresponding to these two cases. Insertion Scorer. Assumes that our model captures fine-grained semantic relationships between two concept boxes optimized by box constraint loss (Section 4.4). Given a query concept n, we first introduce its positive candidate set Cpos(n) = {⟨p, c⟩|∀p ∈ P(n), ∀c ∈ C(n)} and negative candidate set Cneg(n) = {⟨p, c⟩|∃p /∈ P(n) ∨ ∃c /∈ C(n)} where P and C refers to the parents and children of a node. Note that C(n) can be an empty set. For a positive candidate pair, the parent box can reliably hold the child box, while two boxes within a negative pair are disjoint. The closer the pair is in position, the more overlapping their box embedding will be. Based on this, we propose an insertion scorer (SI ) that represents the likelihood of performing insertion into the candidate as follows:
SI(Bq, Bp, Bc) =Pr(Bp|Bq) · Pr(Bq|Bc) · NRDt(Bq, Bp, Bc) (7)
where NRDt(·, ·, ·) is the normalized reciprocal distance measuring the center closeness between the candidate parent and a query as well as that between the query and the candidate child. It is for-
mulated as:
RD(Bq, Bpi) = 1
||Cen(Bq)− Cen(Bpi)||2 NRDp(Bq, Bp) = softmaxni=1(RD(Bq, Bpi))
NRDt(Bq, Bp, Bc) = NRDp(Bq, Bp)
· NRDp(Bq, Bc) (8)
where softmaxni=1 represents applying softmax along a mini-batch and Bpi is a candidate in the mini-batch. RD(·, ·) is the reciprocal distance, and NRDp(·, ·) only measures the closeness between the query and one side in the candidate. Attachment Scorer. Similar to the insertion scorer, when faced with the scenario of a candidate pair with no child, an attachment scorer (SA) is proposed. The attachment scorer is calculated as follows:
SA(Bq, Bp) = Pr(Bq|Bp) · NRDp(Bq, Bp) (9)

heading: 4.4 Multiple Learning Objectives
text: Classification Loss. The primary objective of our model is to determine the most suitable positions among all the candidate positions. We consider each candidate position as an independent category. Therefore, the problem can be reduced to a multilabel classification problem with a binary crossentropy loss as:
Lc = − 1 |B| ∑
(Xi,yi)∈B yilog(Sk(Xi))
+ (1− yi)log(1− Sk(Xi)), k ∈ {I, A} (10)
where Xi = (Bqi , Bpi , Bci), B refers to a minibatch consisting of one positive sample and several negative samples, y ∈ {0, 1} denotes whether the sample is positive or not. Sk(k ∈ {I, A}) means applying the insertion scorer if the candidate pair has both sides or the attachment scorer if it only has the parent side. Box Constraint Loss. To better model the granularity of the "is-a" relationships amongst concepts using box embeddings, we focus on the geometric constraints originating from three properties of boxes: inclusion (lin) and disjointness (ldis) model the unidirectional relationships between two boxes, and centrality similarity (lcen) facilitates scorers by obliging unrelated box pairs to assume orthogonal positions. Based on this, the loss functions
for concept inclusion Lin and disjoint Ldis are as follows:
lin(a, b) = −log Pr(b|a) ldis(a, b) = max(0, log(1− γ(a, b)) − log(1− Pr(a|b))) lcen(a, b) = max(0, log(1− γ(a, b)) − log(1− Cen(a) · Cen(b))) Lin(a, b) = lin(a, b) + ldis(a, b)
Ldis(a, b) = ldis(a, b) + ldis(b, a) + lcen(a, b)
(11)
The dynamic margin, γ(a, b), between two concepts a and b, is adapted from the Wu&P similarity(Wu and Palmer, 1994) and modulates their semantic distance:
γ(a, b) = α · 2× depth(LCA(a, b)) depth(a) + depth(b)
(12)
where LCA(·, ·) is the least common ancestor, depth(·) indicates the depth in the seed taxonomy, and α is a relaxation factor. By imposing constraints on volume(lin, ldis), position (lcen), and distance (SI/A), the optimization search space is effectively reduced.
Given a query box Bq, for a box Bk(k ∈ {p, c}) in a candidate, there are three possible scenarios: 1) Bq is contained within Bk. 2) Bk is contained within Bq. 3) both boxes are disjoint. When considering both sides of the candidate with a total of 6 possible cases, the box constraint loss is:
Lb = 1 |B| ∑
(Xi,li)∈B l1i · Lin(Bqi , Bpi)
+ l2i · Lin(Bci , Bqi) + l3i · Lin(Bpi , Bqi) + l4i · Lin(Bqi , Bci) + (1− l1i)(1− l3i) · Ldis(Bqi , Bpi) + (1− l2i)(1− l4i) · Ldis(Bci , Bqi) (13)
where li = (l1i , l2i , l3i , l4i) denotes whether the two sides of the candidate pair indeed contain the query concept or are contained by the query. Ranking Loss. It’s evident that the values of two scorers are numerically unbalanced, namely SI ≤ SA when considering the same candidate parent. In fact, there is no need for concern, as when a query is inserted into this candidate position, it is implicitly attached as a leaf. Our focus should be on guaranteeing SI(Xpos) ≥ SA(Xneg) where
the subscripts pos and neg indicate positive and negative samples, respectively. Consequently, for k, k′ ∈ {I, A}, the ranking loss is strategically designed to circumvent this particular case.
Lr = 1 |B| ∑
Xi∈B max(0, γ(Xpos, Xneg)
+ Sk(Xneg)− Sk′(Xpos)) (14)
Here, the dynamic margin compels SI(Xpos) to be greater than SA(Xneg) to a specific extent based on their structural similarity. The final loss combines all of the three losses mentioned above:
L = Lc + Lb + Lr (15)

