Neural Collapse Dynamics: Depth, Activation, Regularisation, and Feature Norm Threshold

TL;DR

This paper identifies a critical feature norm value that predicts neural collapse onset in deep networks, revealing how architecture, activation, and regularization influence this process.

Contribution

It uncovers a simple, predictive regularity linking feature norm to neural collapse, and characterizes how network structure and training conditions affect this dynamics.

Findings

Neural collapse occurs when feature norm reaches a dataset-specific threshold.

Crossing the norm threshold reliably predicts neural collapse onset with an average lead of 62 epochs.

Regularities in depth, activation, and regularization significantly influence collapse speed and feature norm threshold.

Abstract

Neural collapse (NC) -- the convergence of penultimate-layer features to a simplex equiangular tight frame -- is well understood at equilibrium, but the dynamics governing its onset remain poorly characterised. We identify a simple and predictive regularity: NC occurs when the mean feature norm reaches a model-dataset-specific critical value, fn*, that is largely invariant to training conditions. This value concentrates tightly within each (model, dataset) pair (CV < 8%); training dynamics primarily affect the rate at which fn approaches fn*, rather than the value itself. In standard training trajectories, the crossing of fn below fn* consistently precedes NC onset, providing a practical predictor with a mean lead time of 62 epochs (MAE 24 epochs). A direct intervention experiment confirms fn* is a stable attractor of the gradient flow -- perturbations to feature scale are self-corrected during training, with convergence to the same value regardless of direction (p>0.2). Completing the (architecture)x(dataset) grid reveals the paper's strongest result: ResNet-20 on MNIST gives fn* = 5.867 -- a +458% architecture effect versus only +68% on CIFAR-10. The grid is strongly non-additive; fn* cannot be decomposed into independent architecture and dataset contributions. Four structural regularities emerge: (1) depth has a non-monotonic effect on collapse speed; (2) activation jointly determines both collapse speed and fn*; (3) weight decay defines a three-regime phase diagram -- too little slows, an optimal range is fastest, and too much prevents collapse; (4) width monotonically accelerates collapse while shifting fn* by at most 13%. These results establish feature-norm dynamics as an actionable diagnostic for predicting NC timing, suggesting that norm-threshold behaviour is a general mechanism underlying delayed representational reorganisation in deep networks.