Recursive Projection Dynamics of Neural Representations: Geometry, Dimensionality, and Task Information

Neural representations evolve through successive transformations across network layers, architectures, and learning paradigms. While representational geometry is conventionally analyzed at individual layer states, the trajectories formed by successive transformations remain less explored. Here, we propose a recursive projection framework in which representation dynamics evolve as hₜ₊₁ = Πₜ(hₜ), where hₜ denotes an observable representation state and Πₜ the corresponding projection operator. This formulation separates geometric transformation from degradation of task-relevant information and motivates their joint analysis along representation trajectories. We evaluate this perspective in a CIFAR-10 ResNet-18 by jointly tracking adjacent-layer geometry, effective dimensionality, and linear accessibility of task information. The largest geometric reorganization occurs from layer3 to layer4 (1 − CKA = 0.411), where effective dimensionality decreases from 19.27 to 9.66 while linear-probe accuracy increases from 88.27% to 92.98%. Thus, substantial geometric reorganization and dimensional compression can coexist with improved task-relevant linear separability. These results support viewing neural representations as trajectories generated by successive transformations rather than as isolated representational states. 4 pages, 1 figure. Submitted to the NeurIPS 2026 Workshop on Symmetry and Geometry in Neural Representations (NeurReps), Extended Abstract Track.