  Modern deep neural networks for classification usually jointly learn a
backbone for representation and a linear classifier to output the logit of each
class. A recent study has shown a phenomenon called neural collapse that the
within-class means of features and the classifier vectors converge to the
vertices of a simplex equiangular tight frame (ETF) at the terminal phase of
training on a balanced dataset. Since the ETF geometric structure maximally
separates the pair-wise angles of all classes in the classifier, it is natural
to raise the question, why do we spend an effort to learn a classifier when we
know its optimal geometric structure? In this paper, we study the potential of
learning a neural network for classification with the classifier randomly
initialized as an ETF and fixed during training. Our analytical work based on
the layer-peeled model indicates that the feature learning with a fixed ETF
classifier naturally leads to the neural collapse state even when the dataset
is imbalanced among classes. We further show that in this case the cross
entropy (CE) loss is not necessary and can be replaced by a simple squared loss
that shares the same global optimality but enjoys a better convergence
property. Our experimental results show that our method is able to bring
significant improvements with faster convergence on multiple imbalanced
datasets.
