From Directions to Regions: Decomposing Activations in Language Models via Local Geometry
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.
698a0c131b2dc6b37d61adeb
Or Shafran
698a0c131b2dc6b37d61adec
Shaked Ronen
698a0c131b2dc6b37d61aded
Omri Fahn
698a0c131b2dc6b37d61adee
Shauli Ravfogel
698a0c131b2dc6b37d61adef
Atticus Geiger
698a0c131b2dc6b37d61adf0
Mor Geva
claimed_verified
2026-03-11T08:46:25.318Z
610b729f9da682cd54ad9adf
https://cdn-avatars.huggingface.co/v1/production/uploads/1628140189042-noauth.jpeg
Mor Geva
mega
user
mega
From Directions to Regions: Decomposing Activations in Language Models via Local Geometry
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.
Mixture of Factor Analyzers (MFA) provides a scalable, unsupervised approach for discovering complex, nonlinear structures in language model activation spaces by modeling local geometric structures through Gaussian regions and their covariance properties.
Mixture of Factor Analyzers activation space Gaussian regions local covariance structure concept discovery model control sparse autoencoders