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arxiv: 2605.22532 · v1 · pith:I6WVSZTWnew · submitted 2026-05-21 · 💻 cs.LG

Relational Linear Properties in Language Models: An Empirical Investigation

Giovanni Valer , Luigi Gresele , Marco Bronzini , Emanuele Marconato This is my paper

Pith reviewed 2026-05-22 07:41 UTC · model grok-4.3

classification 💻 cs.LG
keywords relational linearitylanguage modelsprobing methodsKL divergenceembeddingslinear mapsparaphrased queries
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The pith

A linear map can predict an object's embedding from its subject's for any fixed relation in language models.

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

The paper tests the idea that language model representations encode relations in a linear fashion, so that for a given relation the unembedding of an object can be recovered from the embedding of its subject through a linear transformation. It introduces a probing technique based on Kullback-Leibler divergence to measure the strength of this property and applies it across multiple models and datasets. The results show that the degree of linearity differs between models, follows recognizable patterns from one layer to the next, and changes when the same relation is expressed in different words. A reader would care because this structure could explain how models store and retrieve factual associations internally.

Core claim

Relational linearity holds to varying degrees: for a fixed relation, the unembedding of an object can be predicted from the embedding of its subject by a linear map. The paper demonstrates this using a new probing method based on Kullback-Leibler divergence that compares predicted and observed output distributions. Experiments across four datasets reveal that the property varies across models, exhibits layer-wise patterns, and is affected differently by paraphrased versions of the relational query.

What carries the argument

The KL-divergence probing method, which finds the linear map that minimizes the divergence between the model's actual next-token distribution for the object and the distribution obtained by transforming the subject embedding.

If this is right

  • The strength of relational linearity differs from one language model to another.
  • The property shows systematic changes across successive layers that align with where linguistic information is processed.
  • Paraphrasing the way a relation is stated in the input query alters the measured degree of linearity.
  • The new probing approach evaluates the property more efficiently than methods that rely on Jacobian approximations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the linear maps prove stable, they could be used to edit specific relational facts inside a model by adjusting only the map rather than the full parameters.
  • The same measurement approach might be applied to test linearity for other kinds of structured knowledge beyond binary relations.
  • Layer-wise differences could guide the choice of which internal states to inspect or modify when studying model reasoning.

Load-bearing premise

The KL-divergence probing method accurately captures the degree of relational linearity without introducing its own biases or requiring Jacobian approximations.

What would settle it

A concrete falsifier would be to find that, for a large collection of relations and subject-object pairs, the lowest KL divergence achieved by any linear map is no smaller than the divergence achieved by a random map or by using the subject embedding unchanged.

Figures

Figures reproduced from arXiv: 2605.22532 by Emanuele Marconato, Giovanni Valer, Luigi Gresele, Marco Bronzini.

Figure 1
Figure 1. Figure 1: Relational linearity of final-layer representations for the TENSE dataset. For Llama-3.1 (two left columns) and Gemma-2 (two right columns), we compare the embeddings computed from contexts concatenated with the query q = “What is the tense of the previous sentence?” f(s ⌢ q), with embeddings obtained by KL-based Relational Probe (KL-RP) on the context-only embeddings f(s), for contexts s from the TENSE da… view at source ↗
Figure 2
Figure 2. Figure 2: Relational linear probing (KL-RP) with different para￾phrases. The graphs show F1(LLM) for both models. improvements in middle layers (with major drops in the TENSE and TRUTH datasets). Last-layers often increase in dKL but not sensibly enough to downgrade F1 scores. Overall, middle layers for all datasets show high strong and weak relational linear probing for Llama-3.1. Instead, Gemma-2 representations i… view at source ↗
Figure 3
Figure 3. Figure 3: Relational linear probing across layers (normalized into the range 0–100), comparing Llama-3.1 and Gemma-2. progressively in Gemma-2, indicating that its middle lay￾ers may capture more abstract representations of this latent property, and (ii) Llama-3.1 appears to encode relational linearity earlier, as changes in query surface form have no effect across all layers. 5. Limitations Our work presents some l… view at source ↗
Figure 4
Figure 4. Figure 4: Sensibility of LRE when changing hyperparameters. F1(LLM) scores on TRUTH using different configurations of LRE hyper-parameters, (a) varying ℓ, β, and ρ, or (b) with a fixed layer ℓ. B.3. Randomly permuted embeddings for baseline The random baseline is defined by rearranging the embeddings with π, a random permutation of {1, . . . , N}, so that the randomized training data is: D rand q = f(sπ(i)), p(· | … view at source ↗
Figure 5
Figure 5. Figure 5: Histograms of probabilities for Llama-3.1 and Gemma-2. The x-axis reports max p(y | q˜ ⌢ s ⌢ ˜t) when prompted with the query. The y-axis is the frequency in log scale. We observe that Gemma-2 tends to be more overconfident than Llama-3.1, collapsing output probabilities to high confidence regions two times more in order of magnitude than low confidence intervals. C. Paraphrases [PITH_FULL_IMAGE:figures/f… view at source ↗
read the original abstract

