arxiv: 2604.07886 · v3 · submitted 2026-04-09 · 💻 cs.CL

Recognition: unknown

Linear Representations of Hierarchical Concepts in Language Models

Benjamin Heinzerling, Kentaro Inui, Masaki Sakata, Sho Yokoi, Takumi Ito

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:55 UTC · model grok-4.3

classification 💻 cs.CL

keywords language modelshierarchical conceptslinear representationsprobinginterpretabilitysemantic domainsactivation subspaces

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The pith

Language models encode concept hierarchies like country-region-continent relations as linear transformations in their activations.

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

The paper tests whether hierarchical relations between concepts can be recovered directly from the internal activations of language models using linear methods. The authors train separate linear transformations for each level of a hierarchy and each semantic domain, then compare how these transformations behave across depths and domains. They show that the relations are recoverable within a domain, sit in low-dimensional subspaces, and that these subspaces remain similar even when the domains differ. A sympathetic reader would care because this points to a uniform, editable mechanism inside models for handling nested knowledge, which could affect how models perform on reasoning that requires distinguishing levels of generality.

Core claim

We investigate how and to what extent hierarchical relations are encoded in the internal representations of language models. Building on linear relational concepts, we train linear transformations specific to each hierarchical depth and semantic domain. Experiments show that, within a domain, hierarchical relations can be linearly recovered from model representations. We find that it is encoded in a relatively low-dimensional subspace and that this subspace tends to be domain-specific. Our main result is that hierarchy representation is highly similar across these domain-specific subspaces. Overall, we find that all models considered in our experiments encode concept hierarchies in the form

What carries the argument

Linear transformations trained on model activations, one per hierarchical depth and semantic domain, that recover the inclusion relations from activations across multiple layers and multi-token entities.

If this is right

Within any single semantic domain the hierarchical level of a concept can be read out linearly from the model's hidden states.
The relevant information occupies a low-dimensional subspace of the activation space.
Subspaces learned for different domains are nevertheless highly aligned with one another.
The same linear structure appears across multiple model families and layers.

Where Pith is reading between the lines

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

If the subspaces are as similar as reported, a single set of linear probes trained on one domain might transfer to many others without retraining.
The low-dimensional character suggests that hierarchy information could be edited or suppressed by simple vector operations on activations.
The finding raises the question of whether non-hierarchical relations, such as part-whole or temporal ordering, also admit comparable linear encodings.

Load-bearing premise

The trained linear transformations capture the model's genuine internal encoding of hierarchies instead of patterns introduced by the choice of training examples or the probing procedure itself.

What would settle it

If new hierarchical pairs from the same domains, withheld from the linear training step, cannot be accurately classified by the learned transformations at rates well above chance, the claim of linear recoverability would be refuted.

Figures

Figures reproduced from arXiv: 2604.07886 by Benjamin Heinzerling, Kentaro Inui, Masaki Sakata, Sho Yokoi, Takumi Ito.

**Figure 2.** Figure 2: Overview of Linear Hierarchical Encoding (LHE). Given a hierarchical relation, LHE learns depth-specific and domain-specific linear transformations that map a child concept representation to the representation of its parent concept in LM intermediate layers. Linearity of relation representations in language models. From a methodological perspective, the work most closely related to ours is that on Linear … view at source ↗

**Figure 3.** Figure 3: Results of sweeping the rank of the pseudo-inverse matrix [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Cross-domain evaluation of hierarchical-structure directions in Llama 3.1 8B. We [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Subspace similarity of subject representations for Llama 3.1 8B. We decompose [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: PCA of concept vectors for Llama 3.1 8B in the Location and Organism domains. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Similarity matrix of LM representations based on persistent homology for Llama [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: Accuracy and Causality at each hierarchy depth for all models. The Accuracy [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗

**Figure 9.** Figure 9: Examples of data representing the hierarchical structure of concepts. [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗

**Figure 10.** Figure 10: Prompt format used for both data filtering and the main experiments. [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗

