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

arXiv: https://arxiv.org/abs/2604.00230 · HTML: https://mlsanity.com/paper/2604.00230/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 59 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 30 of 43)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [A Geometric Analysis of Neural Collapse with Unconstrained Features](https://mlsanity.com/paper/2105.02375/) (2021)
- [Exploring Deep Neural Networks via Layer-Peeled Model: Minority Collapse in Imbalanced Training](https://mlsanity.com/paper/2101.12699/) (2021)
- [Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path](https://mlsanity.com/paper/2106.02073/) (2021)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [Neural collapse with unconstrained features](https://mlsanity.com/paper/2011.11619/) (2020)
- [Neural Collapse Inspired Feature-Classifier Alignment for Few-Shot Class Incremental Learning](https://mlsanity.com/paper/2302.03004/) (2023)
- [Extended Unconstrained Features Model for Exploring Deep Neural Collapse](https://mlsanity.com/paper/2202.08087/) (2022)
- [On the Role of Neural Collapse in Transfer Learning](https://mlsanity.com/paper/2112.15121/) (2021)
- [Inducing Neural Collapse in Imbalanced Learning: Do We Really Need a Learnable Classifier at the End of Deep Neural Network?](https://mlsanity.com/paper/2203.09081/) (2022)
- [On the Optimization Landscape of Neural Collapse under MSE Loss: Global Optimality with Unconstrained Features](https://mlsanity.com/paper/2203.01238/) (2022)
- [Neural Collapse: A Review on Modelling Principles and Generalization](https://mlsanity.com/paper/2206.04041/) (2022)
- [Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1709.01953/) (2017)
- [An Unconstrained Layer-Peeled Perspective on Neural Collapse](https://mlsanity.com/paper/2110.02796/) (2021)
- [Are All Losses Created Equal: A Neural Collapse Perspective](https://mlsanity.com/paper/2210.02192/) (2022)
- [Limitations of Neural Collapse for Understanding Generalization in Deep Learning](https://mlsanity.com/paper/2202.08384/) (2022)
- [Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold](https://mlsanity.com/paper/2209.09211/) (2022)
- [Neural Collapse Inspired Attraction-Repulsion-Balanced Loss for Imbalanced Learning](https://mlsanity.com/paper/2204.08735/) (2022)
- [Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model](https://mlsanity.com/paper/2305.13165/) (2023)

## Built on by (most cited first, 2 of 2)

- [The Weight Norm Sets the Grokking Timescale: A Causal Delay Law](https://mlsanity.com/paper/2606.13753/) (2026)
- [Beyond Coordinate Gauge: An Audited Protocol for Detecting Donor-Specific Functional Fingerprints after Neural Collapse](https://mlsanity.com/paper/2607.11967/) (2026)

## Abstract (arXiv)

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.