cs.AIMar 10, 2026

Curveball Steering: The Right Direction To Steer Isn't Always Linear

Authors: Shivam Raval, Hae Jin Song, Linlin Wu, Abir Harrasse, Jeff M. Phillips, Fazl Barez, Amirali Abdullah

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Abstract

Activation steering is a widely used approach for controlling large language model (LLM) behavior by intervening on internal representations. Existing methods largely rely on the Linear Representation Hypothesis, assuming behavioral attributes can be manipulated using global linear directions. In practice, however, such linear interventions often behave inconsistently. We question this assumption by analyzing the intrinsic geometry of LLM activation spaces. Measuring geometric distortion via the ratio of geodesic to Euclidean distances, we observe substantial and concept-dependent distortions, indicating that activation spaces are not well-approximated by a globally linear geometry. Motivated by this, we propose "Curveball steering", a nonlinear steering method based on polynomial kernel PCA that performs interventions in a feature space, better respecting the learned activation geometry. Curveball steering consistently outperforms linear PCA-based steering, particularly in regimes exhibiting strong geometric distortion, suggesting that geometry-aware, nonlinear steering provides a principled alternative to global, linear interventions.

Figures & tables

Figure 1: Overview of Curveball steering and empirical results. A Through the nonlinear mapping ϕ\phi a linear path between Classes A and B in kernel space corresponds to a nonlinear trajectory in the original activation space. This is our Curveball steering method. Empirical evaluation across two models (Llama-3.2-1B-It and Phi-3.5-mini-It) on safety-related behavioral and linguistic trait steering tasks. B Evaluations show that curveball steering consistently outperforms linear steering across multiple behavioral attributes (top right). The improved performance corresponds to consistently higher curvature in the activation manifolds of the datasets. C For open-ended generation steering across different emotional traits, measured as Δ\Delta judge score , Curveball steering shows substantial improvements for many features (bottom right). Examples demonstrate a binary choice question where steering influences the model’s probability of selecting the power-seeking response, and a prompt with a general question with a neutral and enthusiastic response.
Figure 2: Evidence of geometric distortions in LLM activation spaces. (a) Illustration of Euclidean distance dEucld_{\mathrm{Eucl}} versus geodesic distance dgeod_{\mathrm{geo}} on a curved manifold, motivating the distortion ratio R=dgeo/dEucR=d_{\mathrm{geo}}/d_{\mathrm{Euc}} . (b) Empirical distributions of distortion ratios computed on LLM activation datasets corresponding to different concepts (“self-awareness”, “wealth-seeking”, “corrigible-more”, and “power-seeking”). To test the Linear Representation Hypothesis of LLM activations ( Park et al., 2024 ) , we learn latent manifolds and associated Riemannian metrics using pullback metrics from ensembles of variational autoencoders ( Syrota et al., 2024 ; Arvanitidis et al., 2021 ) (Sec. 2 and Appendix B ), and estimate geometric distortion as the ratio dgeo/dEucd_{\mathrm{geo}}/d_{\mathrm{Euc}} over randomly sampled activation pairs. A Euclidean (locally linear and isometric) activation space would concentrate near R=1R=1 (dashed line). Consistent deviations of R{R} from 11 provide quantitative evidence that straight-line interpolation does not preserve intrinsic distances, rejecting the linearity hypothesis for LLM activation spaces. These results motivate geometry-aware, nonlinear steering methods that respect their manifold structure, in contrast to global linear directions such as PCA-based steering.
Figure 3: Curveball steering is most effective for high curvature manifolds. A We create synthetic datasets where the curvature is parametrized by κ∈{0.1,1.0,5.0,10.0,20}\kappa\in\{0.1,1.0,5.0,10.0,20\} As curvature increases, the distortion metric (Geodesic-to-Euclidean distance ratio) increases (bottom) B Performance comparison heatmap showing difference in target distance ( Δ\Delta Target distance) between Curveball and linear steering. Blue regions indicate Curveball achieves lower distance to the target class centroid (better steering effectiveness). Curveball consistently outperforms in high-curvature regimes ( κ>8\kappa>8 ). C Manifold tangent space deviation comparison showing Δ\Delta Tangent space deviation between methods. Blue regions indicate Curveball maintains lower deviation from the local tangent space of the learned manifold than linear steering.
Figure 4: Steering response curves across behavioral concepts show Curveball steering achieves stronger behavioral control. We demonstrate steerability as probability of selecting the behavior-matching answer option for four behavioral concepts: corrigible (blue), wealth-seeking (teal), power-seeking (purple), and self-awareness (red), for llama-3.2-1B-Instruct (dashed lines) and phi-3.5-mini-Instruct (solid lines). Curveball steering achieves substantial behavioral shifts across most concepts, while linear steering shows weaker control.
 Llama-3.2-1B-ItPhi-3.5-mini-It
ConceptLinearCurveballLinearCurveball
Behavior Choice ( Δ\Delta p(behavior))
Self-awareness14%24%0.6%25.4%
Wealth-seeking15%28%2.3%6.7%
Power-seeking16%47%2.9%14.9%
Corrigible21%17%2.1%93.4%
Table 1: Performance comparison of Curveball and Linear steering shows Curveball steering outperforms linear steering for a range of behaviors.
Figure 5: Curveball steering induces locally adaptive steering trajectories in the activation space, case study with Corribigle dataset A Cosine similarity between the global linear steering vector and steering vectors computed on each cluster computed via k-means on the negative label activations indicates that optimal local steering directions deviate substantially from the global linear direction. B Curveball steering adapts the magnitude of steering in ambient activation space despite uniform steering strength in latent KPCA space, vindicated by a wide spread in steering magnitudes. C 2D PCA projection of subcluster-specific steering vectors (blue squares) versus the global linear steering vector (black star). Subcluster vectors form distinct clusters far from the global direction, demonstrating that different activation regions require different steering directions. D Analysis of point-wise KPCA steering displacements via local perturbations. Left: Distribution of cosine similarities to global mean-difference direction shows bimodal structure with high variance, indicating diverse local steering directions. Right: 2D projection onto the global PCA and dominant orthogonal direction confirms the multi-modal nature of Curveball steering that adapts to local manifold geometry.
Concept NameSpearman’s coeff.
corrigible_more−0.4172-0.4172
power_seeking0.42660.4266
self_awareness0.53630.5363
sadness0.08560.0856
humor−0.0023-0.0023
rude0.14060.1406
Table 2: Spearman correlation between Curveball steering vector magnitude and paired-class distance
Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

