Neurons stay mixed, yet tasks can be switched off one at a time: a linear change of coordinates exposes modularity

Modularity is judged at three nested levels rather than by whether single neurons are specialized: unit-level requires distinct computations to occupy disjoint neurons, orthogonal-level only asks whether a rotation reveals functionally independent subspaces, and linear-level allows a bounded invertible linear map to separate the computations. The prevailing practice is to inspect the selectivity of individual neurons or voxels, and by that standard tanh networks and most transformer models show pervasive mixed selectivity and are judged to lack modular structure. On the Yang multitask cognitive tasks, softplus networks reach a double dissociation index of 0.59 at the unit level, while tanh networks reach only 0.03 at the unit level, 0.47 under orthogonal decomposition and 0.70 under linear decomposition, so separation invisible at the unit level appears at the subspace level in the same networks.

Whether a decomposition is genuinely functionally modular is decided by ablation: zeroing a subspace at inference should impair its own function while largely sparing the other, and the reverse must hold as well for a double dissociation. Earlier selectivity measures show that a unit or direction is correlated with a task, not that the computation depends on it; on the Mante task, regression-derived motion and color axes are highly predictive and nearly orthogonal, yet ablating either axis changes accuracy by less than 0.5%. On the Mante context-dependent integration task, the linear decomposition raises the double dissociation index as subspace dimension grows from 8 to 16 and then plateaus, with ablating the motion subspace reducing motion accuracy while leaving color accuracy nearly intact and the color ablation showing the symmetric pattern.

The decompositions are learned with supervision: recurrent networks with explicit input-output tasks use a task-sufficiency objective so that each subspace alone suffices for its own task, while pretrained vision and language models without task labels use contrastive learning on minimal pairs. For vision the minimal pairs are different stylizations of the same scene and different scenes in the same style, and for language they are meaning-preserving paraphrases and sentences sharing a grammatical template with different lexical content. The two contrastive losses act on non-overlapping coordinate blocks of the transformed representation, so gradients update disjoint columns of the transformation and geometric separation holds by construction; training uses 640,000 image pairs built from MS-COCO and Paint by Numbers and 3,900 sentence pairs from the MSR Paraphrase Corpus.

The hierarchy is demonstrated in three settings: multitask recurrent networks on the Yang and Mante tasks, content versus style in pretrained vision models, and syntax versus semantics in pretrained language models, evaluated by causal ablation and nearest-neighbor retrieval respectively. It is aimed at researchers working on mechanistic interpretability, representation decomposition and model-brain alignment. For vision and language, subspace quality is currently measured by retrieval-based selectivity, and the authors propose complementing this with behavioral evaluations, for instance testing whether projecting out the content subspace selectively impairs object recognition while preserving texture judgments.

Learning the decompositions requires non-convex optimization that is sensitive to hyperparameters such as subspace dimensionality, the condition-number bound and the learning rate. The residual subspace is not purely nuisance: it can contain or support task-relevant information, but that information appears shared or redundant rather than selectively tied to either task, and how it is organized remains unclear. The ResNet family is the principal exception, with weak style retrieval across all subspace types. The authors also note that how subspace structure and dimensionality evolve across training and layers, and whether the hierarchy extends naturally to multi-way decompositions beyond two functions, remain open questions.