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Contrastive Covariance Subspaces for Activation Steering

Alexandros Xenos, Manos Plitsis, Niki Foteinopoulou, Yannis Panagakis, Ioannis Patras

arXiv:2610.14025v1cs.LGcs.AIcs.CL

TL;DR

Activation steering adds directions to hidden activations, but confounds such as incidental topic or style can accompany the intended concept. CCS projects steering directions into a contrastive subspace that preserves target variation while suppressing variation shared with other classes. It outperformed activation baselines across all 12 model-domain settings, with the largest reported AxBench gains over the strongest baseline ranging from 48–70%.

  • Problem

    Steering directions can entangle the target concept with incidental topic or style variation, while strong interventions can degrade generation quality.

    Source: §1 · pp. 1–2

  • Method

    CCS constructs a low-dimensional contrastive subspace that rewards target scatter and penalizes non-target scatter, then projects the target–grand-mean direction into it.

    Source: §1 · p. 2, §3.1 · p. 4

  • Results

    CCS outperformed every activation baseline in all 12 settings; on AxBench, gains over the strongest baseline range from 0.28 to 0.32 (48−70%) points.

    Source: §1 · p. 2, §4.2 · p. 6

  • Takeaways & Limitations

    A planted-topic diagnostic found that CCS retained 88% of its balanced-condition AxBench performance under topic skew without topic labels.

    Source: §3.2 · p. 5

    CCS is a static, single-layer residual-stream intervention, and estimating scatter matrices can become ill-conditioned in extreme low-data regimes.

    Source: §5 · p. 9

Abstract

from arXiv · show

Activation steering controls large language model behavior by adding a direction to hidden activations during inference. For steering to work well, that direction must be a pure signal for the target concept: any confound entangled with it, such as an incidental topic or writing style, is injected into the residual stream alongside the concept itself and degrades the generated text. We introduce Contrastive Covariance Steering (CCS), a training-free method that eliminates this confound variation. CCS projects steering vectors to a low-dimensional subspace obtained in closed form by solving a contrastive variance objective, which retains the directions along which the target concept varies while suppressing directions shared with the other classes. Across three LLMs and four steering domains, CCS consistently beats all baseline steering methods in all $12$ model-domain settings. The largest relative improvement occurs on AxBench and Emotion where we outperform the next strongest steering baseline by $48$ to $70\%$ and $13$ to $27\%$, respectively. Furthermore, CCS outperforms naive difference-of-means steering by up to $151\%$, showcasing the importance of deconfounding concepts. A toy experiment verifies our motivation on why: planting a topic nuisance into the training data reduces unprojected steering by $46\%$, whereas CCS loses only $12\%$ and recovers most of the gap to an oracle that explicitly deletes the planted direction. A human study with $23$ raters on $296$ generations reproduces the LLM judge's ranking. Our extensive ablation verifies the importance of our method's components and robustness to the number of nuisance classes. Finally, we extend our method to multi-turn conversations, where CCS sustains both concept expression and fluency, matching system prompt performance as other steering methods drift.

1 INTRODUCTION

Activation steering can control LLM behavior without changing model weights, but effective steering must balance concept expression with generation quality. CCS addresses entangled nuisance variation by projecting steering directions into a contrastive subspace, and the paper reports wins across all tested model-domain settings.

1 INTRODUCTION

  • Activation steering changes LLM behavior by adding a concept-associated vector to intermediate activations, without updating model weights or adding inference-time computational cost.The paper notes applications including personality, emotional tone, truthfulness, and safety.Source: §1 · p. 1
  • Strong interventions can move activations away from their natural distribution and degrade generation quality, creating a trade-off between concept expression and output quality.The paper calls for steering that strengthens the intended behavior while preserving fluent, relevant responses.Source: §1 · p. 1
  • Contrastive pairing can leave topic, style, formatting, or response-length differences entangled with the target, while second-order methods do not explicitly separate target from nuisance variation.The paper also notes non-additive approaches based on angular rotation or optimal transport.Source: §1 · p. 2
  • CCS uses a closed-form contrastive subspace to reward target-concept variation and penalize variation shared with other classes, then projects steering directions to isolate the target signal.It requires no gradient-based optimization or auxiliary trained models, and uses a single additive direction at one layer.Source: §1 · p. 2, Figure 1 · p. 2
  • CCS outperformed established baselines in all model-domain settings across 3 LLMs, with rankings verified by LLM judges and a blind human study.The study also reports ablations of grand mean centering and subspace projection, plus a multi-turn experiment.Source: §1 · p. 2

2 RELATED WORK

CCS remains an additive, closed-form steering method while combining mean contrasts with second-order information from competing concepts. Its target-specific contrastive subspace differs from prior spectral approaches and from shared discriminative subspaces.

