# The Prevalence of Neural Collapse in Neural Multivariate Regression

arXiv: https://arxiv.org/abs/2409.04180 · HTML: https://mlsanity.com/paper/2409.04180/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 39 references that are in the map.
- First posted on arXiv 6 Sep 2024.

## Builds on (most cited first, 30 of 32)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [Revisiting Unreasonable Effectiveness of Data in Deep Learning Era](https://mlsanity.com/paper/1707.02968/) (2017)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [End-to-end Driving via Conditional Imitation Learning](https://mlsanity.com/paper/1710.02410/) (2017)
- [Age Progression/Regression by Conditional Adversarial Autoencoder](https://mlsanity.com/paper/1702.08423/) (2017)
- [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)
- [CORL: Research-oriented Deep Offline Reinforcement Learning Library](https://mlsanity.com/paper/2210.07105/) (2022)
- [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)
- [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)
- [Revealing the Structure of Deep Neural Networks via Convex Duality](https://mlsanity.com/paper/2002.09773/) (2020)
- [On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers](https://mlsanity.com/paper/2012.05420/) (2020)
- [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)
- [A Law of Data Separation in Deep Learning](https://mlsanity.com/paper/2210.17020/) (2022)
- [Perturbation Analysis of Neural Collapse](https://mlsanity.com/paper/2210.16658/) (2022)
- [Jack of All Trades, Master of Some, a Multi-Purpose Transformer Agent](https://mlsanity.com/paper/2402.09844/) (2024)
- [Learning Equi-angular Representations for Online Continual Learning](https://mlsanity.com/paper/2404.01628/) (2024)

## Built on by (most cited first, 21 of 21)

- [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)
- [Parameter Symmetry Potentially Unifies Deep Learning Theory](https://mlsanity.com/paper/2502.05300/) (2025)
- [Neural Collapse is Globally Optimal in Deep Regularized ResNets and Transformers](https://mlsanity.com/paper/2505.15239/) (2025)
- [Neural Collapse in Cumulative Link Models for Ordinal Regression: An Analysis with Unconstrained Feature Model](https://mlsanity.com/paper/2506.05801/) (2025)
- [Uncovering the Latent Potential of Deep Intermediate Representations](https://mlsanity.com/paper/2605.23033/) (2026)
- [Neural Multivariate Regression: Qualitative Insights from the Unconstrained Feature Model](https://mlsanity.com/paper/2505.09308/) (2025)
- [RetroMotion: Retrocausal Motion Forecasting Models are Instructable](https://mlsanity.com/paper/2505.20414/) (2025)
- [Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics](https://mlsanity.com/paper/2512.04006/) (2025)
- [KO: Kinetics-inspired Neural Optimizer with PDE Simulation Approaches](https://mlsanity.com/paper/2505.14777/) (2025)
- [Geometric Analysis of Neural Regression Collapse via Intrinsic Dimension](https://mlsanity.com/paper/2510.01105/) (2025)
- [Deep Neural Regression Collapse](https://mlsanity.com/paper/2603.23805/) (2026)
- [Sharper Convergence Rates for Nonconvex Optimisation via Reduction Mappings](https://mlsanity.com/paper/2506.08428/) (2025)
- [Information Theoretic Perspective on Representation Learning](https://mlsanity.com/paper/2601.11334/) (2026)
- [Emergent Low-Rank Training Dynamics in MLPs with Smooth Activations](https://mlsanity.com/paper/2602.06208/) (2026)
- [DrawMotion: Generating 3D Human Motions by Freehand Drawing](https://mlsanity.com/paper/2605.20955/) (2026)
- [The Implicit Bias of Depth: From Neural Collapse to Softmax Codes](https://mlsanity.com/paper/2605.23087/) (2026)
- [The role of class encoding in neural collapse](https://mlsanity.com/paper/2606.00344/) (2026)
- [Contrastive-Collapsed Loss for Flexible and Geometrically Optimal Embeddings and Faster Convergence](https://mlsanity.com/paper/2607.12916/) (2026)
- [One Sequence, Many Decodings: CAGenMol-2 Recasts Drug Design as Masked Molecular Inference](https://mlsanity.com/paper/2609.34301/) (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)

## Abstract (arXiv)

Recently it has been observed that neural networks exhibit Neural Collapse (NC) during the final stage of training for the classification problem. We empirically show that multivariate regression, as employed in imitation learning and other applications, exhibits Neural Regression Collapse (NRC), a new form of neural collapse: (NRC1) The last-layer feature vectors collapse to the subspace spanned by the n principal components of the feature vectors, where n is the dimension of the targets (for univariate regression, n=1); (NRC2) The last-layer feature vectors also collapse to the subspace spanned by the last-layer weight vectors; (NRC3) The Gram matrix for the weight vectors converges to a specific functional form that depends on the covariance matrix of the targets. After empirically establishing the prevalence of (NRC1)-(NRC3) for a variety of datasets and network architectures, we provide an explanation of these phenomena by modeling the regression task in the context of the Unconstrained Feature Model (UFM), in which the last layer feature vectors are treated as free variables when minimizing the loss function. We show that when the regularization parameters in the UFM model are strictly positive, then (NRC1)-(NRC3) also emerge as solutions in the UFM optimization problem. We also show that if the regularization parameters are equal to zero, then there is no collapse. To our knowledge, this is the first empirical and theoretical study of neural collapse in the context of regression. This extension is significant not only because it broadens the applicability of neural collapse to a new category of problems but also because it suggests that the phenomena of neural collapse could be a universal behavior in deep learning.