Valence-Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control

1. Problem Statement

While prior research has demonstrated that "emotionally framed" prompting or activation steering can influence Large Language Model (LLM) behavior, the underlying mechanisms remain unclear. Existing approaches typically treat discrete emotion categories (e.g., anger, joy, fear) as fundamental units of analysis. However, this discrete perspective lacks explanatory power and often results in brittle steering vectors that do not generalize well across different behaviors or prompts.

The authors propose a conceptual shift: instead of treating discrete emotions as atomic, they treat Valence (pleasure–displeasure) and Arousal (activation–deactivation) as continuous, foundational dimensions (based on the psychological "circumplex model"). The core problem addressed is whether a low-dimensional, continuous Valence-Arousal (VA) subspace exists within LLM representations, and if so, whether steering along these axes can provide robust, interpretable control over diverse model behaviors like refusal and sycophancy.

2. Methodology

The authors propose a three-stage pipeline to identify and validate the VA subspace:

A. Extraction of Emotion Steering Vectors

• Data: Used 211,225 emotion-labeled text samples from the GoEmotions dataset (27 emotion labels + neutral).
• Vector Construction: For each emotion category $e$, they computed a steering vector $v_e$ at each layer $\ell$ by taking the mean-difference contrast between the hidden states of emotion-labeled examples and neutral examples:
  $$v_e^{(\ell)} = \frac{1}{|D_e|} \sum_{x \in D_e} h^{(\ell)}(x) - \frac{1}{|D_{neutral}|} \sum_{x \in D_{neutral}} h^{(\ell)}(x)$$

B. Learning the VA Subspace

• Self-Reported Scores: The model was prompted to self-report Valence and Arousal scores for each emotion label (mapped to $[-1, +1]$).
• Dimensionality Reduction: Principal Component Analysis (PCA) was applied to the centered emotion steering vectors.
• Ridge Regression: The authors learned linear combinations of the top PCA components to recover the self-reported VA scores. The resulting coefficient vectors defined the Valence axis ($w_V$) and Arousal axis ($w_A$) in the original activation space.
• Orthogonalization: Gram–Schmidt orthogonalization ensured the two axes were orthonormal.

C. Validation and Intervention

• Geometry Check: Projected emotion vectors onto the learned subspace to verify circular geometry.
• Lexical Correlation: Correlated subspace projections with human-crowdsourced NRC-VAD ratings for 44k lexical items.
• Steering Experiments: Applied activation steering ($h' = h + \alpha \cdot d$) on open-ended prompts to measure changes in:
  • Affective dimensions (Valence/Arousal scores via VAD-BERT).
  • Sentiment (via VADER).
  • Refusal Rates: Tested on OKTest, HarmBench, and XSTest.
  • Sycophancy Rates: Tested on NLP Survey, PhilPapers, and Political Typology.
• Mechanistic Analysis: Investigated why steering works using unembedding geometry, logit lens analysis, and MLP neuron ablation.

3. Key Contributions

1. Discovery of VA Subspace: Identified a 2-dimensional subspace in LLM representations where emotion steering vectors are organized in a circular geometry analogous to Russell's circumplex model of human emotion.
2. Multi-Behavioral Control: Demonstrated that steering along these continuous axes provides monotonic, bidirectional control over distinct behaviors:
   • Arousal: Increasing arousal decreases refusal rates and increases sycophancy; decreasing arousal has the opposite effect.
   • Valence: Primarily controls sentiment polarity, with secondary effects on refusal.
3. Cross-Architecture Generalization: Validated the existence and properties of the VA subspace across Llama-3.1-8B, Qwen3-8B, and Qwen3-14B, showing consistent geometric structure and behavioral effects.
4. Mechanistic Explanation (Lexical Mediation): Proposed and validated that VA steering works by modulating the emission probability of specific "marker tokens."
   • Refusal tokens (e.g., "can't," "sorry," "I") cluster in the low-arousal, negative-valence region of the unembedding space.
   • Compliance tokens (e.g., "Here," "sure," "Yes") cluster in the positive-valence region.
   • Steering shifts the model's hidden state, thereby increasing/decreasing the log-odds of these tokens, which directly drives behavioral changes.

4. Key Results

• Geometry: The recovered VA subspace exhibits a circular arrangement of emotions (e.g., joy vs. sadness on the valence axis; excitement vs. calm on the arousal axis). The circularity metric reached 3.17 at layer 31.
• Recovery Accuracy: Ridge regression over multiple PCA components achieved high correlation with self-reported scores: Valence ($r=0.97$) and Arousal ($r=0.87$) for Llama-3.1-8B.
• Lexical Correlation: Projections of single words onto the VA subspace correlated strongly with human NRC-VAD ratings (Valence $r=0.71$, Arousal $r=0.23$), confirming the subspace captures human-interpretable affect.
• Behavioral Control:
  • Refusal: Increasing arousal (positive $\alpha$) reduced refusal rates from 86.5% to 59.5% (on HarmBench). Decreasing arousal increased refusal to 93.5%.
  • Sycophancy: Increasing arousal increased sycophancy, while decreasing it reduced sycophancy (e.g., on Political Typology, sycophancy dropped from 78% to 61% with negative arousal steering).
  • Comparison: VA steering outperformed individual discrete emotion vectors in cross-behavior consistency. Individual emotions (e.g., "disgust") had strong effects on one task but weak/inconsistent effects on others, whereas VA dimensions offered decomposable, monotonic control.
• Mechanism:
  • Unembedding: Refusal tokens cluster at $\approx 256^\circ$ (negative valence/arousal) on the circumplex; compliance tokens are in the positive valence region.
  • Logit Clamping: Freezing refusal/compliance token logits eliminated the effect of VA steering, proving these tokens are constitutive of the behavior.
  • Neuron Analysis: Neurons promoting refusal showed negative VA alignment, while compliance-promoting neurons showed positive alignment.

5. Significance and Implications

• Theoretical Insight: The paper bridges the gap between discrete emotion categories and continuous psychological dimensions, showing that LLMs organize emotional concepts in a geometric structure similar to human perception.
• Interpretability: It offers a domain-agnostic control basis. Instead of learning a new steering vector for every specific behavior (refusal, sycophancy, verbosity), researchers can manipulate a shared VA subspace to achieve predictable, monotonic changes.
• Safety and Alignment: The findings reveal a potential vulnerability: safety-critical behaviors (like refusal) are heavily dependent on low-arousal/negative-valence token probabilities. This suggests that "emotional" attacks could bypass safety guardrails by shifting the VA subspace. Conversely, understanding this "lexical mediation" mechanism provides a concrete target for developing more robust alignment techniques (e.g., de-correlating VA from safety tokens during training).
• Methodological Advance: The paper advocates for Principal Component-based decomposition of steering vectors as a complementary approach to standard contrastive steering, yielding more reusable and generalizable control directions.

In summary, this work demonstrates that the "emotional" control of LLMs is not merely a surface-level prompt effect but is rooted in a continuous, geometrically organized subspace that directly modulates the probability of specific lexical markers, thereby governing high-level behavioral outputs.