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 techniques for improving practical deep models. However, most of the existing empirical and theoretical studies in neural collapse focus on the case that the number of classes is small relative to the dimension of the feature space. This paper extends neural collapse to cases where the number of classes are much larger than the dimension of feature space, which broadly occur for language models, retrieval systems, and face recognition applications. We show that the features and classifier exhibit a generalized neural collapse phenomenon, where the minimum one-vs-rest margins is maximized.We provide empirical study to verify the occurrence of generalized neural collapse in practical deep neural networks. Moreover, we provide theoretical study to show that the generalized neural collapse provably occurs under unconstrained feature model with spherical constraint, under certain technical conditions on feature dimension and number of classes.

翻译：神经坍缩为深度分类模型中最后一层隐表示（即特征）与分类器权重提供了优雅的数学刻画。这类结果不仅提供了深刻理论洞见，还催生了改进实际深度模型的新技术。然而，现有神经坍缩的实证与理论研究大多聚焦于类别数量远小于特征空间维度的情形。本文将其扩展至类别数量远大于特征空间维度的情况——此类情形广泛存在于语言模型、检索系统及人脸识别应用中。我们证明，当满足最小化一对多间隔最大化条件时，特征与分类器会呈现广义神经坍缩现象。通过实证研究，我们验证了实际深度神经网络中广义神经坍缩的发生。此外，理论分析表明，在满足特征维度与类别数量的特定技术条件下，基于球形约束的无约束特征模型中必然会出现广义神经坍缩。