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

TL;DR

This paper provides a detailed theoretical analysis of how the effective dimension, measured by rank, oscillates across layers in finite-width ReLU networks with Gaussian weights, revealing geometric decay and revival patterns.

Contribution

It derives explicit formulas and bounds for the expected rank of layer activations, highlighting the finite-width effects and oscillatory behavior absent in infinite-width models.

Findings

Expected rank decays geometrically with depth

Rank revival peaks occur at specific depths related to network parameters

Oscillatory rank behavior is specific to finite-width 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 $\mathbb{E}[EDim(\ell)]=m[1-(1-2/\pi)^\ell]+O(e^{-c m})$ so that the rank deficit decays geometrically with ratio $1-2 / \pi \approx 0.3634$. 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 $\ell_k^*\approx(k+1/2)\pi/\log(1/\rho)$ with height $\approx (1-e^{-\pi/2}) m \approx 0.79m$. 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.