Beyond the Linear Representation Hypothesis: Non-Linear Activation Steering in Text-to-Image Models

Muhammad Atif Butt; Paweł Skierś; Joost Van De Weijer; Kamil Deja

cs.CVOct 3, 2026

Beyond the Linear Representation Hypothesis: Non-Linear Activation Steering in Text-to-Image Models

Authors: Muhammad Atif Butt, Paweł Skierś, Joost Van De Weijer, Kamil Deja

Organizations: Computer Vision Center, Spain · Computer Sciences Department, Universitat Aut`onoma de Barcelona, Spain · Warsaw University of Technology, Poland · IDEAS Research Institute, Poland

Abstract

Mechanistic interpretability often relies on the Linear Representation Hypothesis (LRH), which assumes that high-level concepts are encoded as linear directions in activation space. Yet a natural visual concept does not necessarily require a linear visual transition: between sunny and stormy lies an intermediate weather state such as a sky with a few white clouds, not simply a weaker storm; between a caterpillar and a butterfly, the progression is not a caterpillar with continuously growing wings. This raises the question of whether such true intermediate states are also represented nonlinearly by the model. Indeed, when we prompt text-to-image models directly for intermediate attributes, their activations rarely fall along the straight direction connecting the endpoints. Therefore, we propose KANSteer, which models concept traversal as a curve passing through its intermediate states. Seeking a representation that is both simple and interpretable, we propose to use Kolmogorov-Arnold Networks (KANs), which provide a one-dimensional coordinate whose learned functions define the trajectory. This allows the steering direction to vary along the concept while preserving an interpretable representation. Across several concepts and text-to-image diffusion transformers, we find that their activation trajectories substantially deviate from straight lines, and that KANSteer provide a closer fit and smoother traversal of intermediate attributes than linear steering.

Figures & tables

Figure 1: Many concepts do not vary linearly in activation space. Left: PCA projection of block- $q$ delta activations for three concepts (see more in Appendix 7 ); each point is one seed, with attribute centroids, and the two endpoints outlined. The intermediate attributes bend away from straight line between the endpoints, so a fixed direction cannot pass through these attributes. Right: Sweeping closed-form linear direction ( Linear Steering ) holds the source attribute and then jumps to the target, while prompt-generated attributes it is fitted on ( Prompt Generated ) traverse the concept smoothly.

Figure 2: Overview of KANsteer. Top: Each concept is instantiated at

K=5

ordered attributes (

a_{1},..,a_{k}

), and generated all from same initial noise. Middle: At block

i

, we compute residual activation from the image stream. Every attribute serves as a source

x_{i}

and later one as its target

x_{j}

, giving source–target combinations, each a training example with source

X=x_{i}

and difference

Y_{ij}=x_{j}-x_{i}

. A 1D autoencoder maps

x_{i}

to a coordinate

z_{k}=e_{\theta}(x_{i})

and recovers the direction as a difference of decoded positions, fitted with

\mathcal{L}_{\mathrm{rec}}

and

\lambda\sum_{i<j}\max(0,z_{i}-z_{j})

to ensure reconstruction and monotonicity of the attributes. Bottom: At inference, the coordinate is and shifted by

\alpha

, so the steering vector computed

\nu_{\alpha}(x)=\gamma_{\theta}\left(e_{\theta}(x)+\alpha\right)-\gamma_{\theta}\left(e_{\theta}(x)\right)

Figure 3: Qualitative comparison on FLUX.1-schnell. We sweep the steering strength from unsteered generation, comparing the fixed direction against KANsteer learned trajectory. KANSteer traverses the attribute gradually, with intermediate columns showing intermediate attributes, while the linear direction holds the source attribute and then transitions abruptly.

In-range (fit: 5 attributes, steer: 5)
Model	Method	dir-CLIP $\uparrow$	CLIP-I $\uparrow$	LPIPS $\downarrow$	CLIP-T2I $\uparrow$	FID $\downarrow$	FVU $\downarrow$
FLUX.1 Schnell (1-step)	Linear	0.1079	0.8563	0.2434	0.5367	47.86	0.6187
FLUX.1 Schnell (1-step)	KANSteer	0.1703	0.9017	0.2275	0.5592	28.56	0.4674
FLUX.1 Schnell (4-step)	Linear	0.0855	0.8147	0.2243	0.5056	38.17	0.4906
FLUX.1 Schnell (4-step)	KANSteer	0.1493	0.8625	0.1932	0.5337	28.25	0.4531
FLUX.1 dev	Linear	0.0534	0.7932	0.2121	0.5006	46.73	0.5437

Table 1: Comparison of Linear steering and KANSteer on FLUX.1-schnell (1- and 4-step) and FLUX.1-dev, under two fitting regimes. In-range : the curve is fitted and steered on all ordered attributes. Extrapolation : the curve is fit on the first three attributes and steered over all five; Directional CLIP, CLIP-Image, and CLIP-T2I measure alignment with the target attribute; LPIPS and FID measure fidelity to the unsteered image and to the prompt generated distribution; FVU measures how well the fitted curve reconstructs the model’s own attribute displacement in activation space.

