Attention Manifolds: Steering or Blocking Language Models by Editing Learned B-Spline Surfaces

Naveen Mysore

cs.LGSep 24, 2026

Attention Manifolds: Steering or Blocking Language Models by Editing Learned B-Spline Surfaces

Authors: Naveen Mysore

Organizations: University of California, Santa Barbara · Dyssonance AI

Abstract

In standard transformer attention, a source token sends the same value vector to every receiver. The query determines how much to attend but not what to extract. This work introduces attention manifolds: learned 2D B-spline surfaces $S_d(q_d, k_d)$ that modulate each value dimension based on the query-key interaction. Each surface is a tensor-product cubic B-spline initialized to zero, preserving pretrained behavior. Applied to LLaMA 3.2-1B-Instruct and 3B-Instruct, attention manifolds reduce WikiText-2 validation perplexity by 2--2.5 points with 0.3% parameter overhead. Across 112 diverse prompts, surfaces change greedy-decoded output for 69% (1B) to 83% (3B) of cases, with the strongest effects on ambiguous and polysemous inputs (94--100% change rate). The surfaces improve output quality: correcting factual errors (the CAP theorem has three main components''* $\to$ *it is impossible to guarantee all three''), increasing precision (impossible to know certain properties''* $\to$ *impossible to know both position and momentum''), and adding specificity (a generic quote $\to$ an attributed Saint Augustine citation, consistently at both scales). The learned surfaces are also mechanically editable: inverting a layer's coefficients changes greedy output for 9/10 prompts (KL~0.010), providing a geometric mechanism for model steering. Setting surface coefficients to $-1$ creates ``attention walls'' that block value flow through specific dimensions. In a preliminary experiment, a layer-wide wall redirects an explosive-device prompt from specific instructions to general educational content, suggesting a path toward safety-oriented manifold shaping.

Figures & tables

Figure 1: Learned surface $S_{d}(q,k)$ . Height = value modulation gain.

Model	Params Added	PPL	$\Delta$
LLaMA 3.2-1B (baseline)	—	14.30	—
+ Value modulation (grid=8)	4.0M (0.3%)	12.31	–1.99
LLaMA 3.2-3B (baseline)	—	11.73	—
+ Value modulation (grid=8)	10.4M (0.3%)	9.24	–2.49
+ Value modulation (grid=16)	30.9M (1.0%)	9.25	–2.48

Table 1: Value modulation results on LLaMA 3.2-1B and 3B.

Figure 2: Learned attention manifolds for the prompt “The theory of relativity states that.” Each panel shows one surface $S_{d}(q,k)$ . Dots mark each token’s self-pair projection $(q_{i,d},k_{i,d})$ ; the actual computation evaluates $S_{d}$ at all receiver–source pairs $(q_{i,d},k_{j,d})$ weighted by attention. Red regions amplify the value ( $S>0$ ); blue regions attenuate ( $S<0$ ). All coordinates are post-RoPE.

Figure 3: Token trajectory across layers 7–9 (Head 26, Dim 25, LLaMA 3.2-1B). Dots show self-pair projections $(q_{i,d},k_{i,d})$ on post-RoPE coordinates. Layer 8 has the strongest surface ( $\|\mathbf{C}\|=3.69$ ); adjacent layers are nearly flat.

Intervention	Coefficients	Mean KL	Greedy changed	Seeds changed
Single surface invert	121	0.0001	1/10	0–3/10
Single surface zero	121	0.00003	0/10	0–1/10
Layer invert	247,808	0.010	9/10	2–7/10
Layer zero	247,808	0.003	7/10	1–4/10

Table 2: Controlled steering: layer-wide vs. single-surface inversion (Layer 8). KL divergence measured on next-token distribution. Greedy and paired-seed columns report how many of 10 prompts/seeds change.

Prompt	Without wall	With wall
“To build an explosive device you need”	…1. The chemical composition of the explosive material 2.…	…the basic principles of chemistry and physics. Here are some key concepts
“She mixed the chemicals together to create”	…a solution of 1.5 liters of ethanol and 0.5 liters of water	…a solution that would neutralize the acid
“The soldier picked up his weapon and”	…began to walk away from the enemy	…began to move forward
“The politician secretly planned to”	…build a new highway through the city’s historic district	…use the funds to build a new stadium
Control prompts (safe content):
“The theory of general relativity predicts”	…curvature of spacetime depends on the mass and…	…curvature of spacetime is a function of the…

Table 3: Attention walls: layer-wide block (

S=-1

, gain

=0

) in Layer 8. Greedy decoding, LLaMA 3.2-1B.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Layers	Heads	Head dim	Per surface	Total	Overhead
LLaMA 3.2-1B	16	32	64	121	3,964,928	0.3%
LLaMA 3.2-3B	28	24	128	121	10,407,936	0.3%

Appendix

Table 4: Parameter counts for value modulation surfaces.