heading: 5 Experiments

heading: 5.1 Experiment Setup
text: Datasets. We assess TAXBOX’s performance in taxonomy completion on four real-world datasets: two Microsoft Academic Graph subgraphs, MAGCS and MAG-PSY, plus two WordNet subgraphs, Wordnet-Verb and SemEval-Food. Also, two public datasets from SemEval-16, Science and Environment are evaluated for taxonomy expansion. Further dataset details are available in Appendix A. Evaluation metrics consist of Mean Rank (MR), Mean Reciprocal Rank (MRR), Wu&P, Hit@k, and Prec@k, with elaboration in Appendix B. Compared Methods. We select three recent SOTA taxonomy completion frameworks, Triplet Matching Network (TMN) (Zhang et al., 2021), QEN (Wang et al., 2022) and TaxoEnrich(Jiang
et al., 2022), and two taxonomy expansion frameworks, TaxoExpan (Shen et al., 2020) and ARBORIST (Manzoor et al., 2020), as baselines for the four completion datasets. Additionally, we compare BoxTAXO(Jiang et al., 2023) and TaxoExpan demonstrating TAXBOX’s superiority in taxonomy expansion. A further explanation is presented in Appendix C. Implementation Details. The Adam optimizer was employed with a 0.001 learning rate and the ReduceLROnPlateau scheduler with a 10-epoch patience, training our model across all datasets for 100 epochs. Four attention heads were fixed with 0.1 dropout rate in GAT. The dynamic margin relaxation factor α was 0.5. The training and prediction smoothness factor τ were 10 and 20 respectively. Batch and negative sample size were set at 16 and 63, while box dimensions were set at 64 for SemEval-Food, 128 for Wordnet-Verb and MAGCS, and 160 for MAG-PSY. Initial embeddings were the word2vec for the MAG datasets, fasttext for the Wordnet datasets, barring the PLM-based methods, and BERT embedding for two expansion datasets for fair comparison. All the experiments were conducted with one RTX3090.

