Compressible Representations: Functional Spines in Deep Neural Networks

Deep networks embed inputs in high-dimensional representations, but how much of this capacity actually drives behavior? We introduce a framework for identifying, at each layer, the subspace of the representation that is functionally relevant for the task. Using a matrix analogous to the Fisher information but defined over the hidden representation, we rank directions by their influence on downstream computation and project onto the most task-relevant subspace. We benchmark this behaviorally grounded compression against three geometric alternatives that preserve variance, local neighborhood structure, or global topology. Across pretrained vision models, representations are remarkably compressible: a small fraction of the task-relevant subspace preserves most performance, often requiring far fewer dimensions than geometric alternatives. We show that this decomposition recovers functionally distinct subspaces within the same representation: on cue-conflict stimuli, shape-relevant and texture-relevant directions are largely non-overlapping, and ablating one selectively impairs the corresponding task while leaving the other intact. Task-relevant subspaces also provide a principled foundation for continual learning: constraining gradient updates to lie orthogonal to the task-relevant subspaces of prior tasks outperforms standard variance-based gradient projection methods. Finally, restricting representational comparisons to task-relevant subspaces reveals that inter-model alignment is higher in task-relevant directions than in task-irrelevant directions at later layers, and that conditioning on different tasks over the same stimuli uncovers asymmetries that full-space similarity measures cannot detect. Together, these results establish that deep networks compute through a low-dimensional functional core embedded within a much larger representational space, that these cores differ across tasks, and that they provide a unifying basis for compression, continual learning, and representational comparison.