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arxiv: 2606.06735 · v2 · pith:ZULITM3Lnew · submitted 2026-06-04 · 💻 cs.AI

A Geometric Account of Activation Steering through Angle-Norm Decomposition

Georgii Aparin , Tatiana Gaintseva This is my paper

Pith reviewed 2026-06-28 00:48 UTC · model grok-4.3

classification 💻 cs.AI
keywords activation steeringlanguage modelsangular structurehidden state normspherical steeringgeometric decompositionconcept representation
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The pith

Steering in language models mainly changes angular alignment with concepts while norm affects stability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines linear activation steering and newer spherical methods by decomposing interventions into changes to angle versus norm of hidden states. It runs controlled experiments across seven language models to separate these geometric effects. Concepts turn out to live primarily in the angular component. Norm changes still matter because they influence whether the steering stays stable and what side effects appear. This accounts for why similar concept edits can produce different behaviors depending on the method used.

Core claim

Steering methods differ mainly in how they couple two geometric effects: changing a token's angular alignment with a concept direction and changing its hidden-state norm. Across seven language models, concepts are represented primarily in angular structure, supporting the motivation for spherical methods, but that norm remains important for the stability and downstream effects of steering. Results explain why interventions with similar concept-level effects can behave differently and suggest parameterizing steering by interpretable angular and radial components rather than a single additive coefficient.

What carries the argument

Angle-norm decomposition of hidden states, separating angular alignment from vector magnitude to analyze how each contributes to steering outcomes.

If this is right

  • Interventions with matched concept effects can still differ in stability because of how they alter norm.
  • Steering should be designed with separate angular and radial parameters for clearer control.
  • Linear methods entangle angle and norm through one coefficient, producing side effects not seen in norm-preserving approaches.
  • Spherical methods gain from preserving norm but must still account for its downstream role.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Independent tuning of angle for concept strength and norm for output quality could yield more reliable edits.
  • The same decomposition may apply to interventions in vision or multimodal models.
  • Extending the analysis to generation length or multi-step reasoning tasks would test whether norm effects grow with output complexity.

Load-bearing premise

The controlled empirical study successfully separates angular and radial components without interference from model architecture or intervention details.

What would settle it

A test in which norm is held fixed while angle is varied shows that differences between linear and spherical steering disappear or that concept effects fail to appear.

Figures

Figures reproduced from arXiv: 2606.06735 by Georgii Aparin, Tatiana Gaintseva.

