Linear Representations of Hierarchical Concepts in Language Models

1. Problem Statement

While recent research has established that Large Language Models (LMs) encode various conceptual structures (e.g., colors, spatial relations, time) in their internal representations, the specific encoding of hierarchical concepts (e.g., $Japan \subset Eastern Asia \subset Asia$) remains poorly understood.

Prior work, such as Park et al. (2025), has shown that hierarchical relations exist in the unembedding layer (the final output layer) of LMs. However, this approach has two significant limitations:

1. Layer Restriction: It ignores intermediate layers where the model performs complex reasoning and where many relational computations occur.
2. Token Limitation: It is restricted to single-token concepts, failing to handle multi-token entities (e.g., "Babe Ruth" or "The New York Times Company") which are common in real-world hierarchies.

The paper aims to investigate how and to what extent hierarchical relations are encoded across intermediate layers and for multi-token entities.

2. Methodology: Linear Hierarchical Encoding (LHE)

The authors propose Linear Hierarchical Encoding (LHE), a framework extending Linear Relational Concepts (LRC) (Chanin et al., 2024) to analyze hierarchical structures.

• Data Construction: The authors created a dataset of concept hierarchies across five domains: Locations, Persons, Organizations, Organisms, and Research Topics. These datasets include diverse multi-token entities.
• Filtering: To ensure the model possesses the knowledge, data instances were filtered using a few-shot multiple-choice QA setting; only examples the model answered correctly were retained for analysis.
• Linear Transformation Learning:
  • The framework treats a hierarchy as a chain of parent-child relations.
  • For a specific relation $r$ (e.g., "part-of") at a specific depth $d$ (e.g., depth 2 to 3), the model learns a linear transformation $R_r$ that maps the hidden state of a child node ($s$) to its parent node ($o$).
  • The transformation is defined as: $R_r(s) = W_r s + b_r$.
  • Crucially, separate transformations are learned for each hierarchical depth and each semantic domain.
• Concept Vectors: Using the pseudo-inverse of the learned transformation ($W_r^\dagger$), the authors derive "concept vectors" ($v_{r,c}$) for parent concepts. These vectors represent the direction in the representation space that corresponds to a specific parent concept (e.g., the direction for "part-of: Asia").
• Evaluation Metrics:
  1. Accuracy: Measures if the internal representation of a child node can be linearly decoded to identify its correct parent.
     $$\hat{y} = \arg \max_c (v_{r,c} \cdot a)$$
  2. Causality: Measures if manipulating the representation (adding/subtracting concept vectors) causally changes the model's next-token prediction. For example, changing the hidden state of "Paris" by subtracting the "part-of: France" vector and adding "part-of: Germany" should flip the prediction to "Germany."

3. Key Contributions

1. Extension to Intermediate Layers & Multi-token Entities: Unlike prior work limited to the unembedding layer and single tokens, LHE analyzes hierarchical encoding across all intermediate layers and handles complex multi-token entities.
2. Domain-Specific Subspaces: The study reveals that while hierarchical information is recoverable, the specific subspaces encoding this information are domain-specific (e.g., the subspace for "Locations" differs from "Organisms").
3. Structural Similarity: Despite domain specificity, the geometric structure of these hierarchical representations is highly similar across domains.
4. Low-Dimensional Encoding: Hierarchical information is found to be concentrated in a relatively low-dimensional subspace (approx. 150–250 dimensions) within the high-dimensional hidden states (3000–5000 dimensions).

4. Experimental Results

The authors evaluated four models: Llama 3.2 3B, Llama 3.1 8B, Qwen3 8B, and Qwen3 14B.

• Effectiveness of LHE: LHE significantly outperformed baselines (Input Averaging and SVM) in both Accuracy and Causality across all domains. This confirms that hierarchical relations are linearly decodable and causally relevant in intermediate layers.
• Model Comparison:
  • Llama models generally showed high Accuracy and Causality.
  • Qwen3 14B showed high Accuracy but notably lower Causality in some domains, suggesting that while the information is present, it may not be linearly driving the prediction in the same way as in smaller models.
• Subspace Analysis:
  • Dimensionality: The effective rank of the linear transformations peaked at 150–250 dimensions, indicating a compressed, low-dimensional encoding of hierarchy.
  • Domain Specificity: Cross-domain evaluation showed that while Accuracy (decoding) remained relatively stable across domains, Causality (intervention success) dropped sharply when a transformation trained on one domain (e.g., Locations) was applied to another (e.g., Persons). This implies the mechanism of prediction is domain-specific.
  • Structural Similarity: Using Topological Data Analysis (TDA) and Wasserstein distance on persistence diagrams, the authors found that the shape of the hierarchical representation space is remarkably similar across different domains, distinct from non-hierarchical text datasets.

5. Significance and Conclusion

This paper provides strong empirical evidence that LMs encode concept hierarchies in a highly interpretable, linear manner.

• Theoretical Implication: It supports the "linear representation hypothesis," showing that complex semantic structures like hierarchies are not just emergent statistical artifacts but are explicitly structured in the model's geometry.
• Practical Implication: The discovery of low-dimensional, domain-specific subspaces suggests that hierarchical knowledge can be targeted for editing or steering without affecting unrelated parts of the model's knowledge base.
• Future Directions: The findings open avenues for investigating how these representations emerge during training and how they can be modulated in in-context learning scenarios.

In summary, the paper demonstrates that LMs represent hierarchical concepts not as a monolithic block, but as a collection of domain-specific, low-dimensional linear subspaces that share a common geometric structure, allowing for precise decoding and causal manipulation of hierarchical knowledge.