SMART: Relation-Aware Learning of Geometric Representations for Knowledge Graphs

Andrea Coletta; Bowen Song; Jens Lehmann; Kossi Amouzouvi; Luigi Bellomarini; Sahar Vahdati

REVIEW 3 major objections 4 minor 29 references

SMART: Relation-Aware Learning of Geometric Representations for Knowledge Graphs

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proposes SMART, a knowledge graph embedding framework that ranks four elementary geometric transformations per relation and freezes each relation to its best-matching one, with link prediction comparable to leading models.

desk verdict A useful dimension-transfer result and a genuine framework, but the per-relation "optimal" selection is order-sensitive and the text overclaims on FB15k-237; still deserving of peer review. read the letter →

arxiv 2507.13001 v1 pith:2LM5SSDE submitted 2025-07-17 cs.LG cs.AI

Kossi Amouzouvi , Bowen Song , Andrea Coletta , Luigi Bellomarini , Jens Lehmann , Sahar Vahdati This is my paper · ORCID

classification cs.LGcs.AI

keywords knowledgegraphembeddingsrelation-awareattentiongeometrictransformationslinkpredictiontranslationrotationreflectionscalingelementaryembeddingdimensiontransfer

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Knowledge graph embedding models typically map every relation with the same geometric transformation, such as translation, rotation, reflection, or scaling. SMART claims that relations should instead be matched to the transformation that best fits their observed structure, and it learns this match automatically. The paper introduces a three-phase procedure in which all four elementary transformations are trained equally, then a per-relation attention weight is learned over the four transformations, and finally each relation is frozen to its highest-weighted transformation. The result is an interpretable assignment of one geometric meaning to each relation, with link prediction performance comparable to leading fixed- or composite-transformation models. The paper further shows that the learned relation-to-transformation preference transfers from low to high embedding dimensions, cutting training cost.

What carries the argument

The load-bearing object is the relational attention weight matrix $W=(\omega_{r\tau})$, one row per relation and one column per elementary geometric transformation, constrained so each row sums to one. It drives the scoring function by weighting the four distances $\|r(\tau,h)-t\|$; changes to it are what let a relation concentrate on one transformation. The three-phase schedule, uniform training, adaptive attention learning, and freezing to the arg max, is what turns the matrix into a selection mechanism rather than a soft mixture. A secondary mechanism is the adherence score $\mathrm{adh}(r,\tau)$, the fraction of repeated runs in which relation $r$ is assigned transformation $\tau$; this score is what gets transferred across embedding dimensions.

What would settle it

Run the SMART adaptive phase on the COKG dataset, or on a synthetic knowledge graph with known relation patterns, with the four transformation columns of the attention matrix in each of the 24 possible orders, and compute the fraction of relations whose frozen assignment changes. If that fraction is large, the learned selection reflects column order rather than relation semantics, and the core per-relation selection claim fails.

Watch

Extended reading notes

Core claim

SMART's central discovery is that the affinity between a relation and a candidate geometric transformation can be learned as a trainable attention weight, and that freezing each relation to its arg-max transformation preserves or improves predictive accuracy. Formally, the triple score is the negative weighted sum of distances between each transformed head and the tail, $\Delta(h,r,t)=-\sum_{\tau}\omega_{r\tau}\|r(\tau,h)-t\|$, with weights $\omega_{r\tau}\ge 0$ summing to $1$ per relation. Phase one keeps all weights at $0.25$; phase two makes them trainable; phase three sets the largest weight to $1$ and the rest to $0$, which is equivalent to choosing $r[\tau]=\arg\max_{\tau}\omega_{r\tau}$. The framework also derives a complex-plane expression for reflection, $\mathrm{Ref}(z)=e^{2i\phi}\bar{z}$, and proves that the selected transformations can model symmetry, antisymmetry, inversion, composition, and (non-)commutativity. Empirically, SMART and its majority-vote variant SMART$_m$ match or exceed single-transformation baselines and are competitive with composite models, while producing per-relation transformation assignments that are human-readable.