Linear properties are ubiquitous in the representations of language models; however, testing them experimentally remains a challenging task. This work focuses on relational linearity: the hypothesis that, for a fixed relation (e.g., "plays"), the unembedding of an object (e.g., "trumpet") can be predicted from the embedding of its subject (e.g.,"Miles Davis") by a linear map. We present an experimental method to test the formulation of relational linearity by Marconato et al. (2025). Specifically, we introduce a probing method, based on Kullback-Leibler divergence, to evaluate this property and examine its variation across layers and paraphrased relational queries. It is also more efficient than previous work; for example, it avoids the crude Jacobian approximations used in Linear Relational Embeddings by Hernandez et al. (2024). Our findings across four datasets show that relational linearity varies across models, exhibits layer-wise patterns consistent with prior observations about linguistic information in model representations, and is differently affected by changes in how the relation is phrased.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper empirically investigates relational linearity in language models: the hypothesis that, for a fixed relation, the unembedding of an object can be linearly predicted from the embedding of its subject. It introduces a KL-divergence-based probing method to test this property (building on Marconato et al. 2025), applies it across four datasets and multiple models, and reports that the property varies across models, exhibits layer-wise patterns, and is affected by paraphrased queries. The method is presented as more efficient than prior work by avoiding Jacobian approximations.

Significance. If the central empirical claims hold after validation of the probe, the work would contribute a scalable test for relational structure in LM representations and document its variation, which could inform mechanistic interpretability and the design of relation-aware interventions. The efficiency claim and layer-wise findings align with existing observations about where linguistic information is encoded.

major comments (1)
  1. [Experimental method] The KL-divergence probing method (described in the experimental method section) minimizes KL between the model's output distribution (conditioned on subject + relation) and a target distribution derived from the object while varying the linear map. This approach remains coupled to the full unembedding matrix and softmax geometry; a non-linear perturbation that preserves marginal token probabilities could produce low KL without satisfying the claimed linear relation between subject embedding and object unembedding. No section reports a direct comparison (e.g., least-squares R² on the same embeddings) that would confirm the probe isolates the intended linear property rather than output-distribution match.
minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from a brief explicit statement of the precise mathematical formulation of relational linearity being tested (e.g., the exact linear map and how the target distribution is constructed).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive review of our manuscript on relational linear properties in language models. We address the major comment on the experimental method below and will incorporate additional validation to strengthen the claims.

read point-by-point responses

  1. Referee: [Experimental method] The KL-divergence probing method (described in the experimental method section) minimizes KL between the model's output distribution (conditioned on subject + relation) and a target distribution derived from the object while varying the linear map. This approach remains coupled to the full unembedding matrix and softmax geometry; a non-linear perturbation that preserves marginal token probabilities could produce low KL without satisfying the claimed linear relation between subject embedding and object unembedding. No section reports a direct comparison (e.g., least-squares R² on the same embeddings) that would confirm the probe isolates the intended linear property rather than output-distribution match.

    Authors: We appreciate the referee's point regarding potential confounding factors in our KL-divergence probing method. The approach optimizes a linear map to align the model's output distribution (conditioned on subject plus relation) with a target derived from the object, thereby testing whether a linear transformation suffices for the relational prediction under the model's unembedding and softmax. This is intended to directly evaluate the hypothesis from Marconato et al. (2025) in a manner that is more efficient than Jacobian-based alternatives. That said, we acknowledge that the probe is coupled to the output geometry and that non-linear perturbations preserving token probabilities could in principle yield low KL without a strict linear relation in embedding space. To provide independent confirmation, we will add a direct least-squares comparison (including R² metrics) on the same subject and object embeddings in the revised experimental method section. This will help isolate the linear property from output-distribution effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical test of external hypothesis

full rationale

The paper introduces a KL-divergence probing method as an experimental tool to evaluate the relational linearity hypothesis formulated in prior work (Marconato et al. 2025). No equations, fitted parameters, or results are shown to reduce the measured linearity to a quantity defined by the same experiment or to a self-citation chain that bears the central claim. The method is motivated independently as an efficiency improvement over Jacobian approximations, and findings are reported from direct experiments on four datasets. The derivation chain consists of hypothesis testing rather than self-referential construction, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior formulation of relational linearity from Marconato et al. (2025) and the validity of KL divergence as a linearity metric; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Language model hidden states contain extractable linear relational structure for fixed relations
    This is the hypothesis being tested rather than proved; it is invoked when the probing method is applied to evaluate the linear map prediction.

pith-pipeline@v0.9.0 · 5717 in / 1350 out tokens · 32989 ms · 2026-05-22T07:41:20.245717+00:00 · methodology

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Reference graph

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