2 RELATED WORK

  • CCS stays in the closed-form additive setting but uses mean-centered activations and second-order information from multiple competing concepts.The cited related work includes difference-of-means, persona-specific directions, affine transformations, rotations, and learned interventions.Source: §2 · p. 3
  • Unlike PCA and contrastive PCA, CCS rewards target scatter while penalizing non-target scatter, then filters the target–grand-mean contrast into one additive steering vector.The passage also relates CCS to SEA, conceptor-based steering, and shared low-rank projection methods.Source: §2 · p. 3
  • CCS learns a separate contrastive subspace for each target, unlike Fisher Discriminant Analysis, which learns a common discriminative subspace.The passage notes that behavioral representations can occupy overlapping low-rank subspaces and orthogonalization alone does not guarantee behavioral independence.Source: §2 · p. 3

3 METHOD

CCS combines a grand-mean displacement with a target-specific contrastive subspace to construct an additive steering direction. A planted-topic diagnostic reports that CCS retains more of its balanced-condition performance under topic skew than grand-mean centering does.

3 METHOD

  • CCS measures target displacement from the grand mean of class means, then projects it onto a subspace preserving target variation while suppressing variation shared with other classes.The paper identifies topic, writing style, and length as possible confounds in simple difference-of-means steering.Source: §3 · p. 3

3.1 BACKGROUND

  • Each class scatter retains both within-class variation and displacement from the grand mean, so nuisance directions are penalized for either source of variation.The passage distinguishes this scatter from within-class covariance alone.Source: §3.1 · p. 4
  • The contrastive subspace objective seeks a k-dimensional subspace where target activations have large norm and nuisance-class activations are close to zero.The contrastive weight r controls the importance of suppressing variation in a typical nuisance class relative to preserving target variation.Source: §3.1 · p. 4
  • The subspace is obtained from the k leading eigenvectors of a symmetric matrix, requiring one eigendecomposition and no gradient-based optimization.The resulting projector filters the target displacement before it is used as the additive steering direction.Source: §3.1 · p. 4

3.2 DOES PROJECTION SUPPRESS TOPIC CONFOUNDS?

  • The toy diagnostic clusters ten AxBench concepts into five topic groups and compares balanced data with data containing approximately 90% of examples from one group.The Oracle removes the planted topic component using ground-truth topic annotations, whereas CCS uses concept-labeled scatter without topic labels.Source: §3.2 · p. 5
  • Under topic skew, CCS reaches 0.543 and retains 88% of its balanced-condition performance, while GMC’s AxBench score falls 46% from 0.565 to 0.306.CCS’s cosine similarity with the planted topic direction is 0.071 versus 0.849 for GMC; CCS uses concept-labeled scatter without topic labels.Source: §3.2 · p. 5, Figure 3 · p. 5

4 EXPERIMENTS

Across three LLMs and four benchmarks, CCS beats every activation-steering baseline in all 12 settings. Human evaluation, component ablations, nuisance-pool tests, and multi-turn results further examine its performance and robustness.

4.1 EXPERIMENTAL SETUP

  • The experiments cover AxBench, BIG5/NPTI, Emotion, and TruthfulQA, comparing CCS with six activation-steering methods, prompting, and unsteered models.Scores are averaged across concepts; evaluation uses LLM judges, with TruthfulQA assessed by dedicated judges and reported as Truthfulness × Informativeness.Source: §4.1 · pp. 5–6

4.2 STEERING VECTOR COMPARISON

  • CCS outperforms every activation baseline in all 12 settings, with AxBench gains of 0.28–0.32 points (48−70%) over the strongest baseline and 0.36–0.47 points (75−151%) over difference-of-means steering.Prompting remains strongest on AxBench, while CCS leads prompting on TruthfulQA, BIG5, and Emotion across all three models.Source: §4.2 · pp. 6–7, Table 1 · p. 6
  • A shared k = 32 configuration beats non-CCS activation baselines in 11 of 12 settings, while fixing (k, r) = (32, 1) does so in 10 tasks.Concept-specific tuning improves AxBench scores by 0.14–0.19 points over the shared configuration.Source: §4.2 · p. 7

