The Linear Centroids Hypothesis: Features as Directions Learned by Local Experts

In artificial and biological neural networks, concepts are often encoded as consistent linear directions in representation space. In deep learning, this idea is known as the linear representation hypothesis and underpins many interpretability and control methods based on linear probes, from concept detection to activation steering. Yet while prior work has studied whether such directions should exist $\textit{after}$ training, the dynamics of how they emerge $\textit{during}$ training remain poorly understood. Here, we develop a framework to study the alignment of concept directions during training - a process we call "abstraction". In a minimal linear network setting, we obtain exact solutions for the full trajectory of abstraction. These solutions reveal key analytic principles governing abstraction: (i) data and target geometry jointly determine abstraction at the end-of-learning, (ii) abstraction improves with network depth, and (iii) initialization scale controls the maximum abstraction reached during training. Extending our theory to nonlinear networks, we analyze how the choice of nonlinearity affects abstraction dynamics: erf networks approximate the linear theory, while abstraction in ReLU networks depends less on target geometry and more on input geometry. Across both, we prove a striking attenuation law: both nonlinearities weaken abstraction in activations relative to preactivations. We find evidence for this law in open models (DINOv3, Gemma 4) and apply our theory to improve linear probe generalization in LLMs. Together, our results provide a dynamical theory of abstraction with implications for interpretability and control.

Mechanistic interpretability of vision transformers seeks to decompose model computation into human-readable units, but learned representations entangle many concepts in each neuron. Feature superposition is widely treated as the central obstacle to this decomposition, yet most mitigations (sparse autoencoders, dictionary learning) are post-hoc and leave the underlying network unchanged. We ask whether a spatial-locality training loss (TopoLoss) can act as a lightweight, training-time prior that improves interpretability of standard mech-interp tools. Training ViT on ImageNet-100 across multiple TopoLoss weights $α$, we measure causal sufficiency of topographic clusters via activation patching and feature geometry via sparse autoencoders fit to the same residual stream. At $α=1.0$, topographic clusters are 2.79$\times$ more causally sufficient than random unit sets of the same size, with the effect increasing monotonically in $α$. SAE L0 sparsity decreases by 11% and dead-feature fraction rises 19-fold, yet standard neuron-level monosemanticity scores are unchanged, indicating that topographic pressure acts at circuit level, concentrating causal mass into spatially local structures without disentangling individual neurons. This dissociation suggests current neuron-level monosemanticity metrics are insensitive to a class of real interpretability gains, and positions cheap architectural priors as a viable training-time complement to post-hoc tooling.

Deep ReLU networks are piecewise-affine mappings that partition the input space into cells, each characterized by a distinct activation pattern. This structure motivates fitting a local linear model within each cell to preserve predictive accuracy while improving interpretability. The challenge is to identify regimes that are stable, data-adaptive, and easy to explain. We propose REALM, a mixture of linear models whose regimes are induced by neural activation patterns. Because the number of activation cells in a deep neural network (DNN) can grow rapidly with depth, we first distill a deep teacher into a wide, shallow student network (WSSN), then binarize and cluster its hidden-layer activations to define the regimes and fit a linear model within each regime. Since the regimes are discovered from internal structure, the router does not carry the predictive burden. To make regime assignment interpretable, we train a multiclass logistic regression, the explanatory gate, to reproduce the regime assignments. The two-level structure is interpretable at both stages in terms of raw tabular or learned convolutional features: the gate identifies features that determine regime assignments, while the linear models identify features that drive predictions within each regime. We analyze an idealized setting that illustrates a trade-off between partition complexity and stability: as the number of regimes grows, finer partitions can improve approximation but may reduce regime-assignment stability. Experiments on tabular and image datasets show that REALM achieves competitive predictive performance relative to other DNN-guided mixture surrogates and inherently interpretable models while producing stable regime-level explanations.