  When training deep neural networks for classification tasks, an intriguing
empirical phenomenon has been widely observed in the last-layer classifiers and
features, where (i) the class means and the last-layer classifiers all collapse
to the vertices of a Simplex Equiangular Tight Frame (ETF) up to scaling, and
(ii) cross-example within-class variability of last-layer activations collapses
to zero. This phenomenon is called Neural Collapse (NC), which seems to take
place regardless of the choice of loss functions. In this work, we justify NC
under the mean squared error (MSE) loss, where recent empirical evidence shows
that it performs comparably or even better than the de-facto cross-entropy
loss. Under a simplified unconstrained feature model, we provide the first
global landscape analysis for vanilla nonconvex MSE loss and show that the
(only!) global minimizers are neural collapse solutions, while all other
critical points are strict saddles whose Hessian exhibit negative curvature
directions. Furthermore, we justify the usage of rescaled MSE loss by probing
the optimization landscape around the NC solutions, showing that the landscape
can be improved by tuning the rescaling hyperparameters. Finally, our
theoretical findings are experimentally verified on practical network
architectures.