Figure 4: Extrapolating linear and KANSteer directions, both are fitted on first three attributes ( $a_{1},a_{2},a_{3}$ ) and steered over all five attributes.

Figure 5: Left: PCA of block-18 activation, with per-seed scatter (dots), and model predicted centroids overlaid. KANSteer predictions follow the curved centroid path whereas the linear predictions collapse onto a single direction that misses the curvature. Right: Non-linearity

(1-R^{2})

of KANSteer decoder curve as a function of the z interval, measured by the fraction of decoder variance not captured by the best linear fit. Over the full attribute range the non-linear share rises to

10-25\%

on concepts such as hair curliness and facial expression.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Residual norm $\rho_{k}$ per concept. $\rho_{k}=0$ means the displacement to attribute $k$ lies exactly on the endpoint direction; $\rho_{k}=1$ means it is entirely orthogonal. The dashed line marks the held-out endpoint average. Intermediate attributes (indices 2-4) are far from the line for every concept.

Figure 7: PCA projections of per-seed, per-attribute delta activations. Each point is one seed; large markers are attribute centroids; the dotted line connects the two endpoints. Intermediate attributes consistently bend away from the endpoint line.

Figure 8: Block 18 of FLUX.1 writes the concept; its massive-norm patches do not. Top: Patch-norm maps $\lVert x_{p}\rVert_{2}$ of the block-18 image-stream delta activations for the five ordered attributes of hair curliness, facial expression and season. A handful of patches sit two orders of magnitude above the median, and for a given concept they occupy nearly the same location across attributes. Bottom: Generations for the same seed and prompt template when (col. 1) nothing is removed, (col. 2) the entire block-18 update is removed, (col. 3) only the top-1% patches of the update is removed, (col. 4) only the update at patch $(63,0)$ is removed. Removing the block erases the attribute, whereas removing the massive patches leaves the generation, and the attribute, unchanged.

Table 2: FLUX.1 Schnell — Comparison of linear and KAN directions across ten semantic concepts.

Table 3: FLUX.1 Schnell 4 steps — Comparison of linear and KAN directions across ten semantic concepts.

Table 4: FLUX.1 Dev — Comparison of linear and KAN directions across ten semantic concepts.

Figure 9: Cake decoration (plain $\rightarrow$ decorated): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 10: Human Aging (preteen $\rightarrow$ elderly): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 11: Portion Size (tiny $\rightarrow$ overflowing): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 12: Dog Aging (preteen $\rightarrow$ elderly): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 13: Beard Growth (clean shaven $\rightarrow$ long beard): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 14: Metamorphosis (caterpillar $\rightarrow$ spread butterfly): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 15: Pattern Density (very sparse $\rightarrow$ very dense): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 16: Facial Emotions (sobbing $\rightarrow$ laughing): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 17: River flow (still $\rightarrow$ raging): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 18: Seasons (spring $\rightarrow$ deep winter): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 19: Hair Curliness (straight $\rightarrow$ coily): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Figure 20: Moon Phases (thin crescent $\rightarrow$ full): Comparison the Linear and KANSteer steering on FLUX.1 Schnell.

Activation Steering

Activation steering changes model behavior by intervening directly on internal representations during inference, often without updating weights.

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Steered LLM Activations are Non-Surjective

Aayush Mishra, Daniel Khashabi, Anqi Liu

Johns Hopkins University

Activation Steering LLM Interpretability

Open paper AskPDF arXiv

Activation steering is a popular white-box control technique that modifies model activations to elicit an abstract change in its behavior. It has also become a standard tool in interpretability (e.g., probing truthfulness, or translating activations into human-readable explanations) and safety research (e.g., jailbreakability). However, it is unclear whether steered behavior is realizable by any textual prompt. In this work, we cast this question as a surjectivity problem: for a fixed model, does every steered activation admit a preimage under the model's natural forward pass? Under practical assumptions, we prove that activation steering pushes the residual stream off the manifold of states reachable from discrete prompts. Almost surely, no prompt can reproduce the same internal behavior induced by steering. We also illustrate this finding empirically across three widely used LLMs. Our results establish a formal separation between white-box steerability and black-box prompting. We therefore caution against interpreting the ease and success of activation steering as evidence of prompt-based interpretability or vulnerability, and argue for evaluation protocols that explicitly decouple white-box and black-box interventions.

Abstract

Activation Steering

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