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 $m\times n$ hidden activation matrices. Our main result shows that $\mathbb{E}[EDim(\mathrm{\ell })]=m[1-(1-2\mathrm{/}\pi {)}^{\mathrm{\ell }}]+O(e^{-cm})$ so that the rank deficit decays geometrically with ratio $1-2\mathrm{/}\pi \approx 0.3634$. 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 ${\mathrm{\ell }}_k^{\ast }\approx (k+1\mathrm{/}2)\pi \mathrm{/}\log (1\mathrm{/}\rho )$ with height $\approx (1-e^{-\pi \mathrm{/}2})m\approx 0.79m$. 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.