**Figure 11.** Figure 11: Few-shot prompts in the template-based format. The is-a template is used for the [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Cross-domain evaluation of hierarchical-structure directions for all models. [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗

**Figure 13.** Figure 13: Results of sweeping the rank of the pseudo-inverse matrix [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗

**Figure 14.** Figure 14: Accuracy and Causality obtained by sweeping the subject layer while keeping [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗

**Figure 15.** Figure 15: Accuracy and Causality obtained by sweeping the object layer while keeping the [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗

**Figure 16.** Figure 16: Subspace similarity of subject representations for all models. [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗

**Figure 17.** Figure 17: PCA visualizations of the concept vectors for Llama 3.1 8B. [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗

**Figure 18.** Figure 18: Similarity matrices of LM representations for all models, based on persistent [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗

read the original abstract

We investigate how and to what extent hierarchical relations (e.g., Japan $\subset$ Eastern Asia $\subset$ Asia) are encoded in the internal representations of language models. Building on Linear Relational Concepts, we train linear transformations specific to each hierarchical depth and semantic domain, and characterize representational differences associated with hierarchical relations by comparing these transformations. Going beyond prior work on the representational geometry of hierarchies in LMs, our analysis covers multi-token entities and cross-layer representations. Across multiple domains we learn such transformations and evaluate in-domain generalization to unseen data and cross-domain transfer. Experiments show that, within a domain, hierarchical relations can be linearly recovered from model representations. We then analyze how hierarchical information is encoded in representation space. We find that it is encoded in a relatively low-dimensional subspace and that this subspace tends to be domain-specific. Our main result is that hierarchy representation is highly similar across these domain-specific subspaces. Overall, we find that all models considered in our experiments encode concept hierarchies in the form of highly interpretable linear representations.

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.

Desk Editor's Note private letter to a colleague

The paper shows linear probes recover hierarchies across domains with similar subspaces, but the internal-encoding claim rests on thin evidence without strong controls.

read the letter

The main point is that language models encode concept hierarchies in low-dimensional linear subspaces that turn out similar across different semantic domains. They train separate linear transformations for each hierarchy depth and domain, test recovery on held-out data within a domain, and check transfer across domains. The work also covers multi-token entities and representations from multiple layers, which goes past most prior linear probing on single-token relations or single domains. The headline result is that these domain-specific subspaces align closely, pointing to a reusable linear structure for hierarchies like geographic or taxonomic ones. All tested models show this pattern to some degree. This is a straightforward extension of linear relational concept work, and the cross-domain similarity plus multi-token handling are the concrete additions. The in-domain generalization tests and the low-dimensional subspace observation are useful observations if they hold up quantitatively. The soft spot is the evidential base. The abstract gives no accuracy numbers, no baselines against nonlinear probes or random features, and no details on how the hierarchical data was built or whether entity selection introduced regularities that a linear map could exploit. That leaves the central claim vulnerable to the concern that the probes are fitting data artifacts rather than the model's own geometry. If the full paper has ablations or controls for that, they need to be prominent; otherwise the similarity result risks being circular with the probing setup. This is aimed at people already working on mechanistic interpretability of knowledge in LMs. A reader focused on linear representations of structure would get some ideas from it, but the lack of tight metrics makes it more of a starting point than a settled result. It is coherent enough and engages the literature directly, so it deserves a serious referee who can press on the experimental design and demand clearer quantitative reporting.

Referee Report

2 major / 2 minor

Summary. The paper claims that language models encode hierarchical relations (e.g., Japan ⊂ Eastern Asia ⊂ Asia) as linear structures in their internal representations. Building on linear relational concepts, it trains per-depth and per-domain linear transformations on activations, reports in-domain generalization to unseen data and cross-domain transfer, finds that hierarchies occupy low-dimensional domain-specific subspaces that are nevertheless highly similar across domains, and concludes that all tested models encode concept hierarchies in highly interpretable linear representations, with extensions to multi-token entities and cross-layer analysis.

