--- kind: concept slug: knowledge-graph-embedding title: Knowledge graph embedding summary: >- Knowledge graph embedding maps entities and relations to dense vectors and scores each triple for plausibility, so missing facts can be predicted. Families: tensor factorization, translation (TransE), neural scoring (SME, ConvKB, R-GCN), region-based and Gaussian embeddings, and temporal (TComplEx). Trade-off: static numbers come from one mixed-protocol table, and training details are not stated. tags: [knowledge-graphs, embeddings, link-prediction] papers: ["2303.13948", "2106.01515", "2412.10390", "2105.00696", "2104.04883"] related: [graph-neural-network, attributed-network-embedding, knowledge-graph-completion, knowledge-graph-reasoning, neural-symbolic-integration] status: draft updated_at: 2026-11-01 --- ## Overview A knowledge graph stores facts as triples (head, relation, tail). Knowledge graph embedding places every entity and relation in a dense, low-dimensional space, defines a scoring function that rates how plausible a triple is, and learns the vectors by making observed triples score high [arxiv:2303.13948 §5.1]. The learned scores can then rank candidate triples that are not yet in the graph, which is the link prediction task used in most reported results [arxiv:2303.13948 tab. 3]. The shared part across all variants is the triple score. The variants differ in how that score combines the head, relation and tail vectors, and the temporal variant adds a time argument. The survey gives the scoring functions for SME, ConvKB and R-GCN, but no loss, negative sampling or training settings for any of them [arxiv:2303.13948 §5.1.3]. Five cards on wikillm cover this concept. The 2023 survey gives the static families below and runs no experiments. A 2024 survey on knowledge graph reasoning names the same embedding models, plus region-based and Gaussian embeddings, but reports no numbers [arxiv:2412.10390 §3.2.1]. A 2021 survey of graph learning treats knowledge-graph embedding as one application area among many, and names TransE, RESCAL and DistMult there without numbers [arxiv:2105.00696 §3.5]. A 2021 review of graph representation learning in biomedicine treats knowledge-graph variants as an extension of the node embedding similarity, and gives no equations or results for them [arxiv:2104.04883 §2.1]. The temporal variant comes from a paper on question answering over temporal graphs, which trains its own embeddings and reports results on its own dataset [arxiv:2106.01515 §4] [§6]. ## Variants ### Tensor factorization The graph is written as a binary tensor $\mathcal{X} \in \mathbb{R}^{m \times m \times n}$ for $m$ entities and $n$ relations, one slice per relation. An entry is 1 when the triple is in the graph and 0 otherwise [arxiv:2303.13948 §5.1.1]. The tensor is then represented by embedding matrices for entities and relations. The survey does not specify the factorization itself [arxiv:2303.13948 §5.1.1]. Models the survey files under this family, with Hits@10 on FB15K as reported by their original papers: ComplEx 84%, SimplE 83.8%, RotatE 88.4%, QuatE 90%, HolE 73.9%, RESCAL 44.1% [arxiv:2303.13948 tab. 3]. Card: [Knowledge Graphs: Opportunities and Challenges](/papers/2303.13948). ### Translation-based (TransE and extensions) Each triple is embedded so that the head vector plus the relation vector lands near the tail vector, $\mathbf{h} + \mathbf{r} \approx \mathbf{t}$ [arxiv:2303.13948 §5.1.2]. The survey names TransH and TransR as extensions, and lists TransD, TranSparse, STransE, TransA, KG2E and TransG as further members of the family [arxiv:2303.13948 tab. 3]. The survey gives only the approximate equality above. It does not state the distance norm, the margin or the loss [arxiv:2303.13948 §5.1.2]. Reported Hits@10 on FB15K ranges from 47.1% for TransE to 88.2% for TransG [arxiv:2303.13948 tab. 3]; these come from different original papers. ### Neural scoring (SME, ConvKB, R-GCN) - **SME** [arxiv:2303.13948 §5.1.3]. The linear form projects the head and the tail, each combined with the relation, and