Laguerre Geometry for Interpreting Large Language Models

Technical Summary: Laguerre Geometry for Interpreting Large Language Models

Problem Statement
Despite the ontological clarity of atomic computations within Large Language Models (LLMs), the epistemological mechanisms by which these computations coalesce into emergent intelligence remain opaque. A central mystery concerns the organization of semantic concepts within LLM representation spaces. Existing approaches, primarily grounded in the Linear Representation Hypothesis (LRH), typically define concepts as single points, linear directions, or Gaussian clusters. However, four fundamental questions remain unresolved under these frameworks: (I) What is the precise definition of a concept? (II) In which representation space is it encoded? (III) What concrete geometric structure carries it? and (IV) Why is linearity assumed, and to what extent does it hold? Furthermore, current interpretability tools (e.g., Logit Lens, Tuned Lens) often rely on heuristic decoding or require training, lacking a principled ground truth for the "meaning" of hidden vectors in intermediary layers.

Methodology
The authors propose a geometric framework based on Laguerre Geometry to unify low-level geometric decomposition with high-level semantic organization. The methodology proceeds through three main stages:

1. Unembedding Layer as a Laguerre-Voronoi Diagram (LVD):
   The paper establishes that the final linear projection layer (unembedding layer) of an LLM, parameterized by weights $W$ and biases $b$, naturally partitions the representation space $\mathbb{R}^d$ into a Laguerre-Voronoi Diagram (also known as a Power Diagram).

   • Concept Definition: A concept is redefined not as a vector, but as a specific Laguerre-Voronoi cell $\omega_j$, characterized by a center $c_j$ and a weight $\nu_j$. A category is defined as the union of multiple such cells.
   • Geometric Properties: This formulation allows for the strict definition of linear separability between concepts and categories. Crucially, it introduces a Domination Score derived from the Laguerre weights and centers, enabling the detection of hierarchical relationships (e.g., hyponymy/hypernymy) such as "dog" $\subset$ "mammal" $\subset$ "animal," which are difficult to detect using vector distances alone.

2. Decomposition of Transformer Layers:
   The authors extend this geometric view to the intermediary transformer blocks. By decomposing the network into piecewise-linear operators (MLP blocks) and analyzing the attention mechanism, they demonstrate that the input space is partitioned into piecewise-linear regions.

   • Single-Input vs. Multiple-Input: For single-input tokens, the trajectory follows a static tree of piecewise-linear flow. For multiple inputs, the cross-attention mechanism acts as a dynamic transport term, "hopping" the hidden trajectory across different trees (regions) when attention fires.
   • Region Correspondence: Each linear region in any layer is assigned a semantic label corresponding to the Laguerre cell it would ultimately reach if propagated through the remaining network without further contextual interference.

3. Geometric Lens and Laguerre Autoencoder:

   • Geometric Lens: A training-free, hyperparameter-free method to decode hidden vectors. It isolates a hidden vector $h^{(\ell)}_i$ from subsequent contextual effects by forcing self-attention weights to 1 (zeroing out cross-token updates) and propagating the vector through the remaining piecewise-linear network to identify its final Laguerre cell.
   • Laguerre Autoencoder (LAE): A 2D visualization tool that jointly renders hidden representations and their corresponding LVD regions. It is trained with a loss function that combines reconstruction error and a term ensuring the reconstructed point falls within the same Laguerre cell as the original, preserving the decision geometry.

Key Contributions

• Redefinition of Concepts: The paper moves beyond point-based or direction-based definitions, defining concepts as convex polyhedral regions (Laguerre-Voronoi cells) induced by the unembedding layer.
• Hierarchical Discovery: It demonstrates that Laguerre weights naturally encode conceptual hierarchies and inclusion relationships, allowing for the unsupervised discovery of hyponyms and hypernyms without predefined attribute sets.
• Geometric Lens: A novel, training-free interpretability tool that reveals the exact concept a hidden vector encodes at any layer by tracing its path through the model's intrinsic piecewise-linear partition.
• Visualization: The Laguerre Autoencoder provides a unified 2D view of both the model's decision geometry and the trajectory of token representations.

Experimental Results
The authors validate their framework through several experiments:

• Linear Separability: In linear probing tasks comparing "Animal vs. Plant" and "Event vs. Cognition," Laguerre Geometry (using lifted points from the LVD) achieved lower error rates than Categorical Geometry (whitened vectors) and Lattice Geometry across multiple models (Phi-2, Gemma-2, Pythia).
• Hierarchy Detection: The proposed domination score successfully identified hierarchical relations (e.g., "dog" under "animal") with accuracy significantly outperforming Lattice Geometry and random baselines on Phi-2 and Pythia-70M.
• Trajectory Analysis: Visualizations of the prompt "The capital of France is" show that Geometric Lens traces a more complex and diverse path through the representation space compared to Logit Lens and Patchscopes. It captures the model's internal refinement process, where grammatically incorrect but factually correct tokens (e.g., "Paris") appear in middle layers before the model settles on the grammatically correct token ("_Paris").
• In-Context Interference: In prompts designed to induce interference (e.g., swapping city names in a fictional context), Geometric Lens demonstrated superior robustness. By examining the token rank at intermediate layers (specifically layer 20), Geometric Lens recovered the factually correct token more often than Logit Lens or Patchscopes, suggesting it reveals the model's true reasoning process before it is overwritten by final-layer interference.

Significance and Claims
The paper claims to provide a rigorous geometric foundation for the Linear Representation Hypothesis, moving it from an informal assumption to a strictly defined property of the model's piecewise-linear structure. By establishing that concepts are regions rather than points, the authors argue that this framework offers a principled ground truth for hidden representations, eliminating the need for heuristic decoding or external training.

The significance lies in shifting interpretability from "explanatory geometry" (describing static structures) to "actionable interpretability." The Geometric Lens allows researchers to read out the exact concept encoded in a hidden vector and, as demonstrated, recover factual truths even when the model's final output is corrupted by in-context interference. The authors position this work as a foundational step toward understanding how high-level intelligence emerges from low-level geometric partitions, with future work planned to extend this framework to causal intervention and behavior steering.