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REVIEW 4 major objections 6 minor 32 references

Toward Understanding The Effect Of Loss function On Then Performance Of Knowledge Graph Embedding

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Loss function, not score function, settles TransE's limits

desk verdict Useful reframing of loss functions as shaping a 'region of truth', but the central theorem that margin losses let TransE encode symmetric/reflexive patterns is only proven locally and fails for large symmetric cliques. read the letter →

arxiv 1909.00519 v2 pith:XLYACQ6A submitted 2019-09-02 cs.AI cs.CL

Mojtaba Nayyeri , Chengjin Xu , Yadollah Yaghoobzadeh , Hamed Shariat Yazdi , Jens Lehmann This is my paper · ORCID
classification cs.AIcs.CL
keywords knowledgegraphembeddingTransElossfunctionrelationpatternssymmetricrelationstranslation-basedmodelscomplexembeddingslinkprediction
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

This paper argues that the well-known limitations of the TransE knowledge-graph embedding model—its supposed inability to encode symmetric, reflexive, and transitive relations—are not intrinsic to the scoring function but artifacts of the loss function used in training. The authors show that when the loss imposes only an upper bound on the scores of positive triples (rather than requiring exact equality h+r=t), TransE can represent these relation patterns with a non-zero relation vector. They prove this by defining four 'conditions of truth' for a triple, and they introduce TransComplEx, a translation model in complex space that inherits these benefits and has fewer theoretical limitations. If correct, this reframes a central debate in knowledge-graph embeddings: model capacity depends as much on the loss as on the score function.

What carries the argument

The 'region of truth' is the central object: a hypersphere (or ball) in the score space that defines which triples count as positive. The paper distinguishes four conditions: (a) score exactly zero for positives, (b) score exactly gamma1 on the sphere boundary, (c) score at most gamma1 inside the ball, and (d) triple-specific margins. The loss functions corresponding to these conditions set different upper bounds on positive scores, and the lemmas show that limitations L1-L6 only hold under condition (a), where the region of truth is a single point. The key algebraic step for symmetric relations is deriving the necessity condition ||h + r - t|| = ||t + r - h||, which, after substituting u = h - t, yields ||u||^2 = ($alpha^{2}$ - 1)||r||^2 and forces orthogonality when $\alpha$ > 1.

What would settle it

Take a synthetic knowledge graph with a single symmetric relation r and many entity pairs (a_i, b_i) with both (a_i, r, b_i) and (b_i, r, a_i) as positive triples. Train TransE with loss 6 (inside-ball, gamma1 = alpha||r||, alpha > 1). If the theorem is right, the learned relation vector should be non-zero and each pair should satisfy u_i = a_i - b_i orthogonal to r. If the optimal solution collapses to r = 0, or if the margin constraints cannot be satisfied for all pairs simultaneously, the local-to-global extension fails and the central claim is refuted.

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Extended reading notes

Core claim

The central claim is that existing proofs of TransE's limitations, which conclude that symmetric or reflexive relations force the relation vector to zero, rely on the strict condition that a positive triple must satisfy h + r = t exactly. Under less restrictive conditions—where a positive triple's score is allowed to lie on or inside a hypersphere of radius gamma1, with negative triples kept above a larger threshold gamma2—TransE can encode symmetric relations with non-zero relation vectors. In particular, Lemma 3 shows that for a symmetric relation, setting the radius to gamma1 = alpha||r|| with alpha > 1 permits a solution where the difference vector u = h - t is orthogonal to r, so entities remain distinct. The paper also proves that the other limitations (L4-L6) fail under these conditions, and it introduces TransComplEx, whose conjugate tail term allows symmetric and neither-reflexive-nor-irreflexive patterns even under the strictest condition.

Load-bearing premise

The proofs for encoding symmetric relations construct a solution for a single pair of entities (u perpendicular to r) and assume it can be extended consistently to all entity pairs in the relation graph without violating the margin constraints, but no global construction or argument is provided.

