Coarse-Graining Neural Representations through Layer-Dependent Task-Relevant Subspaces

Deep neural networks transform high-dimensional internal representations through many nonlinear layers, yet it remains unclear whether their task-relevant internal computation can be described by a substantially smaller set of effective variables. We develop an operational coarse-graining framework for testing candidate low-dimensional neural states from three complementary viewpoints: functional sufficiency, whether the state retains enough information to recover the eventual network output; dynamical sufficiency, whether its layerwise evolution admits an accurate reduced description; and causal sufficiency, whether retaining or manipulating it preserves the behavior of the original downstream network. We construct output-aware reference subspaces from the sensitivity of centered logits to hidden activations and use the corresponding resolved coordinates as candidate coarse variables. When the task-relevant subspace changes across depth, we refer to these layer-dependent coordinates as a moving coarse state. Across controlled tasks, MNIST, and CIFAR-10, the framework reveals qualitatively different regimes. On MNIST, a compact state of approximately seven dimensions simultaneously supports high functional fidelity, accurate Markovian closure, and strong causal preservation. On CIFAR-10, these requirements separate: functional information is concentrated at moderate rank, accurate reduced dynamics require nonlinear closure and benefit modestly from finite memory, whereas stronger causal criteria require more resolved dimensions. Conditional reconstruction further shows that naive keep-only interventions can underestimate causal sufficiency by creating off-manifold hidden states. These results suggest that neural representations are better characterized by a sufficiency profile—functional, dynamical, and causal—than by a single intrinsic dimension, and provide an operational starting point for analyzing trained neural networks through coarse variables.