# Beyond Unconstrained Features: Neural Collapse for Shallow Neural Networks with General Data

arXiv: https://arxiv.org/abs/2409.01832 · HTML: https://mlsanity.com/paper/2409.01832/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 44 references that are in the map.
- First posted on arXiv 3 Sep 2024.

## Builds on (most cited first, 30 of 31)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [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)
- [Neural collapse with unconstrained features](https://mlsanity.com/paper/2011.11619/) (2020)
- [Convergence Analysis of Two-layer Neural Networks with ReLU Activation](https://mlsanity.com/paper/1705.09886/) (2017)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [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)
- [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)
- [An Unconstrained Layer-Peeled Perspective on Neural Collapse](https://mlsanity.com/paper/2110.02796/) (2021)
- [Imbalance Trouble: Revisiting Neural-Collapse Geometry](https://mlsanity.com/paper/2208.05512/) (2022)
- [Limitations of Neural Collapse for Understanding Generalization in Deep Learning](https://mlsanity.com/paper/2202.08384/) (2022)
- [On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers](https://mlsanity.com/paper/2012.05420/) (2020)
- [Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold](https://mlsanity.com/paper/2209.09211/) (2022)
- [Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model](https://mlsanity.com/paper/2305.13165/) (2023)
- [Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data](https://mlsanity.com/paper/2301.00437/) (2023)
- [Neural Collapse for Unconstrained Feature Model under Cross-entropy Loss with Imbalanced Data](https://mlsanity.com/paper/2309.09725/) (2023)
- [Explicit regularization and implicit bias in deep network classifiers trained with the square loss](https://mlsanity.com/paper/2101.00072/) (2020)
- [Perturbation Analysis of Neural Collapse](https://mlsanity.com/paper/2210.16658/) (2022)
- [Understanding Deep Representation Learning via Layerwise Feature Compression and Discrimination](https://mlsanity.com/paper/2311.02960/) (2023)
- [On the Implicit Geometry of Cross-Entropy Parameterizations for Label-Imbalanced Data](https://mlsanity.com/paper/2303.07608/) (2023)
- [Neural Collapse for Cross-entropy Class-Imbalanced Learning with Unconstrained ReLU Feature Model](https://mlsanity.com/paper/2401.02058/) (2024)
- [Neural (Tangent Kernel) Collapse](https://mlsanity.com/paper/2305.16427/) (2023)
- [Unifying Low Dimensional Spectra in Deep Learning](https://mlsanity.com/paper/2404.06106/) (2024)

## Built on by (most cited first, 10 of 10)

- [Wide Neural Networks Trained with Weight Decay Provably Exhibit Neural Collapse](https://mlsanity.com/paper/2410.04887/) (2024)
- [The Persistence of Neural Collapse Despite Low-Rank Bias](https://mlsanity.com/paper/2410.23169/) (2024)
- [Neural Collapse is Globally Optimal in Deep Regularized ResNets and Transformers](https://mlsanity.com/paper/2505.15239/) (2025)
- [Flatness is Necessary, Neural Collapse is Not: Rethinking Generalization via Grokking](https://mlsanity.com/paper/2509.17738/) (2025)
- [Linearly Separable Features in Shallow Nonlinear Networks: Width Scales Polynomially with Intrinsic Data Dimension](https://mlsanity.com/paper/2501.02364/) (2025)
- [Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data](https://mlsanity.com/paper/2510.21078/) (2025)
- [Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics](https://mlsanity.com/paper/2512.04006/) (2025)
- [The Implicit Bias of Depth: From Neural Collapse to Softmax Codes](https://mlsanity.com/paper/2605.23087/) (2026)
- [Understanding Head Geometry and Dynamics in Federated Regression through a Natural Solution Selection Rule: An Unconstrained Feature Model Analysis](https://mlsanity.com/paper/2609.39464/) (2026)
- [CAENL: Collapse-Based Active Neural Learning with Objective-Aligned Representation Control](https://mlsanity.com/paper/2610.17509/) (2026)

## Abstract (arXiv)

Neural collapse (NC) is a phenomenon that emerges at the terminal phase of the training (TPT) of deep neural networks (DNNs). The features of the data in the same class collapse to their respective sample means and the sample means exhibit a simplex equiangular tight frame (ETF). In the past few years, there has been a surge of works that focus on explaining why the NC occurs and how it affects generalization. Since the DNNs are notoriously difficult to analyze, most works mainly focus on the unconstrained feature model (UFM). While the UFM explains the NC to some extent, it fails to provide a complete picture of how the network architecture and the dataset affect NC. In this work, we focus on shallow ReLU neural networks and try to understand how the width, depth, data dimension, and statistical property of the training dataset influence the neural collapse. We provide a complete characterization of when the NC occurs for two or three-layer neural networks. For two-layer ReLU neural networks, a sufficient condition on when the global minimizer of the regularized empirical risk function exhibits the NC configuration depends on the data dimension, sample size, and the signal-to-noise ratio in the data instead of the network width. For three-layer neural networks, we show that the NC occurs as long as the first layer is sufficiently wide. Regarding the connection between NC and generalization, we show the generalization heavily depends on the SNR (signal-to-noise ratio) in the data: even if the NC occurs, the generalization can still be bad provided that the SNR in the data is too low. Our results significantly extend the state-of-the-art theoretical analysis of the N C under the UFM by characterizing the emergence of the N C under shallow nonlinear networks and showing how it depends on data properties and network architecture.