Editorial extensions

If this is right

  • TransE trained with a loss that only upper-bounds positive scores (conditions b or c) can encode symmetric and reflexive relations, so practitioners no longer need to abandon TransE for these patterns.
  • The same upper-bound argument applies to TransH, TransR, and other translation-based models, meaning their known limitations are also loss-dependent, not purely architectural.
  • The theoretical comparison predicts that loss 6 (inside-ball) outperforms loss 4 (exact-boundary), which is confirmed by the reported experiments on FB15K and FB15K-237.
  • TransComplEx, by using the conjugate of the tail vector, can represent symmetric relations even under the strictest condition (a), giving it a theoretical advantage over TransE without added computational cost.
  • The reformulation of losses as constrained-optimization problems provides a principled way to select margins (gamma1, gamma2) for any distance-based embedding model.

Reading between the lines

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

  • The 'region of truth' perspective generalizes beyond TransE: any distance-based scoring function, including RotatE and TorusE, likely has its relation-pattern limitations modulated by the loss function, so reported 'inherent' limitations should be re-examined with loss in mind.
  • If loss choice is this influential, then comparisons between embedding models under a single fixed loss are incomplete; fair comparison requires tuning or reporting across the four condition types, which could change leaderboard rankings on benchmark datasets.
  • A testable extension: on a synthetic graph containing only one symmetric relation and many entity pairs, training TransE with condition (b) should converge to a non-zero relation vector with u orthogonal to r for every pair; observing r collapse to zero would refute the global extension of the local proof.
  • The paper leaves 1-N, N-1, and M-N relation handling as future work, but the same upper-bound argument suggests that these patterns can also be mitigated by loss design, potentially reducing the need for relation-specific projection matrices.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper argues that the widely cited limitations of TransE in encoding symmetric, reflexive, and transitive relation patterns are not intrinsic to the scoring function but depend on the choice of loss function. The authors define four conditions on positive and negative triple scores: exact equality at zero (a), equality at a positive margin (b), upper-bound at a positive margin (c), and triple-specific upper-bound (d). They then state lemmas claiming that under conditions (b), (c), and (d), TransE can encode reflexive and symmetric patterns, and that limitations L4, L5, and L6 no longer hold. They also propose TransComplEx, a complex-space translation model, and provide experimental comparisons on FB15K, FB15K-237, WN18, and WN18RR using margin-based losses and relation-pattern regularizers.

Significance. The conceptual point is genuinely valuable: prior impossibility proofs for TransE assume that positive triples must satisfy h+r=t exactly, whereas margin-based losses only require the score to lie below or at a positive bound. If the paper's existence claims were fully established, they would correct an overstatement in the existing literature and redirect attention to the joint role of score function and loss function. The paper also contains a useful reformulation of several losses as constrained optimization problems and proposes a simple new model, TransComplEx, with competitive empirical results across four standard datasets. The experimental section is broad and reports hyperparameter ranges and multiple baselines, which is a strength.

