Beyond Unconstrained Features: Neural Collapse for Shallow Neural Networks with General Data

Neural collapse (NC) is a phenomenon that emerges at the terminal phase of the training (TPT) of deep neural networks (DNNs). The features of the data in the same class collapse to their respective sample means and the sample means exhibit a simplex equiangular tight frame (ETF). In the past few years, there has been a surge of works that focus on explaining why the NC occurs and how it affects generalization. Since the DNNs are notoriously difficult to analyze, most works mainly focus on the unconstrained feature model (UFM). While the UFM explains the NC to some extent, it fails to provide a complete picture of how the network architecture and the dataset affect NC. In this work, we focus on shallow ReLU neural networks and try to understand how the width, depth, data dimension, and statistical property of the training dataset influence the neural collapse. We provide a complete characterization of when the NC occurs for two or three-layer neural networks. For two-layer ReLU neural networks, a sufficient condition on when the global minimizer of the regularized empirical risk function exhibits the NC configuration depends on the data dimension, sample size, and the signal-to-noise ratio in the data instead of the network width. For three-layer neural networks, we show that the NC occurs as long as the first layer is sufficiently wide. Regarding the connection between NC and generalization, we show the generalization heavily depends on the SNR (signal-to-noise ratio) in the data: even if the NC occurs, the generalization can still be bad provided that the SNR in the data is too low. Our results significantly extend the state-of-the-art theoretical analysis of the N C under the UFM by characterizing the emergence of the N C under shallow nonlinear networks and showing how it depends on data properties and network architecture.

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.