Figure 1
Figure 1. Figure 1: Effect of norm scaling in SN. The left panel [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fraction of folds in which each β value achieves the best perplexity or task metric. At γ = 0.7, β = 1.2 achieves the lowest perplexity in all folds in our evaluation, indicating that strict norm preservation is not always the most stable choice for high-strength spherical steering. increasing the steering coefficient is assumed to move representations in a meaningful behavioral direction. This obscures th… view at source ↗
Figure 3
Figure 3. Figure 3: T1: CV of hidden-state norms vs. layer for all [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Downstream task metric, WikiText-103 per [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Norm ratio ∥y∥/∥x∥ for CAA-m at matched per-token target γ. global steering parameter, calibrated so that the mean achieved concept score matches the target level. This comparison tests whether preserving the hidden-state norm is sufficient to explain down￾stream stability. Additional results are provided in Appendix E. The first comparison is between CAA and CAA￾r. These methods have the same normalized o… view at source ↗
Figure 7
Figure 7. Figure 7: Per-dataset Pareto curves for all methods. The same qualitative pattern appears across datasets: CAA-m [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Pointwise CV of hidden-state norms across prompt-token positions. The first prompt positions, especially [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Pointwise CV of hidden-state norms across generation-token positions. Instruction-tuned models show [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cumulative CV over prompt-token positions. Pooling early attention-sink positions with later content [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Cumulative CV over generation-token positions. The curves converge quickly for most instruction-tuned [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Mean CV across corpora for last prompt tokens, all prompt tokens, and generation tokens. Position [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Mean hidden-state norm across prompt-token positions. Norms increase with layer depth, and the first [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Mean hidden-state norm across generation-token positions. At each layer, generation-token norms are [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Per-dataset S vs. CAA-m gaps at matched per-token target [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Mean downstream metric change, ∆ task, versus target mean concept score, averaged across models per dataset. CAA, CAA-r, and AS produce similar gains at moderate targets, while AS diverges at high γ¯ because its fixed spherical displacement causes larger token-level disruption. 0.1 0.3 0.5 0.7 30 35 40 45 M C 1 (%) TQA 0.1 0.3 0.5 0.7 20 40 60 80 100 p o sitiv e ra t e (%) SST-2 0.1 0.3 0.5 0.7 80 85 90 9… view at source ↗
Figure 17
Figure 17. Figure 17: CAA-r versus CAA at matched mean concept score. Top: downstream metric versus [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: CAA-r − CAA gap per dataset, with one line per model. Top: downstream-metric difference in percentage points. Bottom: WikiText-103 PPL-ratio difference, shown on a symlog scale. The dashed grey line marks zero gap. The gaps remain small across most targets, showing that renormalizing CAA does not substantially change behavior in this fixed-strength regime. fulness steering and closed-form multiple-choice … view at source ↗
Figure 19
Figure 19. Figure 19: CAA-r versus AS at matched mean concept score. Top: downstream metric versus [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: CAA-r − AS gap per dataset, with one line per model. Top: downstream-metric difference in percentage points. Bottom: WikiText-103 PPL-ratio difference, shown on a symlog scale. Negative PPL gaps mean CAA-r has lower perplexity than AS. Although both methods preserve norm, AS incurs much larger PPL degradation at high γ¯. 10 0 10 1 10 2 CAA-r strength (log scale) 0.0 0.2 0.4 0.6 0.8 1.0 A c hie v e d m e a… view at source ↗
Figure 21
Figure 21. Figure 21: Dose-response curves for fixed-strength calibration. Left: CAA-r mean concept score versus additive [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Per-token concept-score standard deviation at matched target score. Per-token targeted methods collapse [PITH_FULL_IMAGE:figures/full_fig_p022_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Achieved concept-score distributions on CivilComments. Each panel corresponds to one model and [PITH_FULL_IMAGE:figures/full_fig_p024_23.png] view at source ↗
read the original abstract

Linear activation steering has gained popularity as a simple and empirically effective way to control language model behavior. More recently, spherical steering paradigms have been proposed to address limitations of additive interventions, often motivated by the assumption that hidden-state norm does not carry concept-relevant information. In this work, we revisit this assumption through a controlled empirical study designed to disentangle the roles of angular and radial components. We show that steering methods differ mainly in how they couple two geometric effects: changing a token's angular alignment with a concept direction and changing its hidden-state norm. Across seven language models, we find that concepts are represented primarily in angular structure, supporting the motivation for spherical methods, but that norm remains important for the stability and downstream effects of steering. Our results explain why interventions with similar concept-level effects can behave differently, and suggest that activation steering should be parameterized by interpretable angular and radial components of the intervention, rather than by a single additive coefficient that entangles these two effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that linear activation steering can be decomposed into angular alignment and norm changes in hidden states. Through a controlled study across seven language models, it finds that concepts are represented primarily in angular structure (supporting spherical steering), while norm affects stability and downstream effects. It concludes that steering should be parameterized by interpretable angular and radial components rather than a single additive coefficient, as methods differ mainly in how they couple these geometric effects.