Significance. If substantiated, the result would strengthen evidence that LMs maintain structured, linearly recoverable representations of hierarchies rather than purely distributed or non-linear encodings. It extends prior geometry work by addressing multi-token and cross-layer cases and by quantifying subspace similarity, which could inform interpretability methods and targeted knowledge editing.

major comments (2)

Abstract: the claim of in-domain generalization and cross-domain subspace similarity is stated without any quantitative metrics, baselines, statistical tests, or details on data exclusion criteria, making it impossible to evaluate whether the evidence supports linear recoverability or the main result on subspace similarity.
Probing setup (experiments section): linear transformations are fit to labeled hierarchical relations drawn from the same data distribution used for evaluation; without controls that isolate model-internal geometry from regularities in entity selection or parent-child annotation, both the reported recoverability and the cross-domain similarity could be artifacts of the probing method rather than evidence of the LM's encoding.

minor comments (2)

Clarify the precise procedure and metric used to compare independently trained transformations across domains (e.g., cosine similarity after alignment, Procrustes distance).
Specify the number of models, domains, hierarchical depths, and entity counts, along with any hyperparameter choices for the linear probes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed comments, which highlight important aspects of clarity and methodological rigor. We address each major comment below and outline revisions to the manuscript.

read point-by-point responses

Referee: Abstract: the claim of in-domain generalization and cross-domain subspace similarity is stated without any quantitative metrics, baselines, statistical tests, or details on data exclusion criteria, making it impossible to evaluate whether the evidence supports linear recoverability or the main result on subspace similarity.

Authors: We agree that the abstract should include key quantitative results to support the claims. In the revised version, we will add specific metrics for in-domain generalization (e.g., accuracy on held-out entities), cross-domain transfer performance, subspace dimensionality, and similarity measures (such as average cosine similarity between domain subspaces), along with references to the statistical tests and data exclusion criteria already detailed in the experiments section. revision: yes
Referee: Probing setup (experiments section): linear transformations are fit to labeled hierarchical relations drawn from the same data distribution used for evaluation; without controls that isolate model-internal geometry from regularities in entity selection or parent-child annotation, both the reported recoverability and the cross-domain similarity could be artifacts of the probing method rather than evidence of the LM's encoding.

Authors: We acknowledge this valid concern regarding potential artifacts. Our current setup already evaluates on unseen entities held out from the fitting process, and we compare against random baselines. However, to more directly isolate model-internal geometry, we will add control experiments in the revision, including label permutation tests and analysis of entity selection criteria from the source knowledge bases. These additions will help demonstrate that the linear recoverability and cross-domain similarities reflect the model's representations rather than annotation regularities. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent probes and held-out evaluation

full rationale

The paper trains per-domain, per-depth linear transformations on model activations to recover hierarchical relations, then evaluates in-domain generalization on held-out data and cross-domain transfer. The key claim of similar hierarchy representations across domain-specific subspaces is obtained by comparing these independently trained transformations rather than by any self-definitional reduction, fitted-parameter renaming, or load-bearing self-citation chain. All steps use standard probing with explicit held-out splits and external model activations, keeping the derivation self-contained against the input data and labels.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on fitting linear transformations to model activations from domain-specific data; the key premise is that these fitted probes reveal genuine hierarchical encoding rather than probe-specific artifacts.

free parameters (1)

Linear transformation matrices per hierarchical depth and semantic domain
Trained on model representations to map between hierarchy levels; values are data-dependent and central to recovering the claimed linear structure.

axioms (1)

domain assumption Hierarchical relations between concepts are linearly separable in the model's representation space
This is the foundational assumption of the Linear Relational Concepts framework extended in the paper.

pith-pipeline@v0.9.0 · 5481 in / 1273 out tokens · 72119 ms · 2026-05-10T16:55:52.765327+00:00 · methodology

discussion (0)

Reference graph

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