cs.CLApr 9, 2026

Linear Representations of Hierarchical Concepts in Language Models

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

Organizations: Tohoku University · RIKEN · Langsmith Inc. · NINJAL · 

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

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.

Figures & tables

Figure 1: Overview of our main findings. LMs represent hierarchical concepts (left) in domain-specific subspaces of intermediate layer activations (center). The structure of these internal representations is similar across domains (right). The middle and right panels visualize, using PCA, the representations obtained by applying a linear transformation trained on hierarchical data to the hidden-state vectors corresponding to each concept in intermediate layers of Llama 3.1 8B. See App. F for details.
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.
DepthLocationsResearch TopicPersonsOrganizationsOrganisms
0 (Root)11111
154666
22225352187
3263219224235461
4279714464–––
Table 1: Number of examples by hierarchical depth across domains
 AccuracyCausality
MethodLoc.TopicPers.Org.Orgm.Loc.TopicPers.Org.Orgm.
Input averaging0.410.290.470.350.440.420.050.100.070.18
SVM0.550.360.630.720.550.580.200.320.350.28
LHE0.680.520.890.930.720.670.350.650.570.57
Table 2: Comparison of LHE and baselines in terms of Accuracy and Causality across domains for Llama 3.1 8B. Scores are averaged across hierarchical levels. Results for all models are reported in Table 10 and scores by depth in Figure 8 .
 Accuracy (LHE)Causality (LHE)
ModelLoc.TopicPers.Org.Orgm.Loc.TopicPers.Org.Orgm.
Llama 3.2 3B0.760.690.920.960.900.630.140.690.750.65
Llama 3.1 8B0.700.530.880.960.750.650.120.640.520.53
Qwen3 8B0.770.620.920.980.810.550.190.460.520.35
Qwen3 14B0.760.580.850.900.800.550.120.250.460.12
Table 3: Comparison of model performance using LHE in terms of Accuracy and Causality by domain. The results are computed on child-parent examples shared by all four models after LM-based word-prediction filtering. Scores are averaged across hierarchical levels.
[Location] Cairo All correct Pred: Cairo ⊂\subset Egypt ✓ ⊂\subset Northern Africa ✓ ⊂\subset Africa ✓ Gold: Cairo ⊂\subset Egypt ⊂\subset Northern Africa ⊂\subset Africa
[Person] Lionel Messi All correct Pred: Lionel Messi ⊂\subset Soccer player ✓ ⊂\subset Athlete ✓ Gold: Lionel Messi ⊂\subset Soccer player ⊂\subset Athlete
[Location] Leping Bottom ✗, Top ✓ Pred: Leping ⊂\subset India ✗ ⊄\not\subset Eastern Asia ✓ ⊂\subset Asia ✓ Gold: Leping ⊂\subset China ⊂\subset Eastern Asia ⊂\subset Asia
[Organism] Dolphin Bottom ✓, Top ✗ Pred: Dolphin ⊂\subset Cetacea ✓ ⊄\not\subset Fish ✗ Gold: Dolphin ⊂\subset Cetacea ⊂\subset Mammal
[Organization] The New York Times Company Bottom ✗, Top ✓ Pred: The New York Times Company ⊂\subset Consumer goods company ✗ ⊂\subset Company ✓ Gold: The New York Times Company ⊂\subset Media company ⊂\subset Company
Table 4: Examples for accuracy using Llama 3.1 8B. Each row shows an entity and its predicted parent chain ( Pred ) with step-wise correctness (✓/✗) against the gold chain ( Gold ). Correct and incorrect predicted parents are highlighted in green and red, respectively, and ⊂\subset / ⊄\not\subset indicate inclusion vs. non-inclusion relations between consecutive depths. We group cases by whether errors occur earlier vs. later along the evaluated depth sequence; this table shows representative instances of All correct , Bottom ✗, Top ✓ , and Bottom ✓, Top ✗ . Additional examples are provided in Appendix Table 9 .
Table 5: Causality success and failure cases obtained with Llama 3.1 8B. Before : the LM’s original prediction. Target : the intended prediction after intervention. After : the actual prediction after adding the concept vector difference.
Figure 3: Results of sweeping the rank of the pseudo-inverse matrix Wr†\mathbf{W}_{r}^{\dagger} for Llama 3.1 8B. Scores are averaged across all domains. The vertical dotted line indicates the point at which the average of Accuracy and Causality is maximized. The results for all models are reported in App. E , confirming that the same trend is consistently observed across models.
Figure 4: Cross-domain evaluation of hierarchical-structure directions in Llama 3.1 8B. We train a linear transformation on a domain (x-axis) and evaluate it on test domains (y-axis). Left: Accuracy. Right: Causality (intervention success). Causality drops sharply under domain shift, while Accuracy remains relatively stable. Results for all models in Fig. 12 .
Figure 5: Subspace similarity of subject representations for Llama 3.1 8B. We decompose the learned linear transformation matrix WW via a low-rank SVD, W≈Udiag(s)V⊤W\approx U\mathrm{diag}(s)V^{\top} , and quantify the overlap between the subspaces spanned by the right singular vectors VV as overlap(A,B)=1r∥VA⊤VB∥F2\mathrm{overlap}(A,B)=\frac{1}{r}\|V_{A}^{\top}V_{B}\|_{F}^{2} , where rr denotes the rank and A,BA,B are the two conditions being compared. Results for all models in Fig. 16 .
Figure 6: PCA of concept vectors for Llama 3.1 8B in the Location and Organism domains. Concept vectors are obtained by applying a low-rank pseudo-inverse of the learned transformation to parent representations and averaging over examples grouped by parent concept.
Figure 7: Similarity matrix of LM representations based on persistent homology for Llama 3.1 8B. Each entry shows the Wasserstein distance, computed on persistence diagrams derived from concept vector point clouds, between a pair of domains or datasets. Smaller values indicate more similar representational structure. Results for all models in Fig. 18 .
Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

