Neural Collapse Dynamics: Depth, Activation, Regularisation, and Feature Norm Threshold
This provides an actionable diagnostic tool for researchers and practitioners to predict neural collapse timing in deep networks, though it is incremental as it builds on existing NC theory.
The paper tackles the problem of predicting when neural collapse occurs during training by identifying that it consistently happens when the mean feature norm reaches a model-dataset-specific critical value (fn*), which is largely invariant to training conditions and can predict NC onset with a mean lead time of 62 epochs.
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.