Some Theoretical Results on Layerwise Effective Dimension Oscillations in Finite Width ReLU Networks

We analyze the layerwise effective dimension (rank of the feature matrix) in fully-connected ReLU networks of finite width. Specifically, for a fixed batch of m inputs and random Gaussian weights, we derive closed-form expressions for the expected rank of the m× n hidden activation matrices. Our main result shows that mathbbE[EDim(ℓ)]=m[1-(1-2/π)ℓ]+O(e-c m) so that the rank deficit decays geometrically with ratio 1-2 / π ≈ 0.3634. We also prove a sub-Gaussian concentration bound, and identify the "revival" depths at which the expected rank attains local maxima. In particular, these peaks occur at depths ℓk^*≈(k+1/2)π/log(1/ρ) with height ≈ (1-e-π/2) m ≈ 0.79m. We further show that this oscillatory rank behavior is a finite-width phenomenon: under orthogonal weight initialization or strong negative-slope leaky-ReLU, the rank remains (nearly) full. These results provide a precise characterization of how random ReLU layers alternately collapse and partially revive the subspace of input variations, adding nuance to prior work on expressivity of deep networks.