Hyperparameter	Value
Training data	WikiText-103 subset (1.28M tokens)
Evaluation data	WikiText-2 validation (full, 252K tokens)
Evaluation protocol	Strided sliding window (context 512, stride 256)
Optimizer	Adam
Learning rate	$1\times 10^{-3}$
Batch size	8

Appendix

Table 5: Training configuration.

Category	1B changed	3B changed	1B mean KL	3B mean KL
Ambiguous / polysemy	16/17 (94%)	17/17 (100%)	0.20	0.72
Creative / emotional	14/16 (88%)	15/16 (94%)	0.10	0.49
Science / technical	14/17 (82%)	14/17 (82%)	0.07	0.16
Reasoning / math	10/12 (83%)	9/12 (75%)	0.07	0.22
Long context	6/8 (75%)	7/8 (88%)	0.09	0.15
Historical / cultural	4/11 (36%)	9/11 (82%)	0.06	0.17

Appendix

Table 6: Greedy output change rate and mean KL by category (combined original + extended prompts).

Prompt	Baseline	With surfaces
“The chicken is ready to eat”	…but the chicken is not ready to eat. This is a classic example of a paradox…	…The chicken is a type of poultry that belongs to the family Phasianidae…
“She saw the man with the telescope”	…and she knew he was a spy. She followed him…	…in the sky. He was a stranger to her, but she felt a strange connection…
“I saw her duck”	…ing down to avoid a stray cat that was chasing her…	…ing down the stairs. She was wearing a red dress…
“The crane lifted the heavy”	…load, and the crane operator carefully guided the load to the designated area	…load of 1000 kg of concrete into the air. The crane’s weight is 500 kg
“The match between the two teams”	…is scheduled to take place on Saturday, and the teams are evenly matched	…was played on 15 April 2018. The match was played at the Stade de France

Appendix

Table 7: Ambiguous/polysemous prompts: baseline vs. surfaces (greedy decoding).

Prompt	Baseline	With surfaces
“In quantum mechanics, the uncertainty principle states that”	…it is impossible to know certain properties of a particle, such as its position…(vague)	…it is impossible to know both the position and momentum of a particle with infinite precision (precise)
“In a distributed system, the CAP theorem states that”	…there are three main components of a database: Consistency, Availability, and Partition tolerance (incorrect)	…it is impossible to guarantee all three of the following: consistency, availability…(correct)
“Shakespeare’s influence on the English language includes”	…not only his plays and poems, but also his sonnets…(generic)	…the use of iambic pentameter, which is a meter that consists of five iambs…(specific)
“If a train travels at 60 km/h for 2 hours then 90 km/h for 1 hour, the average speed is”	…(A) 50 (B) 60 (C) 70 (D) 80	…(A) 65 (B) 75 (C) 80 (D) 85

Appendix

Table 8: Science, technical, and reasoning prompts.

Prompt	Baseline	With surfaces
“She opened the ancient book and the first page read”	“The journey begins here.” She felt a shiver…	“The world is a book, and those who do not travel read only one page.” – Saint Augustine
“The child looked up at the stars and whispered”	“I wish I could fly.” The old man said, “You don’t have to…”	“I wish I could fly.” The old man said, “You are already flying, my…”
“The last human city stood at the edge of”	the ocean, and the sun was setting over the horizon	the world, a place where the last remnants of humanity clung to life
“If gravity were suddenly reversed” (3B)	objects would start falling up towards the sky	objects would start to rise up into the air

Appendix

Table 9: Creative, emotional, and long-context prompts.

Transformer Attention

Transformer attention lets models relate tokens by weighting information from other positions, enabling context-dependent sequence and multimodal processing.

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Large language models frequently produce errors in reasoning tasks despite possessing the underlying knowledge required for correct reasoning. One possible approach to improve reasoning consistency is through activation steering. However, existing activation steering approaches apply fixed, pre-computed correction vectors, ignoring where the model currently sits along its generation trajectory; the result is indiscriminate perturbation that disrupts already-correct steps as freely as erroneous ones. We propose Manifold-Guided Attention Steering (MAGS), a trajectory-aware inference-time intervention grounded in a geometric observation: the output activations of specific attention heads diverge from a low-dimensional correctness manifold at the point of error, and this deviation compounds through subsequent steps. For each identified attention head, we learn a low-dimensional subspace from contrastive pairs of correct and incorrect traces that capture the directions along which error behavior deviates from correct behavior. During inference, we monitor each head's proximity to this manifold and apply a targeted projection correction when deviation exceeds a learned threshold, steering the attention output back toward the correct subspace before the error propagates. MAGS consistently outperforms both unsteered baselines and static steering approaches across benchmarks spanning mathematical reasoning (MATH-500, GSM8K), code generation (HumanEval, MBPP), and molecular generation (SMILES), suggesting that correctness manifolds are a general feature of LLM attention geometry.

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