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The modern strategy for training deep neural networks for classificationtasks includes optimizing the network's weights even after the training errorvanishes to further push the training loss toward zero. Recently, a phenomenontermed "neural collapse" (NC) has been empirically observed in this trainingprocedure. Specifically, it has been shown that the learned features (theoutput of the penultimate layer) of within-class samples converge to theirmean, and the means of different classes exhibit a certain tight framestructure, which is also aligned with the last layer's weights. Recent papershave shown that minimizers with this structure emerge when optimizing asimplified "unconstrained features model" (UFM) with a regularizedcross-entropy loss. In this paper, we further analyze and extend the UFM.First, we study the UFM for the regularized MSE loss, and show that theminimizers' features can have a more delicate structure than in thecross-entropy case. This affects also the structure of the weights. Then, weextend the UFM by adding another layer of weights as well as ReLU nonlinearityto the model and generalize our previous results. Finally, we empiricallydemonstrate the usefulness of our nonlinear extended UFM in modeling the NCphenomenon that occurs with practical networks.

Extended Unconstrained Features Model for Exploring Deep Neural Collapse