Relation Geometry in Semantic Space of Language Models

Hiroaki Yamada; Hitomi Yanaka; Kentaro Inui; Simone Teufel; Takenobu Tokunaga; Tatsuya Hiraoka; Zhihan Cao

REVIEW 4 major objections 6 minor 243 references

Relation Geometry in Semantic Space of Language Models

T0 review · 4 major / 6 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Semantic relations do not leave equally clear footprints in language-model vector space; asymmetric ones are clearer than symmetric ones.

desk verdict Solid multi-relation geometry audit with unusually clean anti-leakage design; the asymmetric-vs-symmetric pattern is real, but absolute scores are modest and the distributional-hypothesis punchline outruns the linear-probe evidence. read the letter →

arxiv 2607.26762 v1 pith:YKLCJEPB submitted 2026-07-29 cs.CL

Zhihan Cao , Hiroaki Yamada , Simone Teufel , Tatsuya Hiraoka , Kentaro Inui , Hitomi Yanaka , Takenobu Tokunaga This is my paper · ORCID

classification cs.CL

keywords semanticrelationsrelationgeometrylanguagemodelsprobingdistributionalsemanticshypernymysynonymylexicalvscontextualinformation

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

Language models turn words into vectors, but it is unclear how much of the structure of semantic relations—hypernymy, synonymy, antonymy, and the rest—is actually present as geometry in those spaces. This paper tests that question directly: whether words related to a target by the same relation cluster in a shared region, whether those regions encode classic properties such as asymmetry and transitivity, and whether the signal comes more from word form or from context. Across causal, masked, and diffusion models, asymmetric relations show relatively clearer separable regions than symmetric ones, while directionality and long-distance transitivity are only moderately recovered. The pattern is uneven enough to suggest that not every semantic relation is equally learnable from distributional information alone. A sympathetic reader cares because this is a concrete check on a foundational claim of distributional semantics, not just another leaderboard number.

What carries the argument

A bilinear multi-relation probe that scores whether a target–relatum pair falls into a relation-specific linear “relata region,” evaluated with controlled predictability, directionality, and transitivity metrics, plus ablations that strip lexical form or correct context.

What would settle it

Under the same lemma-held-out, leakage-blocked setup, synonym or antonym pairs becoming as linearly separable and directionally consistent as hypernym or hyponym pairs, or long-distance hypernym chains remaining highly predictable when the probe is trained only on direct pairs.

Watch

Extended reading notes

Core claim

Relation geometry in language-model semantic space is not uniform: relata of asymmetric relations occupy relatively distinct linear regions, while symmetric relations do not, and properties such as directionality and especially long-distance transitivity are only moderately encoded—evidence that semantic relations are not equally recoverable from distributional learning alone.

Load-bearing premise

That how well this linear probe separates held-out word pairs is a faithful readout of whether relation-specific regions and relation properties actually exist in the model’s space.

Editorial extensions

If this is right

Asymmetric relatedness (hypernymy, meronymy and their reverses) should be easier to read out of LM embeddings than synonymy-style similarity.
Probes and applications that assume uniform relational geometry across WordNet-style relations will systematically overestimate what distributional spaces encode for synonyms and antonyms.
Causal models will lean more on surface form for relational geometry, while masked and diffusion models will lean more on context—except for indirect hypernymy, where lexical form matters more across model types.
Long-distance hierarchical inference cannot be treated as a free consequence of local hypernym geometry learned from co-occurrence alone.

Reading between the lines

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

If linear relata regions are weak for similarity but stronger for relatedness, many “analogy” or vector-offset recipes may be succeeding on relatedness structure rather than true synonymy.
The gap between short- and long-distance transitivity suggests hierarchical resources still need explicit structure beyond what next-token or masked training induces.
A natural next test is whether non-linear probes close the synonymy gap or merely confirm that the missing structure is absent, not just hard to read linearly.
Training objectives that mix shared-neighbour and co-occurrence signals may explain why LMs beat static embeddings on asymmetric relations but not on antonymy.