Load-bearing premise

The central assumption is that the attention weights learned in the adaptive phase encode a stable, relation-intrinsic ranking of the four transformations, so that freezing each relation to its highest-weight transformation yields the best choice for that relation; the paper's own column-order experiment shows this ranking can shift when the implementation order of the transformations changes.

Editorial extensions

If this is right

A knowledge graph embedding model can expose, for every relation, which of translation, rotation, reflection, or scaling best matches its structure.
Relation-specific selection lets a single model handle symmetric, antisymmetric, inverse, compositional, and non-commutative patterns without hand-designing a composite transformation.
The learned per-relation preferences can be trained in a low-dimensional space and then loaded into a high-dimensional model, so the expensive search can be done cheaply.
The same three-phase attention-and-freeze procedure can be extended to composite geometric transformations beyond the four elementary ones.
If a relation is equally well explained by two transformations, the framework flags that ambiguity instead of silently forcing one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Read as a diagnostic, the learned adherence matrix could cluster relations by their preferred transformation, offering a data-driven way to discover shared structural patterns across a knowledge graph.
The same attention-and-freeze schedule could be applied to richer transformation families, such as hyperbolic or composite transformations, where the ranking would reveal which family each relation needs.
The cheap low-to-high dimension transfer suggests that adherence could be precomputed once on a small sample and reused for larger versions of the same graph.
A per-relation subset selection, the threshold variant, is a natural middle ground between one fixed transformation and the full soft mixture, and its relative performance is likely dataset-dependent.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real attempt at relation-specific geometric transformation selection, and the dimension-transfer trick is genuinely useful. But the paper overclaims on two fronts: the FB15k-237 numbers contradict the text, and the authors' own order-permutation experiment shows the per-relation "optimal" assignment is an artifact of column order. Still worth a referee.

What's new: SMART runs three phases—uniform training, adaptive attention over four EGTs, then freezing to the argmax. The attention-based mixture per relation is a natural extension of ATTE, but the freezing and the majority-vote variant are new. The paper also reports adherence scores per relation, which gives interpretability (e.g., WN18RR hypernym relations stick to rotation). The dimension transfer result (Tables 5 and 6) is the most solid contribution: learning adherence in d=32 and loading it at d=250/100 saves compute and sometimes improves MRR. That's a useful practical finding.

Soft spots. The biggest one is the order sensitivity. Table 7 shows that swapping the column order (Trans, Rot, Ref, Scal) to (Trans, Scal, Ref, Rot) changes the argmax for several COKG relations; Rel.B flips from Rotation to Reflection. If the EGT preference is relation-intrinsic, relabeling the columns shouldn't matter—the attention weights should be invariant. They aren't. The authors acknowledge the bias, but that acknowledgment undercuts the central claim that Eq. 3 assigns the "most appropriate" transformation. It assigns the one preferred by a particular initialization and column order.

Second, the text in Section 5.4 says SMART and SMARTm surpass the EGT-based models on FB15k-237, but Table 3 shows SMART MRR 274 vs TransE 295 and RotatE 290. Even SMARTm (298) is only marginally above TransE, and the error bars overlap. That's a straightforward overclaim.

Third, the formal properties section (4.3) is not a proof. It lists known properties of the four transformations plus the claim that the framework "infers" patterns because it can select the right EGT. That is true loosely, but it is not a theorem about the scoring function. The conclusion says "we proved that the framework benefits from adaptive early stopping"—that is an empirical observation from Table 8, not a proof.

The code link is anonymous, standard for submission; no issue there.

Bottom line: the framework is coherent and the dimension transfer is worth something. But the headline contribution—optimal per-relation EGT selection—is not established. The authors should test multiple random EGT orders, average over them, or reframe the claim as "order-sensitive but useful in practice." I would send this to peer review; it is a legitimate contribution with a fixable central flaw.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes SMART, a knowledge graph embedding framework that learns a relation-specific elementary geometric transformation (EGT) among translation, rotation, reflection, and scaling. The model first trains all four EGTs with equal attention, then adapts per-relation attention weights, and finally freezes each relation to the EGT with the highest weight (Eq. 3). A majority-vote variant (SMARTm) and a threshold-based variant (SMART>) are also presented. The paper claims a theoretical analysis of relational patterns and reports experiments on WN18RR, FB15k-237, YouTube, and a financial company-ownership KG, including a cross-dimensional transfer study of learned EGT preferences.