takes their inner product [arxiv:2303.13948 eq. 1]. The bilinear form replaces the sum with an element-wise product [arxiv:2303.13948 eq. 2]. Reported Hits@10 on FB15K: 41.3% [arxiv:2303.13948 tab. 3]. - **ConvKB** [arxiv:2303.13948 §5.1.3]. The three vectors of a triple are stacked into a matrix, convolved with a filter set, passed through a ReLU, and projected to a scalar [arxiv:2303.13948 eq. 3]. The filter count and shape are not stated. Reported Hits@10 on WN18RR: 52.5% [arxiv:2303.13948 tab. 3]. - **R-GCN** [arxiv:2303.13948 §5.1.3]. A graph convolution in which each entity sums its neighbours' transformed vectors, normalised per relation, plus a self term [arxiv:2303.13948 eq. 4]. The survey calls it a relation-specific transformation. Reported Hits@10 on FB15K: 84.2% [arxiv:2303.13948 tab. 3]. ### Walk-based and graph-neural methods on knowledge graphs The 2021 graph learning survey covers two routes that apply to knowledge graphs without a triple score. Random walks may follow knowledge-graph paths through heterogeneous networks, and the walks are then fed to a word-embedding model [arxiv:2105.00696 §2.3.3]. Graph neural networks, in the form of GNN-based knowledge base completion, are a separate application [arxiv:2105.00696 §3.5]. The survey names both routes but gives no equations, training settings or results for either [arxiv:2105.00696 §1.3]. Card: [Graph Learning: A Survey](/papers/2105.00696). The biomedical review describes the same shallow idea: a node embedding is trained so that a graph similarity, such as co-occurrence on random walks, matches an embedding similarity. Knowledge-graph variants extend that graph similarity to relation types, and the vectors go to a separate classifier or regressor [arxiv:2104.04883 §2.1]. The review gives no training configuration or metric for it [arxiv:2104.04883 §2.1]. Card: [Graph Representation Learning in Biomedicine](/papers/2104.04883). ### Region-based and Gaussian embeddings The 2024 survey lists region-based embeddings, which represent entities with regions such as boxes and spheres, and Gaussian embeddings as further members of the embedding group [arxiv:2412.10390 §3.2.1]. As a survey with no experiments, it gives no numbers for either group. The card does not give their scoring functions either, so take them from the original papers. Card: [Neural-Symbolic Reasoning over Knowledge Graphs](/papers/2412.10390). ### Temporal embeddings (TComplEx) Each entity and relation is a complex vector, and each timestamp is a complex vector too [arxiv:2106.01515 §4]. The ComplEx score takes the real part of a product of the subject, relation and conjugated object vectors [arxiv:2106.01515 eq. 1]. TComplEx multiplies in a timestamp vector as well, so each fact is scored at a given time [arxiv:2106.01515 eq. 2] [§4.2]. TNTComplEx mixes a time-sensitive and a time-insensitive relation vector [arxiv:2106.01515 §4.2]. The embedding rank is not stated [arxiv:2106.01515 §4.2]. The paper uses TComplEx as the embedding inside a question-answering model, CronKGQA. Swapping ComplEx for TComplEx in that model raises time-answer Hits@1 from 0.057 to 0.549 and overall Hits@1 from 0.288 to 0.647 [arxiv:2106.01515 tab. 7]. These are the authors' numbers on their own dataset and split, and they measure question answering, not link prediction. Card: [Question Answering Over Temporal Knowledge Graphs](/papers/2106.01515). ## When to use which The survey does not give latency, memory or implementation-effort figures for any method, so the table only uses constraints that its text or tables address. The temporal row uses the authors' question-answering results. | Situation | Use | Why | |---|---|---| | Static graph; the task is ranking candidate triples | A tensor factorization model such as ComplEx, or a translation model | The survey lists ComplEx at 84% Hits@10 on FB15K and TransE at 47.1% [arxiv:2303.13948 tab. 3]. The numbers come from original papers under their own protocols. | | Entity types or relation paths carry information you want to keep | A method that uses extra