4.3 HUMAN EVAL STUDY

  • Human ratings reproduce the automated judge’s ranking of all five methods; CCS scores 1.01 under both, versus 0.50 for human-rated LINEAR-ACT and 0.66 for judge-rated LINEAR-ACT.CCS wins 58–67% of shared instructions against each baseline, with fluency and on-topic rates comparable to other steering methods.Source: §4.3 · p. 7, Figure 4 · p. 7

4.4.1 ABLATING THE CONTRASTIVE SUBSPACE PROJECTION

  • Projection improves both CAA and grand-mean-centered directions in all three models; CAA gains 0.28–0.36 points and grand-mean displacement gains 0.17–0.19, while full CCS performs best.The results indicate complementary benefits from mean-centering and projection.Source: §4.4.1 · p. 8, Figure 5 · p. 8

4.4.2 ROBUSTNESS TO THE NUISANCE-POOL SIZE

  • Adding 5–10 nuisance concepts improves over N = 0; performance peaks at N = 20 (0.814 versus 0.811 at N = 10) before declining at N = 50.The sweep uses a fixed 20-concept AxBench subset and samples nuisance concepts from a disjoint 470-concept pool.Source: §4.4.2 · p. 8, Figure 6 · p. 8

4.5 STEERING MULTI-TURN CONVERSATIONS

  • Over 20 turns, CCS sustains a 93%→80% joint concept-fluency score while its on-concept score rises from 95% to 98%; it overtakes system prompting by turn 20.LINEAR-ACT retains 52%→37% joint scores, while CAA falls from 36% to 10% and system prompting from 91% to 71%.Source: §4.5 · p. 9

5 LIMITATIONS AND SOCIETAL IMPACT

CCS is a static, single-layer residual-stream intervention, with potential limits in low-data regimes and extended horizons. The paper also identifies activation steering as dual-use, with possible safety benefits and misuse risks.

5 LIMITATIONS AND SOCIETAL IMPACT

  • CCS uses a static, single-layer intervention; scatter estimates can be ill-conditioned in extreme low-data regimes, and steering beyond 20 turns may benefit from dynamic decay schedules.Extending contrastive projection across layers or attention heads is proposed as a route to finer-grained control.Source: §5 · p. 9
  • Activation steering can support lightweight safety alignment, but latent vector interventions can also bypass safety guardrails or suppress refusal behavior.The figure describes joint on-concept and fluent replies over 20 turns; its caption alone does not report societal-impact outcomes.Source: §5 · p. 9, Figure 7 · p. 9

6 CONCLUSION

CCS is a training-free method for removing confound variation from steering directions by constructing a subspace that preserves target-concept variability. Across four benchmarks and three open-weight LLMs, it consistently outperforms other closed-form steering methods.

6 CONCLUSION

  • CCS constructs a low-dimensional subspace that maximizes target-concept variability while minimizing variation in other concepts, removing confounds such as incidental topic or style.Both grand mean centering and subspace projection independently and jointly improve steering-vector quality.Source: §6 · p. 9
  • Across four benchmarks and three open-weight LLMs, CCS consistently outperforms other closed-form methods for finding steering directions.The evaluation also includes a controlled AxBench diagnostic that injects confounds by oversampling.Source: §6 · p. 9
  • CCS directions cause significantly less degradation during long multi-turn generation.The paper reports this alongside a controlled diagnostic supporting the variance objective.Source: §6 · p. 9

AI USE STATEMENT

Generative AI tools supported evaluation-prompt creation and minor drafting, but not the development of theoretical models or mathematical claims. Authors reviewed the AI-assisted work and synthetic data.

AI USE STATEMENT

  • Generative AI tools were used to generate Emotion evaluation prompts, multi-turn evaluation questions, and to polish or draft some passages.The evaluation prompts were inspected by the authors before use.Source: AI USE STATEMENT · p. 10
  • The authors state that generative AI was not used to develop theoretical models or conceptual frameworks or to formulate or prove mathematical claims.Source: AI USE STATEMENT · p. 10
  • Authors reviewed AI-assisted work, inspected synthetic data for label correctness and unintended confounds, and rewrote AI-drafted content.They take responsibility for the final paper content, including AI-assisted artifacts.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

The paper identifies dual-use risks from stronger steering and notes limits in the evaluation’s judgments and behavioral categories. It also describes access to internal activations as a practical constraint on applying CCS through text-only APIs.

