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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

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

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

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

Neural Collapse is a phenomenon where the last-layer representations of a well-trained neural network converge to a highly structured geometry. In this paper, we focus on its first (and most basic) property, known as NC1: the within-class…

Machine Learning · Computer Science 2025-02-05 Diyuan Wu , Marco Mondelli

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural…

Machine Learning · Computer Science 2023-06-21 Hien Dang , Tho Tran , Stanley Osher , Hung Tran-The , Nhat Ho , Tan Nguyen

The current paradigm of training deep neural networks for classification tasks includes minimizing the empirical risk that pushes the training loss value towards zero, even after the training error has been vanished. In this terminal phase…

Machine Learning · Computer Science 2024-06-07 Hien Dang , Tho Tran , Tan Nguyen , Nhat Ho

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…

Machine Learning · Computer Science 2025-09-30 George Andriopoulos , Zixuan Dong , Li Guo , Zifan Zhao , Keith Ross

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 is a highly symmetric geometric pattern of neural networks that emerges during the terminal phase of training, with profound implications on the generalization performance and robustness of the trained networks. To…

Machine Learning · Computer Science 2022-04-26 Wenlong Ji , Yiping Lu , Yiliang Zhang , Zhun Deng , Weijie J. Su

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 is a phenomenon that helps identify sparse and low rank structures in deep classifiers. Recent work has extended the definition of neural collapse to regression problems, albeit only measuring the phenomenon at the last…

Machine Learning · Computer Science 2026-03-26 Akshay Rangamani , Altay Unal

The modern strategy for training deep neural networks for classification tasks includes optimizing the network's weights even after the training error vanishes to further push the training loss toward zero. Recently, a phenomenon termed…

Machine Learning · Computer Science 2022-10-13 Tom Tirer , Joan Bruna

Deep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a highly symmetric geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical…

Machine Learning · Computer Science 2024-10-08 Arthur Jacot , Peter Súkeník , Zihan Wang , Marco Mondelli

Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on…

Machine Learning · Computer Science 2023-10-27 Vignesh Kothapalli , Tom Tirer , Joan Bruna

Neural collapse, a newly identified characteristic, describes a property of solutions during model training. In this paper, we explore neural collapse in the context of imbalanced data. We consider the $L$-extended unconstrained feature…

Machine Learning · Computer Science 2024-11-27 Haixia Liu

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 (NC) refers to the surprising structure of the last layer of deep neural networks in the terminal phase of gradient descent training. Recently, an increasing amount of experimental evidence has pointed to the propagation of…

Machine Learning · Computer Science 2023-05-23 Peter Súkeník , Marco Mondelli , Christoph Lampert

Recent studies have noted an intriguing phenomenon termed Neural Collapse, that is, when the neural networks establish the right correlation between feature spaces and the training targets, their last-layer features, together with the…

Computer Vision and Pattern Recognition · Computer Science 2025-09-22 Yining Wang , Junjie Sun , Chenyue Wang , Mi Zhang , Min Yang

The notion of neural collapse refers to several emergent phenomena that have been empirically observed across various canonical classification problems. During the terminal phase of training a deep neural network, the feature embedding of…

Machine Learning · Computer Science 2023-04-05 Duc Anh Nguyen , Ron Levie , Julian Lienen , Gitta Kutyniok , Eyke Hüllermeier
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