information, or a neural scoring variant | The survey says weak results come from methods that ignore such information [arxiv:2303.13948 §5.1.4]. Its ranking of the best methods is not a controlled comparison (see Contradictions). | | Facts hold only for certain periods, and answers depend on when | A temporal variant such as TComplEx | With TComplEx, time-answer Hits@1 is 0.549 against 0.057 with ComplEx in the authors' question-answering setup [arxiv:2106.01515 tab. 7]. Time is an argument of the score [arxiv:2106.01515 eq. 2]. | | Questions need several timed facts chained together | Expect weak results and test on your own questions | In the authors' runs, complex questions reach 0.392 Hits@1 with TComplEx, against 0.987 for simple questions [arxiv:2106.01515 tab. 7]. | | Triples are judged true or false, not ranked | Neural scoring evaluated by triplet classification | The survey reports accuracy on WN11 for NTN 86.2%, SLM 76%, RMNN 89.9% [arxiv:2303.13948 tab. 3]. These numbers do not transfer to Hits@10. | | Large graph with many relations | Avoid a dense tensor; check the storage | A dense tensor has $m^2 n$ entries [arxiv:2303.13948 §5.1.1]. The survey does not discuss sparse storage. | | Simple baseline and limited spec to implement from | TransE, after reading its original paper | The survey states only $\mathbf{h}+\mathbf{r}\approx\mathbf{t}$ [arxiv:2303.13948 §5.1.2]. Norm, margin and loss must come from the original work. | | Queries combine several conditions, such as conjunction, union or negation, over embedded entities | Embedding-space query solvers such as GQE or Query2Box, read in their own papers | The 2024 survey names these as neural solvers for tree-form queries, where intersection, union and complement are differentiable operations [arxiv:2412.10390 §4.2.1]. It reports no numbers for them. | | Entities have neighbourhoods that matter for the score | R-GCN style aggregation | R-GCN is the survey's example of neighbour-based relational encoding [arxiv:2303.13948 §5.1.3]. The equation as printed has no relation index on $W_i$; check the original R-GCN paper before implementing. | | You need vectors for a general graph, and the knowledge-graph triple score is not required | Random-walk embedding or a graph neural network, read in their original papers | The 2021 survey groups random-walk and deep graph methods for general graphs [arxiv:2105.00696 §2.3.1] [§2.4]. It reports no numbers, so the choice rests on the cited originals. | | Typed biomedical network where node vectors feed a separate classifier or regressor | A random-walk embedding with relation types in the similarity | The biomedical review describes this design and its use in drug and disease graphs, but reports no results [arxiv:2104.04883 §2.1] [arxiv:2104.04883 §5] [arxiv:2104.04883 §6]. | ## Implementation sketch The common core is a score over triples and a loop over observed facts. The survey gives the score functions but not the loss or the sampling, so the loss is left as a parameter. The temporal scorer adds a fourth argument. The pseudocode is ours. ```pseudocode # Build the binary KG tensor used by factorization methods [arxiv:2303.13948 §5.1.1] def build_kg_tensor(triples, num_entities, num_relations): # one m x m slice per relation; zero means the triple is absent [§5.1.1] tensor = zeros(num_entities, num_entities, num_relations) for head, relation, tail in triples: tensor[head, tail, relation] = 1 return tensor # Generic training loop: maximise plausibility of observed triples [§5.1] # score(h, r, t) is one of the variants below; loss is not stated by the survey def train(triples, entities, relations, score, loss, opt, steps): for step in range(steps): for h, r, t in sample_batch(triples): # plausibility of the observed triple [§5.1] pos = score(E[h], R[r], E[t]) # corrupted triple, sampling scheme not stated [§5.1] neg = score(E[h], R[r], E[sample_entity(entities)]) opt.step(loss(pos, neg)) # SME linear score of one triple [arxiv:2303.13948 eq. 1] def sme_linear_score(h, r, t, W_h1, W_h2, b_h, W_t1, W_t2, b_t): # project