Significance. If the central selection claim were established, SMART would offer a practically useful and interpretable way to assign geometric transformations to relations, with an open-source implementation and experiments on a real financial KG. The cross-dimensional transfer experiment (Tables 5 and 6) is a useful empirical contribution. However, the key claim that attention weights encode a relation-intrinsic ranking of EGTs is not currently supported by the paper's own order-sensitivity experiment, and one performance claim is contradicted by the reported table. The theoretical section mostly restates known properties of the candidate transformations rather than proving properties of the learned selection mechanism.

major comments (3)

[Section 5.4, Eq. (3), Table 7] The central claim that SMART selects the most appropriate EGT per relation is not supported by the order-permutation experiment. When the EGT column order is changed from (Trans, Rot, Ref, Scal) to (Trans, Scal, Ref, Rot), COKG relational adherences change substantially: Rel.B shifts from Rotation to Reflection, Rel.C through Rel.F shift to Scaling, and Rel.H shows equal Rotation/Reflection weights. Because permuting columns only relabels the same learned weight vectors, a relation-intrinsic ranking should be invariant under this operation; the observed sensitivity means the argmax in Eq. (3) can reflect implementation order rather than the geometric fit. The authors acknowledge this bias in the same section. The average-adherence procedure in Section 4.1 averages over runs with the same default order and does not remove the bias. The paper should either demonstrate invariance under multiple orders (for example, by aggregating over all permutations) or substantially weaken the selection claim.
[Section 5.4, Table 3] The text states that SMART and SMARTm surpass the EGT-based models on WN18RR and FB15k-237, but Table 3 reports SMART MRR 0.274 ± 0.004 on FB15k-237, which is lower than TransE's 0.295 and RotatE's 0.290; SMARTm's 0.298 does surpass those two but still trails MuRE's 0.313. This is a direct inconsistency between the stated conclusion and the reported numbers, and it should be corrected.
[Section 4.3] The formal properties section proves properties of the four EGTs, not of the learned per-relation selection. For example, 'SMART infers symmetry through its choice of reflection and rotation transformations' is true only if the attention mechanism happens to select those EGTs for symmetric relations; nothing in the training objective or freezing rule enforces such an alignment. Likewise, the composition and non-commutativity claims are statements about what the candidate transformations can represent, not guarantees about what the framework will select. To support the theoretical contribution, the paper would need a proof or an empirical analysis that the selected EGTs satisfy the claimed pattern constraints for each relation.

minor comments (4)

[Section 1] The introduction contains typos such as 'knowledge grap' and 'such as such as'; proofreading is needed.
[Tables 3-6] Table values are printed without a leading decimal point (e.g., 366 vs 0.366), while Section 5.4 uses MRR values such as 0.274; the notation should be unified.
[Section 4.3] The definition of symmetry as both triples coexisting 'for all h, t ∈ E' is stronger than the standard conditional definition; rephrase to the usual 'whenever (h,r,t) holds, (t,r,h) also holds'.
[References] Reference [1] is listed as 'Anonymous' with a 2025 access date; if the code is publicly available under a persistent link, the repository details should be provided.

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity (score 1): the only self-referential element is the adherence analysis, which by construction reports the model's own fitted attention weights; the paper's own Table 7 permutation experiment flags an order-bias that limits the central selection claim but is a robustness issue, not circularity.

fitted input called prediction [Section 4.1 (Eqs. 3-4) and Section 1, contribution 3; abstract claim that the framework 'evaluates how well each relation fits with different geometric transformations']
"adh(r, τ) = Σ_{j=1}^n δ{r[τ_j]=τ} / n, the average outcome per relation from these n instances of optimal EGTs assigned by the model, is recorded ... Thus, adh(r, τ) quantifies the relational adherence. ... In addition to performance evaluation, we offer an empirical analysis of which transformations are most appropriate for different relations."

