From Directions to Regions: Decomposing Activations in Language Models via Local Geometry

Or Shafran; Shaked Ronen; Omri Fahn; Shauli Ravfogel; Atticus Geiger; Mor Geva

cs.CLFeb 2, 2026

From Directions to Regions: Decomposing Activations in Language Models via Local Geometry

Authors: Or Shafran, Shaked Ronen, Omri Fahn, Shauli Ravfogel, Atticus Geiger, Mor Geva

Organizations: Blavatnik School of Computer Science and AI, Tel Aviv University, Israel · New York University, New York, NY, USA · Goodfire.

Abstract

Activation decomposition methods in language models are tightly coupled to geometric assumptions on how concepts are realized in activation space. Existing approaches search for individual global directions, implicitly assuming linear separability, which overlooks concepts with nonlinear or multi-dimensional structure. In this work, we leverage Mixture of Factor Analyzers (MFA) as a scalable, unsupervised alternative that models the activation space as a collection of Gaussian regions with their local covariance structure. MFA decomposes activations into two compositional geometric objects: the region's centroid in activation space, and the local variation from the centroid. We train large-scale MFAs for Llama-3.1-8B and Gemma-2-2B, and show they capture complex, nonlinear structures in activation space. Moreover, evaluations on localization and steering benchmarks show that MFA outperforms unsupervised baselines, is competitive with supervised localization methods, and often achieves stronger steering performance than sparse autoencoders. Together, our findings position local geometry, expressed through subspaces, as a promising unit of analysis for scalable concept discovery and model control, accounting for complex structures that isolated directions fail to capture.

Figures & tables

Figure 1 : MFA decomposes each activation into a region assignment and a within-region offset. Left: the region structure is modeled by Gaussian components (centroids $\boldsymbol{\mu}_{k}$ ), with complex concepts typically spanning multiple Gaussians – here, the broader Emotions neighborhood is spanned by several interpretable Gaussians. Right: each component is equipped with a low-dimensional subspace that parameterizes structured within-region variation.

Figure 2 : Example MFA Gaussians in the activation space of Llama-3.1-8B, visualized in 3D using three loadings as axes. (Left) A broad region spanning multiple movie genres, where the loadings separate genre-related themes. (Right) A narrow region centered on the token National , where the loadings capture context-dependent usage.

Term	Top promoted tokens
Genres Gaussian (Figure 2 A, broad )
$\boldsymbol{\mu}$	thriller , horror , sitcom , fiction , romance , fantasy , comedy , novel
$\mathbf{w}_{1}$	fantasy , tale , RPG , adventure , realms
$\mathbf{w}_{2}$	sitcom , television , TV , series , show
$\mathbf{w}_{3}$	detective , espionage , spy , thriller , saga
“National” Gaussian (Figure 2 B, narrow )

Table 1 : Centroid vs. loading token promotion. Representative top-promoted tokens under centroid (

\boldsymbol{\mu}

) vs. loading (

\mathbf{w}_{j}

) interventions. In the narrow region, the centroid promotes a general “National” theme, while loadings separate subthemes. In the broad region, the centroid promotes genres, and loadings refine it into subgenres/media-specific patterns.

Model	Task	PCA	SAE	MFA	DBM	DAS
Gemma-2	Continent	70.0	71.7	85.7	69.7	84.2
	Language	56.0	58.9	64.0	58.0	69.7
	Country	53.0	56.1	60.0	65.0	67.7
	MCQA	77.9	64.9	80.2	82.1	92.0
Llama-3.1	Continent	74.0	70.6	81.6	78.1	81.1
	Language	57.0	56.8	67.3	63.1	69.1

Table 2: Localization performance of MFA versus unsupervised and supervised baseline methods on RAVEL and MCQA.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Initialization	Steps	Mean pairwise dist. ( $\pm\;std$ )
Random Point	26136	$143.76\pm 106.18$
K-Means	28008	$108.26\pm 65.41$
Random Init	38880	$71.92\pm 10.33$

Appendix

Table 3 : Convergence and centroid diversity across initializations. We report the convergence rate in number of gradient descent steps and the mean