head and tail, each combined with the relation [eq. 1] left = W_h1 @ h + W_h2 @ r + b_h # (d,) right = W_t1 @ t + W_t2 @ r + b_t # (d,) # inner product of the two projections [eq. 1] return dot(left, right) # TransE-style residual; the survey gives no norm [§5.1.2] def translation_score(h, r, t): # smaller distance means more plausible; L1 or L2 not stated [§5.1.2] return -norm(h + r - t) # TComplEx score of one timed triple [arxiv:2106.01515 eq. 2] def tcomplex_score(s, r, o, t): # complex vectors of one rank; the rank is not stated [§4.2] # sum the elementwise product, with the object conjugated [eq. 2] return real(sum(s * r * conj(o) * t)) ``` ## Contradictions - **Ranking of the best models (within one survey).** The survey names QuatE (90%), RMNN (89.9%) and KBGAN (89.2%) as the only strong results [arxiv:2303.13948 §5.1.4]. The table reports RMNN as triplet classification accuracy on WN11, while QuatE and KBGAN are Hits@10 on FB15K and WN18 [arxiv:2303.13948 tab. 3]. The three numbers do not measure the same thing. The survey does not resolve this. - **Taxonomy differs between the two surveys.** The 2023 survey files ComplEx and RESCAL under tensor factorization and TransE, TransH and TransR under translation [arxiv:2303.13948 §5.1.1] [§5.1.2]. The 2024 survey puts TransE, DistMult, ComplEx, RotatE, TransH, TransR, RESCAL, QuatE and HolE in one embedding group [arxiv:2412.10390 §3.2.1]. Neither survey justifies its grouping here. This page follows the 2023 grouping, because it carries the numbers. The 2021 graph learning survey does not treat these models as embedding methods at all: it lists TransE, RESCAL and DistMult under knowledge-graph applications [arxiv:2105.00696 §3.5]. Its categories are different, so do not map its section numbers onto the 2023 families. The biomedical review places knowledge-graph variants inside shallow node embeddings with relation types added to the similarity [arxiv:2104.04883 §2.1], not in a triple-scoring family. These are different framings of one idea, and this page does not reconcile them. - **Mixed protocols in the literature table.** The table mixes FB15K, WN18, WN18RR and WN11, and Hits@10 with accuracy. The values come from the original papers under their own protocols [arxiv:2303.13948 tab. 3]. - **Equation and text on R-GCN.** The text calls R-GCN a relation-specific transformation [arxiv:2303.13948 §5.1.3], but the printed update uses one weight $W_i^{(l)}$ per neighbour with no relation index [arxiv:2303.13948 eq. 4]. Not resolved here. - **Not comparable across cards.** The survey's Hits@10 values are link prediction on FB15K-style benchmarks [arxiv:2303.13948 tab. 3]. The CronKGQA values are Hits@1 for question answering on the authors' dataset [arxiv:2106.01515 tab. 5]. Do not rank one against the other. ## Papers - [Knowledge Graphs: Opportunities and Challenges](/papers/2303.13948): a 2023 survey of knowledge graphs. It covers the embedding families above, with a literature table of reported link prediction and classification results, and proposes no new method. - [Question Answering Over Temporal Knowledge Graphs](/papers/2106.01515): adds timestamps to the complex-valued embedding (TComplEx) and uses it in a question-answering model, CronKGQA. It also builds a large temporal question dataset, CronQuestions. - [Neural-Symbolic Reasoning over Knowledge Graphs: A Survey from a Query Perspective](/papers/2412.10390): a 2024 survey of knowledge graph reasoning organised by query type. It covers the embedding group, complex logical query solvers and neural-symbolic methods, and reports no experiments. - [Graph Learning: A Survey](/papers/2105.00696): a 2021 survey of graph learning in four families, with a section on knowledge-graph applications. It reports no experiments and proposes no new method. - [Graph Representation Learning in Biomedicine](/papers/2104.04883): a 2021 review of graph representation learning for biomedical networks, including the knowledge-graph variant of shallow node embeddings. It runs no experiments and reports no benchmark numbers.