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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

The useful takeaway is a careful comparative measurement: six WordNet relations plus UNR, bilinear multi-class probes, directionality under order-swap, and distance-stratified transitivity trained only on direct pairs. That package is cleaner than most prior hypernymy probes.

What is actually new is the breadth plus the hygiene. Lemma-held-out folds, removal of property-exhibiting pairs from training, random-representation controls, multi-trial CV with Bonferroni-corrected permutation tests, and the ΔLEX/ΔCTX ablations are done properly. They report low absolute controlled scores honestly (F_r mostly under 0.27, D_r ≤ 0.18, long-distance T to chance). The relative pattern—asymmetric relations clearer than synonymy/antonymy, short-distance transitivity better than long—is consistent across ModernBERT, LLaDA, and LLaMA. The lexical-vs-contextual split by model family is a genuine secondary result.

Soft spots in proportion: absolute effect sizes stay small, so the leap to “counter-evidence that relations are not uniformly learnable from distribution alone” is stronger than the linear final-layer signal supports. They flag the linear-probe limit themselves. The three LMs are unmatched on scale and data, so LLaMA’s edge is confounded; the static baseline winning on antonymy is interesting but under-interpreted. No positive control where known linear geometry should recover high scores. None of this sinks the empirical core.

Math and citation pattern look fine—bilinear scoring is standard, WordNet/SemCor grounding is external, related work covers the right hypernymy and synonymy papers. No circularity problem.

This is for people who care about probing, lexical semantics, and what distributional models actually encode. Worth a serious referee. I would engage: cite the design and the relative pattern, push back on the theoretical framing in revision.

Referee Report

4 major / 6 minor

Summary. The paper asks whether six WordNet semantic relations are encoded as linearly separable “relata regions” in LM token spaces, and whether those regions reflect symmetry/asymmetry and hypernymy transitivity. It trains bilinear multi-relation probes (Eqs. 2–4) on final-layer representations from ModernBERT, LLaDA-8B, LLaMA-3.1-8B, and a fastText baseline, using SemCor/WordNet triplets with lemma-held-out folds, removal of property-exhibiting pairs from training, an unrelated control class, random-representation controls, and multi-trial CV with Bonferroni-corrected tests. It further ablates lexical vs. contextual information via masking/previous-token states and degenerate/egalitarian attention. Controlled scores show clearer (still modest) separability for asymmetric than symmetric relations, only moderate directionality and short-distance transitivity, and model-dependent reliance on lexical vs. contextual cues. The authors read this as evidence that relation geometry is uneven and that semantic relations are not uniformly learnable from distributional information alone.

Significance. The work usefully broadens relation probing beyond hypernymy, introduces property-aware metrics (directionality; leakage-blocked transitivity), a geometrically interpretable bilinear probe, and a quantitative lexical/contextual ablation (ΔLEX/ΔCTX). Methodological care—lemma anti-leakage splits, property-pair removal, random probes, multi-trial CV, and transparent reporting of low absolute controlled scores—is a genuine strength and raises the bar relative to much of the prior probing literature. If the relative asymmetric/symmetric pattern and the information-source differences hold under tighter controls, the paper supplies concrete empirical constraints for distributional-semantics theory and for how different LM objectives encode relational structure. The contribution is primarily empirical and methodological rather than a decisive theoretical refutation.

major comments (4)

