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The neural collapse (NC) phenomenon describes an underlying geometric symmetry for deep neural networks, where both deeply learned features and classifiers converge to a simplex equiangular tight frame. It has been shown that both…

Machine Learning · Computer Science 2023-04-18 Weiyang Liu , Longhui Yu , Adrian Weller , Bernhard Schölkopf

A phenomenon known as ''Neural Collapse (NC)'' in deep classification tasks, in which the penultimate-layer features and the final classifiers exhibit an extremely simple geometric structure, has recently attracted considerable attention,…

Machine Learning · Computer Science 2025-11-05 Chuang Ma , Tomoyuki Obuchi , Toshiyuki Tanaka

The recent work of Papyan, Han, & Donoho (2020) presented an intriguing "Neural Collapse" phenomenon, showing a structural property of interpolating classifiers in the late stage of training. This opened a rich area of exploration studying…

Machine Learning · Computer Science 2022-02-18 Like Hui , Mikhail Belkin , Preetum Nakkiran

Neural Collapse (NC) is a recently observed phenomenon in neural networks that characterises the solution space of the final classifier layer when trained until zero training loss. Specifically, NC suggests that the final classifier layer…

Machine Learning · Computer Science 2024-11-05 Evan Markou , Thalaiyasingam Ajanthan , Stephen Gould

Neural collapse provides an elegant mathematical characterization of learned last layer representations (a.k.a. features) and classifier weights in deep classification models. Such results not only provide insights but also motivate new…

Machine Learning · Computer Science 2023-10-30 Jiachen Jiang , Jinxin Zhou , Peng Wang , Qing Qu , Dustin Mixon , Chong You , Zhihui Zhu

Neural Collapse (NC) refers to the emergence of highly symmetric geometric structures in the representations of deep neural networks during the terminal phase of training. Despite its prevalence, the theoretical understanding of NC remains…

Machine Learning · Computer Science 2026-02-26 Jim Zhao , Tin Sum Cheng , Wojciech Masarczyk , Aurelien Lucchi

Out-of-distribution (OOD) detection and OOD generalization are widely studied in Deep Neural Networks (DNNs), yet their relationship remains poorly understood. We empirically show that the degree of Neural Collapse (NC) in a network layer…

Machine Learning · Computer Science 2025-09-23 Md Yousuf Harun , Jhair Gallardo , Christopher Kanan

Neural collapse (NC) describes the structured geometry that emerges in the features and weights of trained classifiers. Recent theory suggests NC can be suboptimal in deep architectures, attributing this to an explicit low-rank bias from L2…

Machine Learning · Computer Science 2026-05-25 Connall Garrod , Jonathan P. Keating , Christos Thrampoulidis

The empirical emergence of neural collapse -- a surprising symmetry in the feature representations of the training data in the penultimate layer of deep neural networks -- has spurred a line of theoretical research aimed at its…

Machine Learning · Computer Science 2025-05-22 Peter Súkeník , Christoph H. Lampert , Marco Mondelli

Neural Collapse refers to the curious phenomenon in the end of training of a neural network, where feature vectors and classification weights converge to a very simple geometrical arrangement (a simplex). While it has been observed…

Machine Learning · Computer Science 2024-11-14 Jingtong Su , Ya Shi Zhang , Nikolaos Tsilivis , Julia Kempe

Neural Collapse (NC) presents an elegant geometric structure that enables individual activations (features), class means and classifier (weights) vectors to reach \textit{optimal} inter-class separability during the terminal phase of…

Machine Learning · Computer Science 2024-08-15 Enhao Zhang , Chaohua Li , Chuanxing Geng , Songcan Chen

Batch Normalization (BN) has become an out-of-box technique to improve deep network training. However, its effectiveness is limited for micro-batch training, i.e., each GPU typically has only 1-2 images for training, which is inevitable for…

Computer Vision and Pattern Recognition · Computer Science 2020-08-11 Siyuan Qiao , Huiyu Wang , Chenxi Liu , Wei Shen , Alan Yuille

Neural collapse (NC) is a simple and symmetric phenomenon for deep neural networks (DNNs) at the terminal phase of training, where the last-layer features collapse to their class means and form a simplex equiangular tight frame aligning…

Machine Learning · Computer Science 2024-05-03 Sicong Wang , Kuo Gai , Shihua Zhang

When training deep neural networks for classification tasks, an intriguing empirical phenomenon has been widely observed in the last-layer classifiers and features, where (i) the class means and the last-layer classifiers all collapse to…

Machine Learning · Computer Science 2022-03-15 Jinxin Zhou , Xiao Li , Tianyu Ding , Chong You , Qing Qu , Zhihui Zhu

Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a "neural collapse" behavior has been observed: the variability of features…

Machine Learning · Computer Science 2023-05-30 Tom Tirer , Haoxiang Huang , Jonathan Niles-Weed

We provide the first global optimization landscape analysis of $Neural\;Collapse$ -- an intriguing empirical phenomenon that arises in the last-layer classifiers and features of neural networks during the terminal phase of training. As…

Machine Learning · Computer Science 2021-05-07 Zhihui Zhu , Tianyu Ding , Jinxin Zhou , Xiao Li , Chong You , Jeremias Sulam , Qing Qu

With the ever-increasing complexity of large-scale pre-trained models coupled with a shortage of labeled data for downstream training, transfer learning has become the primary approach in many fields, including natural language processing,…

Machine Learning · Computer Science 2024-07-22 Xiao Li , Sheng Liu , Jinxin Zhou , Xinyu Lu , Carlos Fernandez-Granda , Zhihui Zhu , Qing Qu

Weight decay is often used to ensure good generalization in the training practice of deep neural networks with batch normalization (BN-DNNs), where some convolution layers are invariant to weight rescaling due to the normalization. In this…

Machine Learning · Computer Science 2022-06-22 Ziquan Liu , Yufei Cui , Jia Wan , Yu Mao , Antoni B. Chan

When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called "neural collapse" phenomenon. More specifically, for the output features of the penultimate…

Machine Learning · Computer Science 2023-03-09 Can Yaras , Peng Wang , Zhihui Zhu , Laura Balzano , Qing Qu

Training neural networks with batch normalization and weight decay has become a common practice in recent years. In this work, we show that their combined use may result in a surprising periodic behavior of optimization dynamics: the…

Machine Learning · Computer Science 2022-01-19 Ekaterina Lobacheva , Maxim Kodryan , Nadezhda Chirkova , Andrey Malinin , Dmitry Vetrov