\pm

std of pairwise Euclidean distances between learned component means (

\boldsymbol{\mu}

Model	Layer	SAE	MFA-1K	KMeans-1K	MFA-8K	KMeans-8K	MFA-32K	KMeans-32K
Gemma-2	6	0.592	1.098	1.530 ( $\times$ 1.4)	0.896	1.265 ( $\times$ 1.4)	0.800	1.104 ( $\times$ 1.4)
Gemma-2	18	4.595	5.430	7.629 ( $\times$ 1.4)	4.678	6.622 ( $\times$ 1.4)	4.477	5.966 ( $\times$ 1.3)
Llama-3.1	8	0.023	0.005	0.007 ( $\times$ 1.3)	0.004	0.006 ( $\times$ 1.4)	0.004	0.005 ( $\times$ 1.4)
Llama-3.1	22	0.038	0.064	0.086 ( $\times$ 1.3)	0.051	0.074 ( $\times$ 1.5)	0.045	0.066 ( $\times$ 1.5)

Appendix

Table 4 : Reconstruction error reported as MSE for MFAs, SAEs, and a K-Means baseline. For the baseline we additionally provide the error ratio with MFA of the same size. All results were found to be statistically significant with standard error

<1e-4

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
MFA	74.3	77.5	80.1	82.4	76.3	77.1	68.2	68.6	56.0	69.5	71.7	73.5	71.0	75.1	74.2	78.9	82.0	82.6	84.5	84.9	83.9	84.6	85.5	85.4	85.4	85.4
SAE	44.8	47.4	54.9	54.5	60.0	60.4	62.6	66.3	66.7	68.1	66.9	66.3	63.9	66.3	63.1	65.6	71.7	70.2	68.0	68.9	66.1	69.2	65.9	64.1	62.4	66.4
DBM	58.4	58.0	59.8	60.8	60.6	61.3	63.0	65.9	69.0	69.0	68.6	69.1	69.1	68.1	69.6	64.7	68.9	68.7	68.5	69.7	68.4	67.5	67.9	65.8	65.2	61.8

Appendix

Table 5 : RAVEL continent results for Gemma-2 2B

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
MFA	80.0	81.0	79.0	77.0	80.0	77.0	76.0	57.0	62.0	78.0	77.0	74.0	77.0	77.0	81.0	81.0	81.0	82.0	82.0	82.0	82.0	82.0	81.0	82.0	82.0	82.0	82.0	82.0	82.0	82.0	82.0	82.0
SAE	54.4	48.0	57.2	56.1	58.9	61.3	61.2	59.9	63.7	62.3	63.2	63.6	66.6	63.5	59.8	61.6	67.3	70.5	70.6	70.6	68.8	67.9	66.7	63.6	62.4	62.4	62.6	62.6	62.4	62.4	62.6	62.6
DBM	58.2	60.1	59.9	59.2	60.6	63.3	64.3	66.3	66.2	65.5	67.4	68.1	68.0	67.1	65.0	66.5	74.2	73.8	73.9	77.3	76.2	78.1	74.8	77.3	76.1	73.3	72.1	71.6	68.0	67.9	66.1	64.7

Appendix

Table 6 : RAVEL continent results for Llama-3.1 8B

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
MFA	48.4	53.5	54.5	58.0	52.6	52.1	42.7	47.8	41.2	43.1	43.7	46.1	46.2	50.0	46.7	56.1	57.6	56.8	57.5	58.4	58.3	58.9	59.8	59.6	59.6	59.6
SAE	34.0	36.4	42.4	39.1	46.8	46.6	50.4	51.5	52.8	53.5	50.8	49.9	50.4	48.6	48.9	55.7	56.1	53.1	49.1	48.7	49.0	49.3	50.4	50.4	50.5	50.6
DBM	45.1	47.2	48.2	49.0	48.9	49.6	52.3	52.9	56.0	56.9	55.6	55.0	54.5	54.7	54.6	63.1	65.0	65.0	51.2	53.4	51.1	50.2	50.4	50.3	50.3	50.6