[Abstract; §5.1–5.3; §5.5–6; Tables 4–6; Fig. 2] The central theoretical suggestion (abstract; §5.5–6)—that uneven relation geometry shows semantic relations are not uniformly learnable from distribution alone—overreaches the absolute evidence. Controlled F_r never exceeds ~0.27 (asymmetric) / ~0.16 (symmetric) (Table 4); D_r ≤ 0.18 (Table 5); long-distance T falls to chance or below (Fig. 2, Table 6). Relative gaps are real under the authors’ controls, but without a positive control showing that the same bilinear probe recovers high controlled scores when linear relata-region geometry is independently known to be present, modest absolute performance cannot securely license a counter-claim against distributional semantics. Temper the claim to what the relative pattern supports, or add such a control.
[Table 3; §5.1–5.4; §5.5; §6–7] Model comparison confounds scale, data, and objective (Table 3: ModernBERT 395M vs LLaMA/LLaDA 8B; unmatched corpora/context windows). LLaMA’s advantage on asymmetric F/D/T is repeatedly attributed in part to the causal objective and bidirectional vs unidirectional context (§5.5), yet Limitations §7 correctly notes these factors are entangled. As written, the abstract and discussion still invite a family-level conclusion (CLM vs MLM/DLM; lexical vs contextual importance). Either match models more carefully, add ablations that isolate objective/scale, or systematically downgrade causal language about “causal vs masked/diffusion” throughout results and conclusion.
[§3.1 Eqs. 2–5; §5.5–6; §7] The operational definition of “relation geometry” is linear separability under the bilinear probe (Eqs. 2–5, §3.1). Limitations §7 acknowledges that non-linear geometry is untested and that probing ≠ use. Because the negative findings on synonymy/antonymy and weak long-distance transitivity are load-bearing for the “not equally well-represented / not uniformly learnable” claim, the paper should either (i) include at least one non-linear probe baseline as a sensitivity check, or (ii) consistently frame all conclusions as about linear relata regions in final-layer space rather than about relation geometry or distributional learnability in general. The current framing oscillates between these.
[§3.2.3; §4.4; Fig. 2; §5.3; §5.5] Transitivity evaluation trains only on direct pairs and tests indirect pairs (good anti-leakage design), but performance collapse with distance (Fig. 2) is also consistent with sense/representation drift and decreasing semantic overlap, not only with absence of transitive geometry. The discussion (§5.5) notes contextual dissimilarity for long-distance pairs; that alternative should be quantified (e.g., baseline similarity or probe confusion as a function of path length) so that “transitivity not encoded” is distinguished from “indirect pairs are simply harder under the same linear readout.”

minor comments (6)

[Front matter] Placeholder metadata remains in the front matter (“Action editor: {action editor name}”; “Submission received: DD Month YYYY”). Clean before any revision cycle.
[Headers] Running headers still say “Author’s Surnames Here / Running Article Title Here”.
[§4.2; Table 1; Table 2] Table 1 token counts and the 6.8M indirect-triplet figure in the intro should be cross-checked for consistency with the filtering narrative (intra-sentential removal, distance cutoffs).
[Figure 1; §3.1] Figure 1 is schematic only; a 2D illustration of real probe hyperplanes or a qualitative PCA/UMAP of one target’s relata would help readers judge what “region” means empirically.
[§4.5; Appendix C] Appendix C layer analysis is valuable but under-discussed in the main text; a short pointer in §5 on whether upper-layer dominance holds for all metrics would strengthen the final-layer choice.
[§3.2.1; §3.4] Minor wording: “summay relation predictabilities” (§3.2.1); “egalitarian decontextualisation” notation could be introduced once with a short pseudocode block for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

Empirical probing study with external gold labels and held-out evaluation; no derivation reduces to its inputs by construction.

full rationale

The paper operationalizes ‘relation geometry’ via bilinear probe performance (Eqs. 2–9) on LM token pairs, with WordNet/SemCor providing external gold relations, lemma-held-out cross-validation, property-leakage blocking, and random-representation controls. Probe F/D/T scores and lexical/contextual ablations are measurements against that external structure, not quantities fitted from the same data and re-labeled as predictions. Bilinear scoring is imported from knowledge-graph embedding literature (Bordes et al., Nickel et al., etc.), not from a self-citation uniqueness claim. Self-citations (e.g. Cao et al. 2025 on antonymy) are peripheral. There is no self-definitional loop, no fitted-input-as-prediction step, and no load-bearing self-citation chain. Absolute scores are modest and models unmatched—these are validity/scope issues, not circularity. Derivation chain is self-contained empirical evaluation; score 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

Load-bearing content is methodological and empirical rather than axiomatic physics-style theory. The claim rests on standard distributional-semantics background, WordNet as gold taxonomy, the assumption that bilinear separability indexes “relata regions,” and several experimental design choices (linear probes, final layer, noun-only SemCor, leakage filters). No new physical entities; free parameters are ordinary ML hyperparameters and design thresholds, not constants fitted to prove a law.