major comments (4)
  1. [Section 4.2, Supplementary Lemma 1] Lemma 1(1) is false as stated for TransComplEx under condition (a). From h+r−\bar{h}=0, the proof derives Re(r)=0 and Im(r)=−2Im(h), but this only fixes the imaginary part of every entity to be −Im(r)/2; the real parts of the entity embeddings are unconstrained. Choosing r=(0,i c) with c≠0 and h=a−i c/2 gives h+r−\bar{h}=0, so a non-zero relation vector encodes a reflexive relation in TransComplEx. The conclusion that "all entities will have same embedding vectors" is not implied by the equations. This error directly affects the lemma's central claim that TransComplEx cannot infer reflexive patterns with non-zero relation vectors.
  2. [Section 4.2, Eqs. (8)-(10) and Supplementary Lemma 3] Lemma 3(3) is not established by the given proof. The derivation shows that for a single pair (h,t), the two positive triple constraints imply u=h−t is orthogonal to r and satisfies ‖u‖^2=(α^2−1)‖r‖^2. This is a local necessary condition, not a construction of a global embedding. The proof never shows that the equalities ‖h+r−t‖=γ1 and ‖t+r−h‖=γ1 can be satisfied simultaneously for all pairs in a symmetric relation, nor that all corrupted negative triples satisfy the score constraint ≥γ2. The gap is load-bearing: for a clique of d+1 entities in dimension d, all pairwise differences must be mutually equidistant and lie in the (d−1)-dimensional subspace orthogonal to r, which is impossible (e.g., a triangle in d=2). Thus the unqualified claim that "TransE can infer a relation pattern which is symmetric with conditions (b)" is false for general symmetric patterns; at best a restricted family of symmetric relations is demonstrated.
  3. [Supplementary Lemmas 4-6] The proofs of Lemmas 4, 5, and 6 exhibit the same local-to-global gap. For example, the proof of Lemma 6 verifies only that a single inequality ‖e2+r−s2‖>γ1 can be consistent with three equalities for a particular four-tuple of entities; it does not construct embeddings for all entities in the relation graph that simultaneously satisfy all positive and negative constraints. Similarly, the proof of Lemma 5 derives an inequality ⟨e2,(e1+e2+e3)⟩<0 as a non-contradiction condition but never shows a global assignment. These lemmas are used to conclude that limitations L4, L5, and L6 "are not valid" under conditions (b)-(d), so the missing global consistency argument is essential to the paper's central claim that the previous limitation theories are inaccurate.
  4. [Supplementary Lemma 3 statement] The supplementary statement of Lemma 3(3) says TransE can infer a relation pattern which is "symmetric and reflexive" under conditions (b), (c), and (d), while the main-text Lemma 3(3) claims only "symmetric." The proof only treats the symmetric pair constraints and does not show reflexivity. Additionally, the supplementary proof says α≥1 whereas the main proof requires α>1; if α=1 then ‖u‖=0, which collapses the two entities and does not give a non-trivial symmetric encoding. The intended claim and its exact conditions should be stated consistently and proved completely.
minor comments (6)
  1. [Title] The title contains a typo: "Then Performance" should read "the Performance."
  2. [Section 4.2, Eq. (10)] There is a typographical error in the derivation following Eq. (10): "−(|u‖2+(1−α2)‖r‖2)" should be "−(‖u‖2+(1−α2)‖r‖2)."
  3. [Supplementary Lemma 1] In the proof of Lemma 1, condition (b) is claimed to give ‖r‖=γ1 for TransComplEx; the correct residual is Re(r)+i(Im(r)+2Im(h)), so the norm is not generally ‖r‖. The existence claim may still be true with a different assignment, but the stated derivation is incorrect.
  4. [Section 5, Discussion] The subsection heading "Dissuasion of Results" should be "Discussion of Results," and the text contains the typo "PRTransComplEx" for "RPTransComplEx."
  5. [References] The references list two entries, Zhou et al. 2017a and Zhou et al. 2017b, with the same title and venue; it appears the same paper is cited twice with different year labels. The duplicate should be removed and the in-text citations unified.
  6. [Section 4.3] The algebraic formula listed for the symmetric pattern, RS=‖Re(h)−Re(t)‖=0, appears to omit the relation vector and is not derived in the text; please provide the intermediate steps connecting fr(h,t)=fr(t,h) to that formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper proves existence results from explicit score-bound equations, and its self-citations are not load-bearing.

full rationale

The paper's central derivation is a set of mathematical lemmas showing that, under relaxed positive-score bounds corresponding to specific loss functions, TransE can encode symmetric and reflexive relation patterns. These proofs solve the stated equations (e.g., Eqs. 8-10 and 20-22) directly; they are not fitted to data and do not rename any empirical result. The four conditions in Eq. 2 are modeling choices tied to real losses (Eqs. 4, 6, 7), and the paper proves consequences from those choices, so the conclusions are not equivalent to the premises by construction. The self-citations to Nayyeri et al. (2019) for the soft-margin loss formulations are not load-bearing because those losses are fully restated in this paper and the new theorems about them are proved here. The proof's reliance on local pair-wise constructions (e.g., Lemma 3 showing a single u perpendicular to r) is a possible mathematical gap regarding global consistency, but that is a correctness concern, not circularity. Accordingly, no circular step is identified.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central proof structure depends on newly introduced margin conditions and on global consistency assumptions not fully proven. The free parameters are the margins and slack weights used in the loss, which are tuned on validation data. No novel physical or mathematical entities are introduced.

