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

Mangadoddi Srikar Vardhan, Lekkala Sai Teja|arXiv (Cornell University)|Jan 19, 2026

Neurobiology of Language and Bilingualism0 citations

TL;DR

The paper introduces L2-matched perturbation analysis to disentangle the roles of direction and magnitude in transformer representations, revealing a cross-over dissociation: angular perturbations harm language modeling more, while magnitude perturbations harm syntactic processing more, with causal pathways through attention and LayerNorm.

ABSTRACT

Transformer hidden states encode information as high-dimensional vectors, yet whether direction (orientation in representational space) and magnitude (vector norm) serve distinct functional roles remains unclear. Studying Pythia-family models, we discover a striking cross-over dissociation: angular perturbations cause up to 42.9 more damage to language modeling loss, while magnitude perturbations cause disproportionately more damage to syntactic processing (20.4% vs.1.6% accuracy drop on subject-verb agreement).This finding is enabled by L2-matched perturbation analysis, a methodology ensuring that an gular and magnitude perturbations achieve identical Euclidean displacements. Causal intervention reveals that angular damage flows substantially through the attention pathways (28.4% loss recovery via attention repair), while magnitude damage flows partly through the LayerNorm pathways(29.9% recovery via LayerNorm repair). These patterns replicate across scales within the Pythia architecture family. These findings provide evidence that direction and magnitude support partially distinct computational roles in LayerNorm based architectures. The direction preferentially affects attentional routing, while magnitude modulates processing intensity for fine-grained syntactic judgments. We find different patterns in RMSNorm-based architectures, suggesting that the dissociation depends on architectural choices. Our results refine the linear representation hypothesis and have implications for model editing and interpretability research

Motivation & Objective

• Investigate whether direction (orientation) and magnitude (norm) of transformer hidden states play distinct functional roles.
• Disentangle their contributions to language modeling and syntactic processing.
• Develop and validate a controlled perturbation methodology that equates perturbation size for direction and magnitude.
• Provide mechanistic insight into how perturbations propagate through attention and normalization pathways.

Proposed method

• Introduce L2-matched perturbation analysis to equalize Euclidean displacement for angular and magnitude perturbations.
• Perturb hidden states at layers 8–15 in Pythia models during forward passes.
• Measure effects on language modeling loss and BLiMP subject-verb agreement accuracy.
• Perform causal interventions that repair attention outputs and LayerNorm to identify mechanistic pathways.
• Replicate findings across Pythia-410M and Pythia-1.4B scales.

Figure 1 : Cross-over dissociation. (A) Loss damage across $\delta$ values: angular perturbations (red) cause up to 42.9 $\times$ more damage than magnitude perturbations (blue) at small displacements. (B) BLiMP accuracy drop: the pattern reverses, with magnitude perturbations causing 12.8 $\times$

Experimental results

Research questions

• RQ1 Do angular (directional) and magnitude perturbations have equalized displacement effects on downstream tasks?
• RQ2 Which downstream tasks (language modeling vs syntactic judgments) are more affected by each perturbation type?
• RQ3 What mechanistic pathways (attention vs LayerNorm) mediate the observed damages?

Key findings

• Angular perturbations cause dramatically more loss damage than magnitude perturbations at matched displacement, with up to 42.9× higher loss impact at the smallest delta.
• Magnitude perturbations cause substantially larger drops in BLiMP subject-verb agreement than angular perturbations (up to 21.1% greater at delta=10).
• A cross-over dissociation emerges: angular perturbations are loss-critical while magnitude perturbations are syntax-critical.
• Attention repair recovers 28.4% of angular-induced loss (410M) and 23.8% (1.4B), suggesting angular damage flows substantially through attention pathways.
• LayerNorm repair recovers 29.9% of magnitude-induced damage (410M) and 2.0% for angular (1.4B pattern), indicating magnitude damage is mediated via LayerNorm pathways.
• The dissociation generalizes within the Pythia family and amplifies with larger scale, but shows architecture-dependent variation across OPT-T/TinyLlama families.

Figure 2 : Dissociation replicates and amplifies at scale. (A) Angular/magnitude loss damage ratio (log scale) across perturbation magnitudes. The effect is 2 to 4 $\times$ stronger in Pythia-1.4B (light blue) than Pythia-410M (dark blue). (B) Causal attention repair pattern replicates across scales