  Modern practice for training classification deepnets involves a Terminal
Phase of Training (TPT), which begins at the epoch where training error first
vanishes; During TPT, the training error stays effectively zero while training
loss is pushed towards zero. Direct measurements of TPT, for three prototypical
deepnet architectures and across seven canonical classification datasets,
expose a pervasive inductive bias we call Neural Collapse, involving four
deeply interconnected phenomena: (NC1) Cross-example within-class variability
of last-layer training activations collapses to zero, as the individual
activations themselves collapse to their class-means; (NC2) The class-means
collapse to the vertices of a Simplex Equiangular Tight Frame (ETF); (NC3) Up
to rescaling, the last-layer classifiers collapse to the class-means, or in
other words to the Simplex ETF, i.e. to a self-dual configuration; (NC4) For a
given activation, the classifier's decision collapses to simply choosing
whichever class has the closest train class-mean, i.e. the Nearest Class Center
(NCC) decision rule. The symmetric and very simple geometry induced by the TPT
confers important benefits, including better generalization performance, better
robustness, and better interpretability.