free parameters (3)

Attention temperature τ for egalitarian decontextualisation = 100
Set by hand to 100 after informal search so attention is effectively uniform; enters all ΔCTX comparisons.
Probe training hyperparameters (Adam, 5 epochs, batch 1024, 4-fold × 10 trials) = 5 epochs, bs=1024, 4×10 CV
Chosen experimental settings that determine absolute probe fit; controlled scores mitigate but do not remove dependence.
UNR sample size and max hypernymy distance cutoff = 500000 UNR; distance ≤9 kept
500k unrelated triplets sampled; distances >10 and non-unique distances discarded (~5% indirect loss), shaping class balance and T metrics.

assumptions (5)

domain assumption Harris-style distributional semantics: regular differences in environments correspond to semantic relations that could appear as geometry in embedding space.
Frames the entire research question in §1; the paper tests rather than proves it.
domain assumption WordNet synset relations on SemCor-disambiguated nouns are an adequate gold standard for the six relations and for hypernym path distance.
All triplets and transitivity distances are defined from WordNet (§4); known WordNet DAG ambiguities are only partly filtered.
ad hoc to paper A softmax over bilinear scores e_w^T W_r e_v measures linear separability of relation-specific relata regions in the LM’s token space.
Core operationalisation in §3.1; alternative combination ops or nonlinear probes could yield different geometry conclusions (acknowledged in §7).
ad hoc to paper Random vectors drawn Uniform[0,1] yield an appropriate chance baseline so that controlled performance isolates geometry rather than label skew.
§3.3 control procedure; baseline distribution choice is conventional but not uniquely justified.
ad hoc to paper Removing the surface token (mask or previous-token state) vs forcing degenerate/egalitarian attention cleanly isolates lexical vs contextual contributions.
§3.4 ablation design; entanglement of form and context in real LMs means isolation is approximate.

invented entities (2)

Relata regions
purpose: Name the hypothesized linear regions occupied by words in a fixed relation to a target.
Central geometric construct illustrated in Fig. 1 and tested via probes; descriptive rather than a new ontological object in nature.
Controlled performance / ΔLEX / ΔCTX loss metrics
purpose: Quantify geometry beyond raw probe accuracy and importance of lexical vs contextual information.
Paper-defined evaluation quantities; useful instrumentation, not entities with external existence claims.

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Cite this review

Pith. "Pith review of Relation Geometry in Semantic Space of Language Models." pith.science (2026). https://pith.science/paper/YKLCJEPB

@misc{pith2026260726762,
  author       = {Pith},
  title        = {Pith review of: Relation Geometry in Semantic Space of Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YKLCJEPB}},
  note         = {Machine review of arXiv:2607.26762}
}

read the original abstract

When it comes to generating vector representations of words, current language models are achieving high-quality results. However, what is not known is the extent to which knowledge about semantic relations is represented in the geometry of the semantic spaces created in this way. In order to answer this question, we study the relation geometry of such semantic spaces from three perspectives. We first examine whether words standing in a particular relation to a target word~(called relata) occupy the same region in semantic space, and whether the regions corresponding to different relations are distinct from each other. We then verify to what extent semantic spaces reflect certain well-known properties of relations, such as symmetry, asymmetry, and transitivity. Finally, we consider which information about the target words and relata is more important for relation geometry: their surface forms, or their contexts. We conduct experiments on six semantic relations using causal, masked, and diffusion language models. The results show that relata in asymmetric relations relatively clearly occupy a distinct region in semantic space. Asymmetric relations' properties are only moderately well encoded in the semantic space, yet better than those of symmetric ones. Furthermore, when considering the question which information source has the strongest impact on results amongst the models we evaluated, we find that lexical information tends to be more important for the causal language model, whereas contextual information is more important for the masked and diffusion language models. Our results empirically show that relation geometry is not equally well-represented for all relations in semantic space, suggesting that there is a difference in how well semantic relations might be learned from distributional information alone.

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