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In the context of classification problems, Deep Learning (DL) approaches represent state of art. Many DL approaches are based on variations of standard multi-layer feed-forward neural networks. These are also referred to as deep networks.…

Machine Learning · Computer Science 2023-11-21 Andrea Apicella , Francesco Isgrò , Roberto Prevete

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

Neural Collapse (NC) is a well-known phenomenon of deep neural networks in the terminal phase of training (TPT). It is characterized by the collapse of features and classifier into a symmetrical structure, known as simplex equiangular tight…

Machine Learning · Computer Science 2023-10-13 Peifeng Gao , Qianqian Xu , Yibo Yang , Peisong Wen , Huiyang Shao , Zhiyong Yang , Bernard Ghanem , Qingming Huang

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 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

In some studies \citep[e.g.,][]{zhang2016understanding} of deep learning, it is observed that over-parametrized deep neural networks achieve a small testing error even when the training error is almost zero. Despite numerous works towards…

Machine Learning · Statistics 2022-02-25 Yue Xing , Qifan Song , Guang Cheng

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

The success of deep learning is usually accompanied by the growth in neural network depth. However, the traditional training method only supervises the neural network at its last layer and propagates the supervision layer-by-layer, which…

Computer Vision and Pattern Recognition · Computer Science 2022-07-13 Linfeng Zhang , Xin Chen , Junbo Zhang , Runpei Dong , Kaisheng Ma

Today, deep neural networks are widely used since they can handle a variety of complex tasks. Their generality makes them very powerful tools in modern technology. However, deep neural networks are often overparameterized. The usage of…

Machine Learning · Computer Science 2024-12-20 Zhu Liao , Nour Hezbri , Victor Quétu , Van-Tam Nguyen , Enzo Tartaglione

Supervised classification has a theoretical optimum, Neural Collapse (NC), yet neither of its two dominant paradigms reaches it in practice. Cross entropy (CE) leaves radial degrees of freedom unconstrained and converges to a degenerate…

Machine Learning · Computer Science 2026-05-22 Panagiotis Koromilas , Theodoros Giannakopoulos , Mihalis A. Nicolaou , Yannis Panagakis

Neural Collapse refers to the remarkable structural properties characterizing the geometry of class embeddings and classifier weights, found by deep nets when trained beyond zero training error. However, this characterization only holds for…

Machine Learning · Computer Science 2022-08-12 Christos Thrampoulidis , Ganesh R. Kini , Vala Vakilian , Tina Behnia

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

Neural collapse describes the geometry of activation in the final layer of a deep neural network when it is trained beyond performance plateaus. Open questions include whether neural collapse leads to better generalization and, if so, why…

Machine Learning · Computer Science 2024-06-28 Siwei Wang , Stephanie E Palmer

Modern neural network architectures for large-scale learning tasks have substantially higher model complexities, which makes understanding, visualizing and training these architectures difficult. Recent contributions to deep learning…

Machine Learning · Computer Science 2024-10-30 Jayadeva , Himanshu Pant , Mayank Sharma , Abhimanyu Dubey , Sumit Soman , Suraj Tripathi , Sai Guruju , Nihal Goalla

Deep convolutional classifiers linearly separate image classes and improve accuracy as depth increases. They progressively reduce the spatial dimension whereas the number of channels grows with depth. Spatial variability is therefore…

Machine Learning · Computer Science 2022-03-22 Florentin Guth , John Zarka , Stéphane Mallat

In this paper, we provide an overview of a common phenomenon, condensation, observed during the nonlinear training of neural networks: During the nonlinear training of neural networks, neurons in the same layer tend to condense into groups…

Machine Learning · Computer Science 2026-04-14 Zhi-Qin John Xu , Yaoyu Zhang , Zhangchen Zhou

Generalization of deep neural networks remains one of the main open problems in machine learning. Previous theoretical works focused on deriving tight bounds of model complexity, while empirical works revealed that neural networks exhibit…

Machine Learning · Computer Science 2022-01-31 James Wang , Cheng-Lin Yang

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

While cross entropy (CE) is the most commonly used loss to train deep neural networks for classification tasks, many alternative losses have been developed to obtain better empirical performance. Among them, which one is the best to use is…

Machine Learning · Computer Science 2022-10-11 Jinxin Zhou , Chong You , Xiao Li , Kangning Liu , Sheng Liu , Qing Qu , Zhihui Zhu

Deep neural networks are widely used prediction algorithms whose performance often improves as the number of weights increases, leading to over-parametrization. We consider a two-layered neural network whose first layer is frozen while the…

Machine Learning · Computer Science 2023-04-10 Roman Worschech , Bernd Rosenow