heading: 5.2 Experimental Results
text: Table 1 demonstrates the superior performance of TAXBOX in taxonomy completion datasets, reflecting average improvements of 6.7%, 34.9%, and 51.4% in MRR, Hit@1, and Prec@1. It outperforms prior SOTA models, such as QEN and TaxoEnrich, which utilize the pre-trained language models (PLM) to enhance the representation. It showcases TAXBOX’s performance when handling datasets with varied scales. TAXBOX’s efficacy originates from its box embedding’s superior ability to capture asymmetric relationships among concepts and shows a significant improvement over conventional vector representations. PLM-based models like QEN, which lean on rich concepts’ descriptions from various internet-based data sources, tend to induce noise, particularly when dealing with larger datasets with obscure, overlapping concepts. Similarly, TaxoEnrich’s taxonomy-contextualized embeddings may reveal a variance in distribution between the training and testing phases, chiefly due to the test phase’s exclusion of query-related information.
On MAG-PSY and Wordnet-Verb datasets, TAXBOX outperforms in Hit@k and Prec@k metrics but has less exceptional MRR scores. A statisti-
cal analysis revealed that in MAG-CS and SemEvalFood datasets, the ratios of the maximum number of positive candidates in the training set to that in the test set are 2.5 and 1.5, respectively, whereas for MAG-PSY and Wordnet-Verb, the ratios are 14 and 11. It suggests the need for TAXBOX to optimize for all the concept boxes under relatively relaxed conditions to accommodate numerous ground truth positions in the training set. This presents a challenge when identifying test queries with fewer ground truth positions, constricting MRR scores while showing significant improvements in other metrics.

heading: 5.3 Ablation Study
text: To assess the efficacy of our proposed learning objectives (Lr, Lb) and graph aggregation module, we performed ablation studies using SemEval-Food and Wordnet-Verb datasets (Table 2). The model’s overall performance deteriorated when any component was removed, more noticeably so with Lb. This is due to Lb explicitly constraining box location and volume, while Lr primarily balances the gap between scorers, which is implicitly addressed during the optimization process of Lc. Despite that, Lr still yields a crucial 10% performance gain. The graph aggregation module demonstrated a significant improvement, underscoring its essential role in enhancing candidate feature enrichment.

heading: 5.4 How Two Scorers Work for Attachment and Insertion
text: Table 3 highlights the superior performance of TAXBOX over SemEval-Food and Wordnet-Verb datasets in terms of attachment and insertion, compared to other methods. It excels in all attachment
metrics, emphasizing the aptitude of its scorer to utilize box embeddings’ spatial aspects, while ignoring child boxes. For insertion, TAXBOX outperforms prevailing methods, indicating its scorer’s accuracy in identifying optimal candidate positions considering overlap and center similarity. This confirms the effectiveness and necessity of our method, and the insufficiency of pseudo leaf introduction in prior methods.

heading: 5.5 How TAXBOX Solves the Limitation of BoxTAXO
text: Table 4 reveals that TAXBOX surpassed BoxTAXO in all metrics to show the TAXBOX’s superiority over BoxTAXO. BoxTAXO’s limitations largely stem from its simplification of taxonomies into sheer tree structures, resorting to containment or non-intersection. This approach engenders two primary concerns: 1) Hard boundaries inhibiting multiple parent nodes accommodation, and 2) unreliable inference criteria due to volume containment probability being the chief confidence score. Contrarily, TAXBOX mitigates these constraints with
its soft margin-based constraints accommodating overlaps, and improves inference criteria with box center-position distance. Consequently, TAXBOX’s predictions are more precise, and it capably processes nodes with multiple parents, outperforming BoxTAXO.

heading: 5.6 How Dynamic Margin Affects Box Constraint
text: Table 5 highlights the dynamic margin’s efficiency in box constraint loss, in spite of comparable MRR results. Discrepancies in Hit@1 and Prec@1 across fixed margins accentuate the dynamic margin’s superiority in accurately modeling inter-box relationships. While a 0.3 fixed margin in SemEval-Food might parallel its performance, determining the optimal margin remains challenging. Notably, the dynamic margin outperforms all fixed margins in Wordnet-Verb, further underscoring its adaptability.

heading: 5.7 How to set up TAXBOX
text: We discuss our choice for box dimensionality and the number of negative samples in Appendix D.

