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The training dynamics of deep neural networks often defy expectations, even as these models form the foundation of modern machine learning. Two prominent examples are grokking, where test performance improves abruptly long after the…

Machine Learning · Computer Science 2026-01-28 Keitaro Sakamoto , Issei Sato

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

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

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

A recent study has shown a phenomenon called neural collapse in that the within-class means of features and the classifier weight vectors converge to the vertices of a simplex equiangular tight frame at the terminal phase of training for…

Computer Vision and Pattern Recognition · Computer Science 2023-01-04 Zhisheng Zhong , Jiequan Cui , Yibo Yang , Xiaoyang Wu , Xiaojuan Qi , Xiangyu Zhang , Jiaya Jia

Recent studies on Neural Collapse (NC) reveal that, under class-balanced conditions, the class feature means and classifier weights spontaneously align into a simplex equiangular tight frame (ETF). In long-tailed regimes, however, severe…

Machine Learning · Computer Science 2025-12-10 Jinping Wang , Zhiqiang Gao , Zhiwu Xie

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

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

A longstanding goal in the theory of deep learning is to characterize the conditions under which a given neural network architecture will be trainable, and if so, how well it might generalize to unseen data. In this work, we provide such a…

Machine Learning · Computer Science 2020-07-14 Lechao Xiao , Jeffrey Pennington , Samuel S. Schoenholz

This paper explores the connection between two recently identified phenomena in deep learning: plasticity loss and neural collapse. We analyze their correlation in different scenarios, revealing a significant association during the initial…

Machine Learning · Computer Science 2024-04-04 Guglielmo Bonifazi , Iason Chalas , Gian Hess , Jakub Łucki

While deep learning models and techniques have achieved great empirical success, our understanding of the source of success in many aspects remains very limited. In an attempt to bridge the gap, we investigate the decision boundary of a…

Neural and Evolutionary Computing · Computer Science 2019-01-03 Yu Li , Lizhong Ding , Xin Gao

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

When training a neural network for classification, the feature vectors of the training set are known to collapse to the vertices of a regular simplex, provided the dimension $d$ of the feature space and the number $n$ of classes satisfies…

Machine Learning · Computer Science 2026-03-24 James Alcala , Rayna Andreeva , Vladimir A. Kobzar , Dustin G. Mixon , Sanghoon Na , Shashank Sule , Yangxinyu Xie

We study how different output layer parameterizations of a deep neural network affects learning and forgetting in continual learning settings. The following three effects can cause catastrophic forgetting in the output layer: (1) weights…

Machine Learning · Computer Science 2022-08-19 Timothée Lesort , Thomas George , Irina Rish

Recent findings reveal that over-parameterized deep neural networks, trained beyond zero training-error, exhibit a distinctive structural pattern at the final layer, termed as Neural-collapse (NC). These results indicate that the final…

Machine Learning · Computer Science 2024-03-01 Tina Behnia , Christos Thrampoulidis

Why do neural networks trained with large learning rates for a longer time often lead to better generalization? In this paper, we delve into this question by examining the relation between training and testing loss in neural networks.…

Machine Learning · Computer Science 2024-01-23 Yinuo Ren , Chao Ma , Lexing Ying

Topological deep learning is a formalism that is aimed at introducing topological language to deep learning for the purpose of utilizing the minimal mathematical structures to formalize problems that arise in a generic deep learning…

Machine Learning · Computer Science 2021-02-17 Mustafa Hajij , Kyle Istvan

Deep Equilibrium Model (DEQ), which serves as a typical implicit neural network, emphasizes their memory efficiency and competitive performance compared to explicit neural networks. However, there has been relatively limited theoretical…

Machine Learning · Computer Science 2024-12-05 Haixiang Sun , Ye Shi

Modern deep learning models employ considerably more parameters than required to fit the training data. Whereas conventional statistical wisdom suggests such models should drastically overfit, in practice these models generalize remarkably…

Machine Learning · Statistics 2020-08-18 Ben Adlam , Jeffrey Pennington

This work investigates the generalization behavior of deep neural networks (DNNs), focusing on the phenomenon of "fooling examples," where DNNs confidently classify inputs that appear random or unstructured to humans. To explore this…

Machine Learning · Computer Science 2025-08-22 Yen-Lung Lai , Zhe Jin