free parameters (3)
  • gamma1 = 0.4, 1.5, 1, 0.6, 1.6 (per dataset)
    Upper bound for positive triple scores; the proofs require gamma1 = alpha * ||r|| with alpha > 1, and the value is tuned on validation data.
  • gamma2 = 0.5, 2, 2, 1.7, 2.7 (per dataset)
    Lower bound for negative triple scores; must exceed gamma1, tuned on validation data.
  • lambda0 = 1, 10, 100 (per dataset)
    Weight on the slack variable in the soft-margin loss (12), tuned from {0.01, 0.1, 1, 10, 100}.
assumptions (5)
  • standard math Score functions are norms in real or complex vector spaces.
    Both TransE and TransComplEx use norm-based distances; the proofs rely on properties of inner products and norms.
  • domain assumption Relation vectors are assumed to be non-zero.
    Section 4.2 explicitly assumes relation vectors should not be null, since a null vector would collapse entity roles in condition (a).
  • ad hoc to paper Positive triples can be defined by score upper bounds (conditions b, c, d) rather than exact equality.
    The paper introduces these conditions as a modeling choice and derives all limitation results under them; this is not an externally established fact.
  • ad hoc to paper Solutions for individual triple pairs can be combined into a global embedding satisfying all constraints simultaneously.
    The proofs for symmetric relations only construct vectors for a single pair and assume extension to the full relation graph without explicit justification.
  • domain assumption Negative samples are generated by corrupting head or tail of positive triples.
    Standard negative sampling in TransE (Bordes et al., 2013) is assumed throughout the theoretical and experimental sections.

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Cite this review

Pith. "Pith review of Toward Understanding The Effect Of Loss function On Then Performance Of Knowledge Graph Embedding." pith.science (2026). https://pith.science/paper/XLYACQ6A

@misc{pith2026190900519,
  author       = {Pith},
  title        = {Pith review of: Toward Understanding The Effect Of Loss function On Then Performance Of Knowledge Graph Embedding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLYACQ6A}},
  note         = {Machine review of arXiv:1909.00519}
}
read the original abstract

Knowledge graphs (KGs) represent world's facts in structured forms. KG completion exploits the existing facts in a KG to discover new ones. Translation-based embedding model (TransE) is a prominent formulation to do KG completion. Despite the efficiency of TransE in memory and time, it suffers from several limitations in encoding relation patterns such as symmetric, reflexive etc. To resolve this problem, most of the attempts have circled around the revision of the score function of TransE i.e., proposing a more complicated score function such as Trans(A, D, G, H, R, etc) to mitigate the limitations. In this paper, we tackle this problem from a different perspective. We show that existing theories corresponding to the limitations of TransE are inaccurate because they ignore the effect of loss function. Accordingly, we pose theoretical investigations of the main limitations of TransE in the light of loss function. To the best of our knowledge, this has not been investigated so far comprehensively. We show that by a proper selection of the loss function for training the TransE model, the main limitations of the model are mitigated. This is explained by setting upper-bound for the scores of positive samples, showing the region of truth (i.e., the region that a triple is considered positive by the model). Our theoretical proofs with experimental results fill the gap between the capability of translation-based class of embedding models and the loss function. The theories emphasise the importance of the selection of the loss functions for training the models. Our experimental evaluations on different loss functions used for training the models justify our theoretical proofs and confirm the importance of the loss functions on the performance.

Figures

Figures reproduced from arXiv: 1909.00519 by the authors.

Figure 1
Figure 1. The region of truth for a triple: A triple is [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Necessity condition for encoding symmet [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Investigation of L6 with condition (c): The [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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