The reported 'relational adherence' - the paper's empirical finding about which EGT fits each relation - is defined by Eqs. (3)-(4) as the argmax of the trained attention weights ωrτ, and those weights are the fitted parameters of the score function (Section 4.2) optimized directly on the training triples. Hence the conclusion 'which transformations are most appropriate for different relations' is, by construction, a summary of the model's own fitted weight distribution rather than an independent measurement of relation-EGT fit; the abstract's claim that the framework 'evaluates how well each relation fits with different geometric transformations' is equivalent to 'trains relation-specific attention weights.' This is descriptive, not independent evidence.

full rationale

The derivation chain is otherwise self-contained. The core mechanism (uniform training, adaptive attention, freezing to argmax in Eq. 3) is exactly what it claims to be: ωrτ is a learnable parameter inside the score function, and Table 8 shows that each phase changes the externally validated MRR/H@N metrics against independent baselines (TransE, RotatE, ComplEx, MuRE, etc.). The formal properties in Section 4.3 are restatements of known algebraic facts about translation, rotation, reflection, and scaling (reflection is symmetric, rotation is antisymmetric unless the angle is a multiple of π, scaling commutes with all but translation); they are presented as capabilities conditional on the model's selection, not as novel first-principles derivations, and they do not feed back into the fitted parameters. The reflection expression Ref(z)=e^{2iφ} z̄ is derived from the standard 2x2 reflection matrix, not imported by citation. Self-citations ([2] COKG data, [17] 5*E, [26] ensemble, [1] code) are related-work or data citations and are not load-bearing for any uniqueness or equivalence claim; no 'uniqueness theorem' is invoked. The order-permutation experiment in Section 5.4 (Table 7) is an acknowledged limitation - 'this fixed ordering introduces a bias in SMART, favoring earlier EGTs' and 'EGT order can influence relational preferences and, consequently, model behavior' - which undermines the robustness of the central per-relation selection claim, but that is a validity/correctness risk rather than circularity. The one self-referential element is the adherence analysis itself: adh(r,τ) is by construction the argmax of the trained attention weights, so the reported 'empirical analysis' describes the model's own fitted behavior. This is honestly operationalized and does not affect the benchmark comparisons, warranting a low circularity score of 1.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The framework relies on standard KGE mathematics and several unproven domain assumptions, most notably that attention weights are stable indicators of relation-EGT fit and that one EGT per relation suffices. The free parameters are conventional hyperparameters plus the SMART> threshold. No new physical or abstract entities are introduced; 'adherence' is a descriptive statistic, not an invented entity.

free parameters (9)

margin gamma = 1, 9, 24 (per dataset)
Loss margin selected by grid search on best MRR; not derived from data.
self-adversarial temperature alpha = 0, 0.5, 1
Controls negative sample weighting; tuned per dataset.
regularizer rho = 0, 0.0001, 0.01, 0.1
Regularization strength; tuned per dataset.
embedding dimension d = 32 (low), 100/250/300 (high)
Dimensionality changes across datasets and affects expressivity.
negative sample count eta = 128, 256, 512, 1024
Number of negative samples per positive; tuned per dataset.
batch size beta = 128, 256, 512, 1024
Tuned per dataset.
learning rate lambda = 0.0001, 0.001
Adam learning rate; tuned per dataset.
phase step sizes sigma = 120000 (T), 50000 (TA), 90000 (F)
Maximum training steps per phase; chosen by hand.
SMART> threshold epsilon = 0.25, 0.35
Threshold for selecting multiple EGTs per relation; only used in the SMART> variant.

assumptions (5)

standard math Complex-valued embeddings combined with Hadamard product implement translation, rotation, reflection, and scaling as in Eqs. (1)-(4).
Standard construction used by TransE, RotatE, and related models; not proved in this paper.
domain assumption A weighted sum of distances to the tail is a valid scoring function that can rank true triples above negatives.
Standard KGE assumption; no convergence or separation guarantee is given.
ad hoc to paper Each relation can be represented by a single geometric transformation chosen by arg max attention after freezing.
SMART's core design; relations with mixed patterns may be poorly captured, as the SMART> multi-EGT variant only tests a thresholded alternative.
ad hoc to paper Learned attention weights reflect the intrinsic fit between a relation and an EGT.
Used to interpret adh(r,tau) as relational adherence; Table 7's ordering sensitivity challenges this.
domain assumption Relational adherence learned in low dimension transfers to high dimension.
Tested in Section 5.4, but no theoretical justification.

