A Closed Form Depth-Width Law for Neural Collapse in Deep Linear Networks

Pankhil Gawade, Kirsten Fischer, Guy Wolf, Ewa Szczurek

How depth and width of neural networks shape learned representations is usually studied only after training. We show that, in a tractable feature-learning setting, this dependence can instead be obtained in closed form. Using neural collapse as a probe of representation geometry, we express NC1--NC3 directly in terms of the learned feature kernel, extending an identity known for NC1 to the angular part of NC2 and to NC3 under a ridge readout. For deep linear networks with exchangeable class structure, the resulting kernel equations yield a geometric law, exact within the theory, for the layerwise evolution of within-class variability. At large width, the accumulated compression becomes a function of the depth-to-width ratio $L/N$, with its rate determined self-consistently by the data, prior and architecture. Numerical experiments support the derived dependence on depth, width and signal-to-noise ratio. The result is a representation-level scaling law derived from feature-learning theory rather than fitted after training.