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

Abstract

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 $\mathbb{E}[EDim(\ell )]=m[1-(1-2/\ddot{I}€{)}^{\ell }]+O(e^{-cm})$ so that the rank deficit decays geometrically with ratio $1-2/\ddot{I}€\approx 0.3634$. 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 ${\ell }_k^{\ast }\approx (k+1/2)\ddot{I}€/\log (1/\ddot{I})$ with height $\approx (1-e^{-\ddot{I}€/2})m\approx 0.79m$. 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.

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