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Cite this review

Pith. "Pith review of SMART: Relation-Aware Learning of Geometric Representations for Knowledge Graphs." pith.science (2026). https://pith.science/paper/2LM5SSDE

@misc{pith2026250713001,
  author       = {Pith},
  title        = {Pith review of: SMART: Relation-Aware Learning of Geometric Representations for Knowledge Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2LM5SSDE}},
  note         = {Machine review of arXiv:2507.13001}
}

read the original abstract

Knowledge graph representation learning approaches provide a mapping between symbolic knowledge in the form of triples in a knowledge graph (KG) and their feature vectors. Knowledge graph embedding (KGE) models often represent relations in a KG as geometric transformations. Most state-of-the-art (SOTA) KGE models are derived from elementary geometric transformations (EGTs), such as translation, scaling, rotation, and reflection, or their combinations. These geometric transformations enable the models to effectively preserve specific structural and relational patterns of the KG. However, the current use of EGTs by KGEs remains insufficient without considering relation-specific transformations. Although recent models attempted to address this problem by ensembling SOTA baseline models in different ways, only a single or composite version of geometric transformations are used by such baselines to represent all the relations. In this paper, we propose a framework that evaluates how well each relation fits with different geometric transformations. Based on this ranking, the model can: (1) assign the best-matching transformation to each relation, or (2) use majority voting to choose one transformation type to apply across all relations. That is, the model learns a single relation-specific EGT in low dimensional vector space through an attention mechanism. Furthermore, we use the correlation between relations and EGTs, which are learned in a low dimension, for relation embeddings in a high dimensional vector space. The effectiveness of our models is demonstrated through comprehensive evaluations on three benchmark KGs as well as a real-world financial KG, witnessing a performance comparable to leading models

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Pytorch implementation of: Representation learning over knowledge graphs with relation-guided geometric transformations

Anonymous. Pytorch implementation of: Representation learning over knowledge graphs with relation-guided geometric transformations. https://anonymous.4open.science/r/smart-kgc-6666. Accessed: 2025- 05-05

work page 2025

[2] [2]

Atzeni, L

P. Atzeni, L. Bellomarini, M. Iezzi, E. Sallinger, and A. Vlad. Weaving enterprise knowledge graphs: The case of company ownership graphs. In EDBT, pages 555–566, 2020

work page 2020

[3] [3]

Balazevic, C

I. Balazevic, C. Allen, and T. Hospedales. Multi-relational poincaré graph embeddings. In Advances in Neural Information Processing Sys- tems, pages 4465–4475, 2019

work page 2019

[4] [4]

Bordes, N

A. Bordes, N. Usunier, A. Garcia-Duran, J. Weston, and O. Yakhnenko. Translating embeddings for modeling multi-relational data. In Ad- vances in neural information processing systems , pages 2787–2795, 2013

work page 2013

[5] [5]

Y . Cen, X. Zou, J. Zhang, H. Yang, J. Zhou, and J. Tang. Representation learning for attributed multiplex heterogeneous network. In Proceed- ings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, pages 1358–1368, 2019

work page 2019

[6] [6]

Chami, A

I. Chami, A. Wolf, D.-C. Juan, F. Sala, S. Ravi, and C. Ré. Low- dimensional hyperbolic knowledge graph embeddings. arXiv preprint arXiv:2005.00545, 2020

arXiv 2005

[7] [7]

Chang, I

D. Chang, I. Balaževi ´c, C. Allen, D. Chawla, C. Brandt, and R. A. Tay- lor. Benchmark and best practices for biomedical knowledge graph embeddings. In Proceedings of the conference. Association for Com- putational Linguistics. Meeting , volume 2020, page 167. NIH Public Access, 2020

work page 2020

[8] [8]

Dettmers, P

T. Dettmers, P. Minervini, P. Stenetorp, and S. Riedel. Convolutional 2d knowledge graph embeddings. In Thirty-Second AAAI Conference, 2018

work page 2018