A Mixture of Factor Analyzers decomposes real LLM activation space into local Gaussian-region subspaces (centroid + local covariance), more interpretable than global directions

- Shafran, Ronen, Fahn, Ravfogel, Geiger & Geva train 12 Mixtures of Factor Analyzers (K = 1K, 8K, 32K components, latent rank R = 10) on 100M residual-stream activations from The Pile at layers 6 and 18 of Gemma-2-2B and layers 8 and 22 of Llama-3.1-8B; each component is a Gaussian with a centroid mu_k and a low-rank local subspace span(W_k), so an activation is decomposed into a region assignment plus a within-region offset rather than a combination of global directions. [shafran-etal-2026-directions-to-regions] - Of 600 annotated Gaussians, some are narrow (concentrated on specific tokens or contexts) and some broad (thematic); narrow Gaussians skew syntactic and broad ones semantic, larger K makes more Gaussians narrow in Gemma-2-2B but not in Llama-3.1-8B, and nearby components form semantic neighborhoods (e.g. an "emotions" neighborhood of several Gaussians), which the authors read as concepts being realized by constellations of nearby Gaussians. [shafran-etal-2026-directions-to-regions] - Under the authors' own interpretability-fraction protocol (human annotators for MFA, an LLM judge for SAE features) MFA scores 0.96 versus 0.29 for Gemma Scope/Llama Scope SAEs; centroid-based steering beats SAE features and difference-in-means in most settings; on MIB RAVEL/MCQA localization (Table 2) MFA beats SAEs in all 8 model-task cells and PCA in 7 of 8 (Llama-3.1-8B MCQA is the exception: PCA 74.3 vs MFA 70.5), and is competitive with the supervised DBM and DAS baselines (above DAS only on Continent for both models, below DAS on the other tasks). Steering uses centroids only; the loadings are used for localization and in qualitative token-promotion examples. [shafran-etal-2026-directions-to-regions]