Appendix

Table 7 : RAVEL country results for Gemma-2 2B

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
MFA	60.0	61.0	59.0	56.0	57.0	54.0	51.0	45.0	47.0	54.0	53.0	51.0	53.0	56.0	60.0	60.0	62.0	63.0	63.0	63.0	63.0	63.0	62.0	63.0	63.0	63.0	63.0	63.0	63.0	63.0	63.0	63.0
SAE	45.0	38.6	45.5	40.6	47.6	47.4	49.5	50.9	52.0	49.9	49.4	50.1	51.7	52.4	57.6	56.1	53.7	51.8	52.2	55.0	55.1	55.9	54.3	54.7	50.4	50.9	50.5	50.5	50.5	50.4	50.5	50.7
DBM	49.2	48.2	45.6	49.1	48.9	48.9	50.6	52.1	52.6	54.2	55.1	55.3	56.5	58.2	59.8	60.4	56.3	58.3	58.3	57.8	56.4	57.2	55.4	54.9	50.5	51.5	50.4	50.4	50.5	50.5	50.6	50.7

Appendix

Table 8 : RAVEL country results for Llama-3.1 8B

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
MFA	57.2	58.7	61.1	61.9	56.2	55.4	48.6	47.8	44.5	50.2	48.6	52.8	54.2	54.9	54.1	58.2	60.5	61.1	62.4	62.9	63.4	63.1	63.2	63.1	63.0	63.0
SAE	50.8	43.1	49.9	44.0	50.9	48.6	50.8	53.7	54.3	53.1	53.7	53.0	53.3	52.8	53.0	52.9	53.4	55.7	58.9	57.0	56.0	56.3	54.8	54.6	54.6	54.6
DBM	53.4	54.1	54.0	53.8	53.0	53.5	54.1	53.6	53.8	54.4	53.8	54.3	54.3	53.4	54.1	53.8	51.6	53.3	58.0	56.8	55.8	56.2	54.6	54.6	54.7	54.6

Appendix

Table 9 : RAVEL language results for Gemma-2 2B

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
MFA	65.0	65.0	64.0	63.0	63.0	57.0	58.0	46.0	50.0	57.0	59.0	58.0	60.0	62.0	63.0	63.0	66.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0	67.0
SAE	50.2	52.2	48.3	53.0	54.2	54.3	54.2	52.6	54.5	54.8	54.9	55.4	55.5	55.2	55.3	53.9	55.1	56.5	56.0	55.4	55.6	56.8	55.4	55.7	55.6	55.6	55.6	55.7	55.7	55.7	55.7	55.9
DBM	54.2	53.8	54.5	53.5	54.2	54.4	55.2	56.0	56.4	56.0	56.2	56.8	57.8	57.5	57.8	59.5	63.0	63.1	61.2	61.8	62.8	61.9	59.6	60.1	56.7	56.8	55.7	55.7	55.8	55.8	55.8	55.9

Appendix

Table 10 : RAVEL language results for Llama-3.1 8B

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
PCA	52.2	46.2	47.9	48.2	45.1	42.7	41.9	48.7	50.6	49.2	50.2	51.4	50.6	52.3	48.5	50.3	36.6	77.9	63.3	68.9	70.7	59.3	60.4	61.7	59.7	67.1
SAE	50.0	41.9	53.9	47.8	44.0	52.9	49.6	51.4	49.0	50.2	49.1	47.8	46.6	46.4	45.4	47.4	44.4	64.8	60.1	54.7	57.4	57.3	54.7	54.7	50.8	54.0
MFA	46.1	44.3	45.9	45.2	45.3	47.0	40.0	40.2	40.9	41.5	41.5	43.7	41.9	40.6	41.2	39.1	36.1	80.3	68.7	66.9	68.7	68.7	62.4	66.2	61.3	58.6
DBM	58.7	59.8	57.1	58.5	56.7	56.7	58.9	59.9	56.4	55.8	57.0	60.3	61.2	65.6	68.8	63.9	53.5	82.1	74.9	66.9	66.6	70.0	65.7	63.6	63.2	64.9

