# Neural Collapse versus Low-rank Bias: Is Deep Neural Collapse Really Optimal?

arXiv: https://arxiv.org/abs/2405.14468 · HTML: https://mlsanity.com/paper/2405.14468/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 63 references that are in the map.
- First posted on arXiv 23 May 2024.

## Builds on (most cited first, 30 of 50)

- [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)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [A Geometric Analysis of Neural Collapse with Unconstrained Features](https://mlsanity.com/paper/2105.02375/) (2021)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [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)
- [The Low-Rank Simplicity Bias in Deep Networks](https://mlsanity.com/paper/2103.10427/) (2021)
- [Neural collapse with unconstrained features](https://mlsanity.com/paper/2011.11619/) (2020)
- [Extended Unconstrained Features Model for Exploring Deep Neural Collapse](https://mlsanity.com/paper/2202.08087/) (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)
- [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)
- [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)
- [No Fear of Classifier Biases: Neural Collapse Inspired Federated Learning with Synthetic and Fixed Classifier](https://mlsanity.com/paper/2303.10058/) (2023)
- [Implicit Regularization Towards Rank Minimization in ReLU Networks](https://mlsanity.com/paper/2201.12760/) (2022)
- [On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers](https://mlsanity.com/paper/2012.05420/) (2020)
- [Rank Diminishing in Deep Neural Networks](https://mlsanity.com/paper/2206.06072/) (2022)
- [Implicit Regularization in Tensor Factorization](https://mlsanity.com/paper/2102.09972/) (2021)
- [Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model](https://mlsanity.com/paper/2305.13165/) (2023)
- [Gradient Descent for Deep Matrix Factorization: Dynamics and Implicit Bias towards Low Rank](https://mlsanity.com/paper/2011.13772/) (2020)
- [Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data](https://mlsanity.com/paper/2301.00437/) (2023)
- [Generalized Neural Collapse for a Large Number of Classes](https://mlsanity.com/paper/2310.05351/) (2023)
- [Implicit Bias of Large Depth Networks: a Notion of Rank for Nonlinear Functions](https://mlsanity.com/paper/2209.15055/) (2022)
- [Sharpness-Aware Minimization Leads to Low-Rank Features](https://mlsanity.com/paper/2305.16292/) (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)
- [A Law of Data Separation in Deep Learning](https://mlsanity.com/paper/2210.17020/) (2022)

## Built on by (most cited first, 17 of 17)

- [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)
- [On the Local Complexity of Linear Regions in Deep ReLU Networks](https://mlsanity.com/paper/2412.18283/) (2024)
- [Learning to Compress: Local Rank and Information Compression in Deep Neural Networks](https://mlsanity.com/paper/2410.07687/) (2024)
- [Neural Collapse in Cumulative Link Models for Ordinal Regression: An Analysis with Unconstrained Feature Model](https://mlsanity.com/paper/2506.05801/) (2025)
- [MaxSup: Overcoming Representation Collapse in Label Smoothing](https://mlsanity.com/paper/2502.15798/) (2025)
- [Unpacking Softmax: How Temperature Drives Representation Collapse, Compression, and Generalization](https://mlsanity.com/paper/2506.01562/) (2025)
- [Decoupling Dynamical Richness from Representation Learning: Towards Practical Measurement](https://mlsanity.com/paper/2410.04264/) (2024)
- [Neural Multivariate Regression: Qualitative Insights from the Unconstrained Feature Model](https://mlsanity.com/paper/2505.09308/) (2025)
- [Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics](https://mlsanity.com/paper/2512.04006/) (2025)
- [The Implicit Bias of Logit Regularization](https://mlsanity.com/paper/2602.12039/) (2026)
- [Neural Collapse by Design: Learning Class Prototypes on the Hypersphere](https://mlsanity.com/paper/2605.20302/) (2026)
- [The Implicit Bias of Depth: From Neural Collapse to Softmax Codes](https://mlsanity.com/paper/2605.23087/) (2026)
- [Feature Lottery? A Bifurcation Theory of Concept Emergence](https://mlsanity.com/paper/2605.24057/) (2026)
- [The role of class encoding in neural collapse](https://mlsanity.com/paper/2606.00344/) (2026)
- [Aligned but Flattened: Analyzing the Trade-off between Cultural Alignment and Diversity in LLMs](https://mlsanity.com/paper/2609.00565/) (2026)

## Abstract (arXiv)

Deep neural networks (DNNs) exhibit a surprising structure in their final layer known as neural collapse (NC), and a growing body of works has currently investigated the propagation of neural collapse to earlier layers of DNNs -- a phenomenon called deep neural collapse (DNC). However, existing theoretical results are restricted to special cases: linear models, only two layers or binary classification. In contrast, we focus on non-linear models of arbitrary depth in multi-class classification and reveal a surprising qualitative shift. As soon as we go beyond two layers or two classes, DNC stops being optimal for the deep unconstrained features model (DUFM) -- the standard theoretical framework for the analysis of collapse. The main culprit is a low-rank bias of multi-layer regularization schemes: this bias leads to optimal solutions of even lower rank than the neural collapse. We support our theoretical findings with experiments on both DUFM and real data, which show the emergence of the low-rank structure in the solution found by gradient descent.