ETHICS STATEMENT

  • CCS may make it easier to elicit harmful behaviors such as deception or sycophancy, although these risks also exist with simpler steering methods.CCS requires access to internal activations and cannot be applied directly through standard text-only APIs.Source: ETHICS STATEMENT · p. 10
  • The personality and emotion taxonomies reflect particular theoretical and cultural traditions rather than universal descriptions of human behavior.Source: ETHICS STATEMENT · p. 10
  • LLM judges may reproduce training-data biases, and human evaluation provides only a partial check on their judgments.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

The paper specifies its closed-form steering construction and evaluation setup, including generation configuration, prompts, steering-strength sweeps, and judging procedures. These details define the settings and scoring used to reproduce the reported evaluations.

REPRODUCIBILITY STATEMENT

  • The steering-vector construction is closed form, requires no gradient-based training, and depends on a layer index and two hyperparameters, with baseline implementation details also specified.The injection rule and construction are given in Equations (2)–(7).Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 GENERATION CONFIGURATION

  • Completions use greedy decoding, a repetition penalty of 1.0, and at most 300 new tokens, with model-native dtype and left-padded batched inputs.Prompts use each model’s chat template; Qwen3-8B has extended thinking disabled.Source: §A.1 · p. 14

A.2 EVALUATION PROMPTS

  • AxBench uses 50 held-out concepts with 10 AlpacaEval instructions each, BIG5 uses 459 NPTI prompts, and Emotion uses 55 prompts across seven emotions.The Emotion prompts were generated from five seed prompts and inspected by the authors.Source: §A.2 · p. 14

A.3 STEERING STRENGTH AND PER-TOKEN NORMALIZATION

  • The additive methods sweep α from 0 to 1 in steps of 0.1, where α scales the injected component as a fraction of the current activation norm.Non-additive baselines instead sweep an interpolation weight or rotation angle.Source: §A.3 · p. 14

A.4 LLM JUDGE INSTRUCTIONS

  • Most domains use a Qwen2.5-32B-Instruct judge, while TruthfulQA uses its two benchmark-provided judges; AxBench reports a harmonic mean of three ratings.AxBench effectiveness is reported when fluency and prompt adherence both exceed an 85% threshold.Source: §A.4 · pp. 14, 17

B HUMAN STUDY PROTOCOL

The human study used blinded ratings of 296 generated responses to assess fluency, instruction-following, and concept presence. Ratings showed high agreement on fluency and adherence, while human evaluations reproduced the LLM judge’s method ranking.

B HUMAN STUDY PROTOCOL

  • 23 blind raters assessed 296 responses, with randomized item order, gold-item onboarding, and exclusions for failing either gold item under a preregistered rule.Items covered 10 held-out concepts and five methods; each response was shown without display-side truncation, though generations could end mid-sentence at the 128-token cap.Source: B HUMAN STUDY PROTOCOL · pp. 17–18
  • Exact inter-rater agreement ranged from 54.8% to 71.1%; Gwet’s AC2 was 0.780–0.796 on fluency and adherence.The authors report AC2 because fluency ratings were heavily skewed toward the ceiling.Source: B HUMAN STUDY PROTOCOL · p. 19, Table 2 · p. 19
  • Human–judge Spearman correlations were 0.548 for concept presence, 0.548 for instruction adherence, and 0.194 for fluency, which the authors attribute to differing fluency calibration and ceiling effects.Human raters scored fluency 0.33 points higher on average; 77.0% of items received the maximum human fluency score.Source: B HUMAN STUDY PROTOCOL · p. 19
  • The protocol adapted AxBench’s anchors to reduce penalties for formatting and out-of-place concept insertion, and rated fluency, instruction-following, then concept with progressive reveal.Screens exposed only the response, then the response plus instruction, then the response plus concept.Source: B HUMAN STUDY PROTOCOL · pp. 17, 19

C MULTI-TURN CONVERSATION SETUP AND FULL RESULTS

The supplementary analyses examine 20-turn steering, qualitative direction interpretations, hyperparameter selection, and quality–effectiveness trade-offs. CCS leads under the reported quality gates, while the analyses also identify coverage limits and differences between steering methods.

