Disentangling Direction and Magnitude in Transformer Representations: A Double Dissociation Through L2-Matched Perturbation Analysis

1. Problem Statement

Transformer models encode information as high-dimensional vectors. While the Linear Representation Hypothesis posits that semantic concepts are encoded as specific directions (orientations) in activation space, the functional role of vector magnitude (L2 norm) remains unclear.

• The Gap: Existing literature largely treats magnitude as noise or a secondary property, yet vector norms vary significantly across tokens and layers.
• The Confounding Problem: Previous attempts to compare direction and magnitude were flawed because perturbing direction (rotation) and magnitude (scaling) by "small amounts" results in vastly different Euclidean displacements in the vector space. Without controlling for displacement size, it is impossible to determine if observed effects are due to the property being tested or simply the magnitude of the perturbation.

2. Methodology: L2-Matched Perturbation Analysis

The authors introduce a rigorous framework to isolate the functional contributions of direction and magnitude by ensuring identical Euclidean displacements ($\delta$).

• Decomposition: A hidden state vector $h$ is decomposed into magnitude $\|h\|$ and unit direction $\hat{h}$.
• Perturbation Types:
  • Magnitude Perturbation: Scales the vector ($h' = \alpha h$) while preserving direction.
  • Angular Perturbation: Rotates the vector within a random orthogonal subspace ($h' = \|h\| \cdot \hat{h}'$) while preserving the norm.
• L2 Matching: The scaling factor $\alpha$ and rotation angle $\theta$ are mathematically constrained such that the Euclidean distance from the original vector is identical for both types: $\|h - h'_{mag}\| = \|h - h'_{ang}\| = \delta$.
• Experimental Protocol:
  • Models: Pythia-410M and Pythia-1.4B (LayerNorm-based).
  • Intervention: Perturbations applied to hidden states in middle layers (8–15).
  • Metrics: Language modeling loss (next-token prediction) and syntactic accuracy (BLiMP subject-verb agreement).
  • Causal Interventions: "Repair" experiments where clean attention outputs or LayerNorm outputs are injected back into perturbed forward passes to trace mechanistic pathways.

3. Key Contributions

1. Empirical Discovery: Identification of a Crossover Dissociation (Double Dissociation) between direction and magnitude in LayerNorm-based transformers.
2. Methodological Innovation: The L2-matched perturbation framework, enabling the first fair, controlled comparison of geometric properties.
3. Mechanistic Insight: Causal evidence mapping direction to Attention pathways and magnitude to LayerNorm pathways.
4. Theoretical Refinement: A refined Linear Representation Hypothesis suggesting partial functional specialization of geometric properties.

4. Key Results

A. The Crossover Dissociation

The study reveals that direction and magnitude support distinct computational roles, evidenced by a reversal in damage patterns:

• Language Modeling Loss (Next-Token Prediction):
  • Angular (Direction) perturbations cause catastrophic damage. At small displacements ($\delta=1.0$), angular perturbations cause 42.9× more loss increase than magnitude perturbations.
  • Interpretation: Direction is critical for information routing and token prediction.
• Syntactic Accuracy (Subject-Verb Agreement):
  • Magnitude perturbations cause significantly more damage. At $\delta=5$, magnitude perturbations drop accuracy by 20.4%, whereas angular perturbations drop it by only 1.6%.
  • Interpretation: Magnitude is critical for fine-grained syntactic judgments.

B. Mechanistic Pathways (Causal Repair)

The authors traced how these perturbations affect the model:

• Direction $\rightarrow$ Attention: Repairing attention patterns recovers 28.4% of angular-induced loss, compared to only 15.2% for magnitude. This suggests direction primarily governs attentional routing (which tokens attend to which).
• Magnitude $\rightarrow$ LayerNorm: Repairing LayerNorm outputs recovers 29.9% of magnitude-induced damage, compared to only 13.7% for angular. This suggests magnitude modulates processing intensity via LayerNorm.
• Propagation: Angular errors amplify significantly more (3.18×) than magnitude errors as they propagate through layers, indicating directional errors cascade through the attention mechanism.

C. The Encoding vs. Computation Paradox

• Probing: Linear probes show that direction encodes syntactic categories (POS) much better (81.6% accuracy) than magnitude (29.7%).
• Perturbation: Despite direction encoding the syntax, magnitude perturbations destroy syntactic performance more.
• Resolution: Direction encodes what a token is (category), while magnitude encodes where it is structurally (parse tree depth) and modulates how that information is processed. Disrupting magnitude destroys the structural context needed for agreement, even if the category label remains intact.

D. Architecture Dependence (Critical Limitation)

The dissociation is not universal:

• Pythia (LayerNorm): Shows the standard dissociation (Direction $\to$ Loss, Magnitude $\to$ Syntax).
• TinyLlama (RMSNorm): The pattern reverses entirely. Magnitude perturbations cause more loss damage than angular ones.
• Conclusion: The functional roles of direction and magnitude depend heavily on the normalization layer (LayerNorm vs. RMSNorm).

5. Significance and Implications

• Refining Interpretability: The paper argues that analyzing full vectors is insufficient. Separating direction and magnitude can reveal distinct information streams (semantic routing vs. structural intensity).
• Model Editing: Techniques for editing model knowledge should be more surgical:
  • Preserve direction to maintain language modeling quality and factual knowledge.
  • Preserve magnitude to maintain syntactic coherence and structural reasoning.
• Theoretical Impact: Challenges the view of magnitude as mere noise. It suggests that in LayerNorm architectures, magnitude is a functional carrier of structural position information (parse depth) that modulates computation.
• Future Directions: Highlights the need to investigate how normalization schemes (LayerNorm vs. RMSNorm) fundamentally alter the geometric utilization of representations in neural networks.

Summary

This paper provides the first causal evidence that in LayerNorm-based transformers, direction and magnitude are not interchangeable. They serve a double dissociation: direction drives attentional routing and language modeling, while magnitude encodes structural position and modulates syntactic processing via LayerNorm. These findings are contingent on architectural choices, specifically the normalization method, offering a new lens for understanding and editing neural language models.