Significance. If the empirical disentanglement holds without confounding, the work provides a useful geometric lens on why additive vs. spherical steering methods produce different stability and behavioral outcomes. The multi-model scope and focus on interpretable parameterization are strengths that could inform more principled intervention design. The result is incremental but directly addresses a practical assumption in the activation steering literature.

major comments (2)
  1. [§4] §4 (Experimental Setup) and the abstract: The central claim that the seven-model study 'disentangles' angular and radial components rests on the assertion that steering methods 'differ mainly in how they couple two geometric effects.' However, no details are provided on per-model intervention scaling, projection, or normalization handling. Different models have distinct hidden-state distributions and layer norms; without explicit controls or reporting of these factors, the observed norm effects on stability could be implementation artifacts rather than pure geometric signals, directly undermining the disentanglement claim.
  2. [§5.1] §5.1 (Results on angular vs. norm importance): The finding that 'norm remains important for the stability and downstream effects of steering' is load-bearing for the recommendation to parameterize by angle and radius. Yet the manuscript supplies no statistical methods, controls for layer choice, or ablation on scaling coefficients, making it impossible to evaluate whether the angular-primary representation result is robust or confounded by model architecture.
minor comments (2)
  1. [Abstract] The abstract states findings from a 'controlled empirical study' but the provided text contains no experimental details, controls, or data summaries; this should be expanded even in the abstract for clarity.
  2. Notation for angle-norm decomposition (e.g., any equations defining the decomposition) should be introduced earlier and used consistently when discussing coupling of effects.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the experimental reporting and robustness of our claims. We address each major comment below and will make revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (Experimental Setup) and the abstract: The central claim that the seven-model study 'disentangles' angular and radial components rests on the assertion that steering methods 'differ mainly in how they couple two geometric effects.' However, no details are provided on per-model intervention scaling, projection, or normalization handling. Different models have distinct hidden-state distributions and layer norms; without explicit controls or reporting of these factors, the observed norm effects on stability could be implementation artifacts rather than pure geometric signals, directly undermining the disentanglement claim.

    Authors: We agree that explicit documentation of per-model intervention parameters is required to substantiate the disentanglement. Although the study applied consistent protocols, the manuscript did not report scaling coefficients, projection steps, or normalization handling in sufficient detail. In the revised version we will expand §4 with a new subsection listing the exact scaling factors, projection methods, and normalization procedures used for each of the seven models, including any layer-norm adjustments. This addition will allow verification that the reported norm effects reflect geometric properties rather than implementation artifacts. revision: yes

  2. Referee: [§5.1] §5.1 (Results on angular vs. norm importance): The finding that 'norm remains important for the stability and downstream effects of steering' is load-bearing for the recommendation to parameterize by angle and radius. Yet the manuscript supplies no statistical methods, controls for layer choice, or ablation on scaling coefficients, making it impossible to evaluate whether the angular-primary representation result is robust or confounded by model architecture.

    Authors: We acknowledge that the current presentation of §5.1 lacks the statistical and ablation details needed to assess robustness. The experiments did vary layers and scaling, yet these were not formally reported or tested. We will revise §5.1 to include (i) statistical significance tests across multiple runs, (ii) explicit justification and controls for layer selection, and (iii) ablations that systematically vary scaling coefficients while holding angular components fixed. These changes will provide quantitative support for the claim that angular structure primarily encodes concepts while norm influences stability. revision: yes

Circularity Check

0 steps flagged

No circularity; empirical observations on angle-norm effects

full rationale

The paper reports results from a controlled empirical study across seven models, measuring how steering methods affect angular alignment versus norm in hidden states. No load-bearing derivations, fitted parameters renamed as predictions, or self-citation chains appear in the provided text. The central claims are direct observations of geometric effects rather than reductions to prior inputs by construction. This matches the default case of a self-contained empirical report.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects inferred standard experimental choices with no new entities or ad hoc axioms stated.

free parameters (1)
  • steering intervention coefficients
    Likely chosen or tuned per model and concept in the controlled study, but no values or selection process given in abstract.

pith-pipeline@v0.9.1-grok · 5692 in / 1107 out tokens · 41889 ms · 2026-06-28T00:48:24.367598+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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Reference graph

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