Appendix

Table 11 : MCQA results for Gemma-2 2B

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
PCA	37.5	33.6	33.3	36.4	37.8	39.2	35.6	40.6	43.8	42.0	44.5	44.9	46.3	48.4	47.7	53.5	33.3	70.0	68.9	69.6	71.4	71.8	73.6	72.2	74.3	58.2	58.5	61.1	64.3	58.6	61.4	68.3
SAE	43.1	44.5	51.7	53.5	53.5	55.6	58.1	54.5	60.3	59.9	60.7	62.8	59.9	59.9	54.2	56.0	33.7	65.6	64.9	65.6	63.8	64.5	65.6	63.5	63.8	58.8	53.9	55.2	60.3	63.1	54.2	43.5
MFA	41.0	53.3	46.3	52.5	47.1	53.1	56.0	61.4	59.1	56.0	55.3	56.8	59.8	59.1	56.9	61.5	34.0	65.4	65.2	69.7	68.2	69.0	68.2	69.0	69.8	68.9	70.6	66.7	65.2	71.4	67.6	58.2
DBM	57.4	61.0	59.9	57.8	60.3	63.5	64.6	67.5	70.7	69.7	68.2	71.1	70.0	67.1	63.9	61.7	41.9	72.5	70.7	75.0	74.0	73.2	72.2	73.6	69.6	67.1	65.6	66.0	65.3	64.9	67.1	66.5

Appendix

Table 12 : MCQA results for Llama-3.1 8B

Model	Method	Early layer	Late layer
Gemma-2-2B	SAE	$0.967\pm 0.111$	$1.004\pm 0.031$
	DiffMeans	$1.002\pm 0.033$	$1.003\pm 0.035$
	Prompting	$1.010\pm 0.045$	$1.010\pm 0.045$
	ReFT-r1	$0.927\pm 0.152$	$0.958\pm 0.089$
	MFA-32K	$0.841\pm 0.234$	$0.855\pm 0.197$
	MFA-8K	$0.844\pm 0.224$	$0.847\pm 0.196$

Appendix

Table 13: Fluency scores computed over all evaluated units, measured at the intervention setting selected by the best final score. We report mean

\pm

standard deviation for each method, model, and evaluated layer.

Term	Descriptor	Top promoted tokens
(1) Superheroes / comics universe
$\boldsymbol{\mu}$	superheroes	Superman , Batman , Joker , Gotham , Deadpool , superhero
$\mathbf{w}_{1}$	Superman	Superman , Clark , krypton , rypton , SUPER , restoring
$\mathbf{w}_{2}$	DC film studio	Batman , Warner , WB , Nolan , Snyder , films
$\mathbf{w}_{3}$	Batman	Joker , Gotham , Commissioner , Mafia , inmates , profiler
(2) Vascular procedures / clinical devices

Appendix

Table 14 : More centroid vs. loading examples (Llama-3.1-8B). Each centroid shows broad theme promotion, while loadings isolate sharper subthemes.

Term	Descriptor	Top promoted tokens
(1) continents/countries
$\boldsymbol{\mu}$	continents	continent , Europe , EU , France , Parlamento , Countries
$\mathbf{w}_{1}$	continent names	Europe , Asia , Africa , continents , European
$\mathbf{w}_{2}$	Africa	Africa , Ebola , malaria , Angola , regions
$\mathbf{w}_{3}$	SE Asia	Nang , Cambodian , Ceylon , Nepali , Indon
(2) Sleep / tiredness

Appendix

Table 15 : More centroid vs. loading examples (Gemma-2-2B). Each centroid shows broad theme promotion, while loadings isolate sharper subthemes. Additionally, we label narrow Gaussians to show that local variation is often meaningful.

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A Geometric Account of Activation Steering through Angle-Norm Decomposition

Georgii Aparin, Tatiana Gaintseva

Huawei Noah’s Ark Lab · Queen Mary University of London

Linear Activation Steering Steering Model Activations Hidden States Latent Feature Interventions

Open paper AskPDF arXiv

Linear activation steering has gained popularity as a simple and empirically effective way to control language model behavior. More recently, spherical steering paradigms have been proposed to address limitations of additive interventions, often motivated by the assumption that hidden-state norm does not carry concept-relevant information. In this work, we revisit this assumption through a controlled empirical study designed to disentangle the roles of angular and radial components. We show that steering methods differ mainly in how they couple two geometric effects: changing a token's angular alignment with a concept direction and changing its hidden-state norm. Across seven language models, we find that concepts are represented primarily in angular structure, supporting the motivation for spherical methods, but that norm remains important for the stability and downstream effects of steering. Our results explain why interventions with similar concept-level effects can behave differently, and suggest that activation steering should be parameterized by interpretable angular and radial components of the intervention, rather than by a single additive coefficient that entangles these two effects.

Abstract

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