DepthLocationsResearch TopicPersonsOrganizationsOrganisms
Model: Llama 3.2 3B
0 (Root)11111
153656
22112311440
315179188157354
4208201302–––
Appendix
Table 6: Number of LM-filtered examples by hierarchical depth across domains and models, where parent-child relations are retained only when the model correctly predicts the parent category
ModelsHidden dim.#Layer#Head
Llama 3.2 3B30722824
Llama 3.1 8B40963232
Qwen3 8B40963632
Qwen3 14B51204040
Appendix
Table 7: Hyperparameters of each model’s architecture.
Figure 8: Accuracy and Causality at each hierarchy depth for all models. The Accuracy reported in this figure is chance-corrected to account for the depth-specific chance rate and to enable fair comparison across depths.
Figure 9: Examples of data representing the hierarchical structure of concepts.
Figure 10: Prompt format used for both data filtering and the main experiments.
Figure 11: Few-shot prompts in the template-based format. The is-a template is used for the Person, Organization, Organism, and Research Topic domains, whereas the part-of template is used for the Location domain. Results obtained with these prompts are shown in Tbl. 11 .
ModelSubject LayerObject LayerRank
Llama 3.2 3B1215128
Llama 3.1 8B1221160
Qwen3 8B Base2027192
Qwen3 14B Base2528256
Appendix
Table 8: Hyperparameters used for LRC training and inference for each model.
[Organism] panther All correct Pred: panther ⊂\subset Carnivora ✓ ⊂\subset Mammal ✓ Gold: panther ⊂\subset Carnivora ⊂\subset Mammal
[Organization] Toyota All correct Pred: Toyota ⊂\subset Automotive company ✓ ⊂\subset Company ✓ Gold: Toyota ⊂\subset Automotive company ⊂\subset Company
[Research Topic] Information Retrieval and Search Behavior Bottom ✓, Top ✗ Pred: Information Retrieval and Search Behavior ⊂\subset Information Systems ✓ ⊂\subset Computer Science ✓ ⊄\not\subset Social Sciences ✗ Gold: Information Retrieval and Search Behavior ⊂\subset Information Systems ⊂\subset Computer Science ⊂\subset Physical Sciences
Appendix
Table 9: Qualitative examples for accuracy using Llama 3.1 8B. Each row shows an entity and its predicted parent chain ( Pred ) with step-wise correctness (✓/✗) against the gold chain ( Gold ). Correct and incorrect predicted parents are highlighted in green and red, respectively, and ⊂\subset / ⊄\not\subset indicate inclusion vs. non-inclusion relations between consecutive depths. We group cases by whether errors occur earlier vs. later along the evaluated depth sequence; this table shows representative instances of All correct , Bottom ✗, Top ✓ , and Bottom ✓, Top ✗ .
Figure 12: Cross-domain evaluation of hierarchical-structure directions for all models.
Figure 13: Results of sweeping the rank of the pseudo-inverse matrix Wr†\mathbf{W}_{r}^{\dagger} for all models. Scores are averaged across all domains. The vertical dotted line indicates the point at which the average of Accuracy and Causality is maximized.
Figure 14: Accuracy and Causality obtained by sweeping the subject layer while keeping the object layer fixed.
Figure 15: Accuracy and Causality obtained by sweeping the object layer while keeping the subject layer fixed.
Figure 16: Subspace similarity of subject representations for all models.
Figure 17: PCA visualizations of the concept vectors for Llama 3.1 8B.
Figure 18: Similarity matrices of LM representations for all models, based on persistent homology. Each entry shows the Wasserstein distance (computed on persistence diagrams derived from the concept vector point clouds; log1p scale) between a pair of domains/datasets. Smaller values indicate more similar representational structure.
More in this topic

Hierarchical Representation Learning

Hierarchical representation learning organizes data into nested, coarse-to-fine features that capture parts, patterns, and global structure.

See all 224 papers

Explore similar work

CardsList

  1. H-Probes: Extracting Hierarchical Structures From Latent Representations of Language Models

    Apr 15, 2026Cutter Dawes, Aryan Sharma, Angelos Ioannis Lagos +1Representation Geometry in Language ModelsHierarchical Representation Learning

    Supervised Program for Alignment Research · Yale University · Harvard University

  2. Do Language Models Encode Semantic Relations? Probing and Sparse Feature Analysis

    Mar 18, 2026Andor Diera, Ansgar ScherpLLM Interpretability

    Ulm University, Ulm, Germany

  3. The Answer-Basin Representation Hypothesis: We Are Not Probing or Steering Concepts

    Sep 21, 2026Manjiang Yu, Hongji Li, Zihan Wang +5LLM InterpretabilityLinear Representation Hypothesis

    The University of Queensland · Mohamed bin Zayed University of Artificial Intelligence · Institute of Science Tokyo