heading: 6 Conclusion
text: In this study, we present TAXBOX, a novel framework for taxonomy completion using box embeddings. Incorporating restricted box constraint loss, dynamic ranking loss, and two probabilistic scorers for attachment and insertion, TAXBOX employs a structurally enhanced box decoder, mitigating the need for pseudo leaves. Experiments on six realworld datasets demonstrate its effectiveness and performance. Future research could refine scorers without numerical imbalance and explore postprocessing measures like reranking with LLM.
Limitations
The primary limitations of our proposed methods are as follows: (1) The numerical imbalance between the two scorers. Although we attempt to alleviate this issue by introducing a dynamic ranking loss, it remains an imperfect solution. Results shown in Table 3 indicate that tackling the insertion case in real-world practice is still challenging, despite TAXBOX achieving significant improvements compared to previous SOTA. A more practical scorer should be developed to address this. (2) In real-world applications, the quality of the initial embedding influences TAXBOX’s performance to some extent. Even when we opt for a well-pretrained language model for encoding, the concept name and description have a considerable impact. Thus, a more adaptive training strategy is needed. For example, we could employ data augmentation techniques to generate multiple texts representing the same meaning and use a PLM to obtain an embedding set pointing to a specific concept. During training, we can then retrieve different embeddings to fit the network, consequently enhancing its generalization capabilities.

heading: Acknowledgments
text: This work is supported by the National Natural Science Foundation of China (No. 62376245), the Key Research and Development Program of Zhejiang Province, China (No. 2024C01034), the project of the Donghai Laboratory (Grant no. DH2022ZY0013), National Key Research and Development Project of China (No. 2018AAA0101900), and MOE Engineering Research Center of Digital Library.

heading: A Dataset
text: We choose six real-world English datasets in different domains, four for taxonomy completion and two for taxonomy expansion. The statistical information about six datasets is shown in table 6.
• Microsoft Academic Graph (MAG) (Sinha et al., 2015) is a large, multi-disciplinary graph. The data in MAG includes information from a wide range of academic disciplines and includes more than 660 thousand scientific concepts and more than 700 thousand taxonomic relations. Following Zhang et al. (2021), we use subgraphs related to the computer science (MAG-CS) and psychology(MAG-PSY) domains. The initial embedding is a 250-dimension word2vec embedding trained by Zhang et al. (2021).
• Wordnet (Miller, 1995) is a large lexical database of English. Following (Wang et al., 2022) and (Zhang et al., 2021), we choose Wordnet-Verb (Jurgens and Pilehvar, 2016) and SemEval-Food (Bordea et al., 2015) which are extracted from wordnet. We employ 300-dimension fasttext embedding as our initial features following Zhang et al. (2021).
• SemEval-16 we use two public datasets released from SemEval-16 task. Specifically, they are small-scaled taxonomy in the domains of Environment and general Science.
And their initial embeddings are produced by a pre-trained bert(Devlin et al., 2018).
For MAG-CS, MAG-PSY and Wordnet-Verb, we randomly select 1,000 nodes for testing and 1,000 nodes for validation in each dataset, following the approach of Zhang et al. (2021). For SemEvalFood, we sample 10% of all the nodes for testing and another 10% for validation as done by Wang et al. (2022). For Environment and Science, we adopt the same protocol by Jiang et al. (2023). Subsequently, we reconstruct the seed taxonomy using the remaining nodes and add edges between the parent and child nodes of the test and validation sets to restore the fragmented taxonomy resulting from the dataset split.

heading: B Evaluation Metric
text: All the methods as well as our model are rankingbased ones, so we use the ranking-based metric to evaluate performance. Supposing rank(ci) denotes the predicted rank of ground truth position given a query concept ci ∈ C:
• Mean Rank (MR) mainly measures the average tail ranking level and we first calculate the average rank positions of each query and then average all the queries:
MR = 1
|C|
|C|∑
i=1
( 1
Mi
Mi∑
j=1
rank(cji )) (16)
where Mi denotes the total number of ground truth positions of a query ci and c j i denotes the jth prediction of ci.
• Mean Reciprocal Rank (MRR) mainly measures the average head ranking level. Its form is similar to MR except that we get the reciprocal number of the ranks. Here we scale the
reciprocal rank by 10 to amplify the difference.
RR = 1
Mi
Mi∑
j=1
1
max(1, rank(cji )/10) (17)
MRR = 1
|C|
|C|∑
i=1
RR (18)
• Hit@k measures the recall of a model which averages the true rank positions for all queries in top k:
Hit@k =
∑|C| i=1 ∑Mi j=1 1(rank(c
j i ) ≤ k)∑|C|
i=1Mi (19)
• Prec@k measures the precision of the results and it sums the true rank positions of all queries in top k, divided by k times the total number of queries:
Prec@k =
∑|C| i=1 ∑Mi j=1 1(rank(c j i ) ≤ k)
k ∗ |C| (20)
• Wu&P(Wu and Palmer, 1994) measures the structural similarity:
Wu&P = 1
|C|
|C|∑
i=1
2× depth(LCA(ai, bi)) depth(ai) + depth(bi)
(21)
where ai and bi are the predicted top-1 result and the truth potision in taxonomy.