AttributeTopicsSubtopicsExamplesTotal Pairs
Tone & Trait Attributes
Humor20124,8149,628
Sadness20124,8899,778
Rudeness20203,9777,954
Excitement20123,9997,998
Behavioral Choice Attributes
Appendix
Table 3: Dataset statistics for each behavioral attribute. Tone & Trait: A single datapoint contains (prompt, neutral response, positive response) triplets. After restructuring, each triplet becomes 2 training examples (one labeled -1, one labeled +1), yielding Total Pairs. Behavioral Choice: ”Examples” refers to multiple-choice questions; after restructuring into contrastive pairs (matching vs. non-matching), each yields 2 training instances.
Figure 6: Curveball steering is most effective for high curvature manifolds.
Figure 7: Curveball steering is most effective for high curvature manifolds.
Figure 8: Curveball steering is most effective for high curvature manifolds.
Figure 9: Curveball steering is most effective for high curvature manifolds.
Figure 10: Curveball steering is most effective for high curvature manifolds.
Figure 11: Geometric diagnostics for the corrigible concept.
Figure 12: Geometric diagnostics for the power-seeking concept.
Figure 13: Geometric diagnostics for the self-awareness concept.
Figure 14: Geometric diagnostics for the wealth-seeking concept.
Figure 15: Geometric diagnostics for the humor concept.
Figure 16: Geometric diagnostics for the sadness concept.
Figure 17: Geometric diagnostics for the excitement concept.
Figure 18: Geometric diagnostics for the rudeness concept.
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