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学习到的最后一层表示（即特征）与分类器权重提供了优雅的数学刻画。此类结果不仅揭示了深层机理，更激发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关注类别数远小于特征空间维度的情形。本文将其拓展至类别数远大于特征空间维度的场景——这在语言模型、检索系统与人脸识别应用中广泛存在。我们证明了特征与分类器呈现广义神经坍缩现象，此时最小化一对多间隔达到最大值。通过实证研究验证了实际深度神经网络中广义神经坍缩的存在性。此外，我们通过理论研究表明：在球面约束的无约束特征模型下，当特征维度与类别数满足特定技术条件时，广义神经坍缩现象可被严格证明。