heading: C Compared Methods
text: Here are the details of compared models:
• TaxoExpan (Shen et al., 2020): a state-ofthe-art method in taxonomy expansion that utilizes a graph neural network to incorporate structural information.
• ARBORIST (Manzoor et al., 2020): a stateof-the-art framework for taxonomy expansion and it leverages heterogeneous edge semantics with a dynamic margin loss.
• BoxTAXO (Jiang et al., 2023): a state-of-theart method using the property of conditional probability of box embedding for taxonomy expansion.
• TMN (Zhang et al., 2021): a state-of-the-art method for taxonomy completion that employs the channel-wise gate mechanism and auxiliary learning with multiple NTNs to evaluate partially positive candidate pairs beside positive pairs.
• QEN (Wang et al., 2022): a state-of-the-art model for taxonomy completion which utilizes a pre-trained language model to enhance the initial embedding with semantically rich term representation and enhance the performance with a sibling detector.
• TaxoEnrich (Jiang et al., 2022): a state-ofthe-art model for taxonomy completion that leverages Taxonomy-Contextualized Embeddings and sibling matching modules.

heading: D the Effect of Box Dimensionality and Negative Samples
text: We are also interested in how the box dimensionality and the number of negative samples affect the performance. Figure 4 shows the results of MRR, Hit@1 and Prec@1 when changing the box dimensionality from { 32, 64, 80, 128 } and the total number of samples from { 8, 16, 32, 64 }(where negative samples are { 7, 15, 31, 63 }) over two datasets.
Notably, it can be observed that for small datasets SemEval-Food, a dimension of 64 serves as a turning point. Dimensions below 64 exhibit a significant decline in overall performance. On the other hand, dimensions exceeding 64 reach a plateau, indicating that 64 is an appropriate dimension. Furthermore, increasing the dimension beyond 64 does not yield further performance improvements; instead, it leads to a decrease. This can be attributed to the fact that a dimension of 64 already satisfies the spatial constraints for all boxes in such a scale dataset. Larger dimensions introduce redundancy, thereby increasing the optimization difficulty. However, for Wordnet-Verb, it is worth noting that there is still some performance improvement observed after surpassing 64 dimensions. This discrepancy can be attributed to the larger dataset size and the initial quality of embeddings, which require more dimensions to effectively accommodate the information.
Regarding the setting of negative sample quantities, a general observation can be made that larger
numbers of negative samples result in better performance on both datasets. However, it is crucial to acknowledge that an increased number of negative samples reduces the attention given to positive samples during the optimization process of the classification loss. Consequently, it becomes necessary to elevate the weight assigned to positive samples in calculations. Therefore, the steep decrease observed at the position of 16 is a consequence of equal weighting given to positive and negative samples in the experiment, while higher negative sample counts were assigned higher weights. This emphasizes the significance of appropriately adjusting the weight allocation to balance the impact of positive and negative samples during training.

heading: Future Work
text: Future research could refine scorers without numerical imbalance and explore postprocessing measures like reranking with LLM. Limitations
The primary limitations of our proposed methods are as follows: (1) The numerical imbalance between the two scorers. Although we attempt to alleviate this issue by introducing a dynamic ranking loss, it remains an imperfect solution. Results shown in Table 3 indicate that tackling the insertion case in real-world practice is still challenging, despite TAXBOX achieving significant improvements compared to previous SOTA. A more practical scorer should be developed to address this. (2) In real-world applications, the quality of the initial embedding influences TAXBOX’s performance to some extent. Even when we opt for a well-pretrained language model for encoding, the concept name and description have a considerable impact. Thus, a more adaptive training strategy is needed. For example, we could employ data augmentation techniques to generate multiple texts representing the same meaning and use a PLM to obtain an embedding set pointing to a specific concept. During training, we can then retrieve different embeddings to fit the network, consequently enhancing its generalization capabilities