Neural Collapse for Unconstrained Feature Model under Cross-entropy Loss with Imbalanced Data

Recent years have witnessed the huge success of deep neural networks (DNNs) in various tasks of computer vision and text processing. Interestingly, these DNNs with massive number of parameters share similar structural properties on their feature representation and last-layer classifier at terminal phase of training (TPT). Specifically, if the training data are balanced (each class shares the same number of samples), it is observed that the feature vectors of samples from the same class converge to their corresponding in-class mean features and their pairwise angles are the same. This fascinating phenomenon is known as Neural Collapse (N C), first termed by Papyan, Han, and Donoho in 2019. Many recent works manage to theoretically explain this phenomenon by adopting so-called unconstrained feature model (UFM). In this paper, we study the extension of N C phenomenon to the imbalanced data under cross-entropy loss function in the context of unconstrained feature model. Our contribution is multi-fold compared with the state-of-the-art results: (a) we show that the feature vectors exhibit collapse phenomenon, i.e., the features within the same class collapse to the same mean vector; (b) the mean feature vectors no longer form an equiangular tight frame. Instead, their pairwise angles depend on the sample size; (c) we also precisely characterize the sharp threshold on which the minority collapse (the feature vectors of the minority groups collapse to one single vector) will take place; (d) finally, we argue that the effect of the imbalance in datasize diminishes as the sample size grows. Our results provide a complete picture of the N C under the cross-entropy loss for the imbalanced data. Numerical experiments confirm our theoretical analysis.