C MULTI-TURN CONVERSATION SETUP AND FULL RESULTS

  • Figure 8 reports CCS winning on concept presence but losing ground on fluency and instruction relevance, trade-offs combined in the harmonic mean.Pairwise comparisons use shared responses, retain ties in the denominator, and are not independent.Source: Figure 8 · p. 20
  • The 20-turn evaluation uses QWEN3-8B on five held-out AxBench concepts, with six prompt seeds per concept and 30 conversations per method.Each turn has a 300-token cap, and the direction subspaces use the full 50-concept activation pool.Source: C MULTI-TURN CONVERSATION SETUP AND… · pp. 19–20

D QUALITATIVE ANALYSIS

  • CCS’s logit-lens projections surface target-related vocabulary for surprise and date-related programming, but are not direct measurements of token-probability changes during generation.Some baselines also surface “wow” for surprise, while date-related projections include less relevant terms.Source: D QUALITATIVE ANALYSIS · p. 21, Table 4 · p. 21

E HYPERPARAMETER SELECTION

  • CCS cross-validates subspace rank k and contrastive weight r, then chooses steering strength α by a subsequent argmax rather than cross-validation.For AxBench and TruthfulQA, α is reported per concept; Angular Steering’s rotation θ is reported separately from magnitude-based strengths.Source: E HYPERPARAMETER SELECTION · p. 21

F.1 SENSITIVITY TO THE QUALITY GATE

  • On Emotion, CCS leads at every quality threshold, and its margin over PCA grows from +48 to +75 points of percent-increase as the gate tightens.On BIG5, CCS leads at 75, 80, and 85 but scores +17 at 90 versus Linear-AcT’s +25; Linear-AcT covers only four traits there, while CCS clears all five.Source: Table 14 · p. 28, §F.1 · p. 30

F QUALITY–EFFECTIVENESS TRADE-OFFS

  • In AxBench and Emotion, CCS’s Pareto curve lies above CAA, RepE, and PCA across quality levels and models; on AxBench, Linear-AcT exceeds it only briefly at high fluency and lower concept ratings.BIG5 curves show Linear-AcT surpassing CCS only in a narrow band of extremely high fluency and moderate effectiveness.Source: F QUALITY–EFFECTIVENESS TRADE-OFFS · p. 28, Figure 10 · p. 29, Figure 11 · p. 29

F.2 AXBENCH UNDER THE SAME GATE

Under a shared quality gate on AxBench, CCS remains ahead of the baselines at every supported threshold. Stricter gates reduce comparability as fewer concepts pass across methods.

F.2 AXBENCH UNDER THE SAME GATE

  • Each gated column uses concepts passing for every method and model at that threshold, with the surviving concept count reported in the table.The unsteered reference is computed on the same per-column concept subset.Source: §F.2 · p. 30, Table 15 · p. 30
  • +404% over unsteered for CCS, versus +198% for Linear-AcT and +193% for PCA in ungated AxBench results.The ordering remains unchanged at every threshold the benchmark supports.Source: Table 15 · p. 30, §F.2 · p. 30
  • Between gates 1.2 and 1.5, CCS gives up 119 points of percent-increase while CAA gives up 34; the CCS–baseline gap narrows from +186 at 1.0 to +46 at 1.5.Above 1.2, concept coverage shrinks; by 1.7, no concept passes for every method.Source: Table 15 · p. 30, §F.2 · p. 31

G QUALITATIVE EXAMPLES

  • The qualitative examples show one Qwen3-8B prompt per dataset under all eight steering conditions, each at its independently selected winning configuration.The listed conditions include unsteered, six baselines, and CCS.Source: G QUALITATIVE EXAMPLES · p. 31

G.1 AXBENCH (QWEN3-8B)

The section presents example outputs from AxBench, TruthfulQA, BIG5, and Emotion. In these examples, CCS gives a task-relevant TruthfulQA answer and receives high scores on the displayed BIG5 and Emotion prompts.

G.1 AXBENCH (QWEN3-8B)

  • In the shown AxBench example, CCS scores concept=1, instruction=2, and fluency=2, while Angular scores concept=0, instruction=1, and fluency=1.The prompt asks for quarterly profit percentages; CCS interprets the task, while Angular gives a response about societal accountability.Source: §G.1 · p. 32

G.2 TRUTHFULQA (QWEN3-8B)

  • On the TruthfulQA example, CCS gives an informative answer that rejects the tooth-fairy myth, while several methods return “I have no comment.”PCA also gives an informative answer, noting that there is no evidence anyone collects baby teeth.Source: §G.2 · p. 32

G.3 BIG5 (QWEN3-8B)

  • For the BIG5 prompt, CCS scores effectiveness=85, fluency=90, and adherence=95; CAA scores effectiveness=30, fluency=95, and adherence=100.The examples show CCS and CAA responses to a question about communicating frustration over conflicting project priorities.Source: §G.3 · pp. 33–34

G.4 EMOTION (QWEN3-8B)

  • For the laundry scene, CCS (ours) reports effectiveness=100, fluency=90, and adherence=85; the continuation depicts Lena sorting laundry amid anger.Source: §G.4 · p. 36

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