  In this paper, we introduce the \textit{Layer-Peeled Model}, a nonconvex yet
analytically tractable optimization program, in a quest to better understand
deep neural networks that are trained for a sufficiently long time. As the name
suggests, this new model is derived by isolating the topmost layer from the
remainder of the neural network, followed by imposing certain constraints
separately on the two parts of the network. We demonstrate that the
Layer-Peeled Model, albeit simple, inherits many characteristics of
well-trained neural networks, thereby offering an effective tool for explaining
and predicting common empirical patterns of deep learning training. First, when
working on class-balanced datasets, we prove that any solution to this model
forms a simplex equiangular tight frame, which in part explains the recently
discovered phenomenon of neural collapse \cite{papyan2020prevalence}. More
importantly, when moving to the imbalanced case, our analysis of the
Layer-Peeled Model reveals a hitherto unknown phenomenon that we term
\textit{Minority Collapse}, which fundamentally limits the performance of deep
learning models on the minority classes. In addition, we use the Layer-Peeled
Model to gain insights into how to mitigate Minority Collapse. Interestingly,
this phenomenon is first predicted by the Layer-Peeled Model before being
confirmed by our computational experiments.
