  The recently discovered Neural Collapse (NC) phenomenon occurs pervasively in
today's deep net training paradigm of driving cross-entropy (CE) loss towards
zero. During NC, last-layer features collapse to their class-means, both
classifiers and class-means collapse to the same Simplex Equiangular Tight
Frame, and classifier behavior collapses to the nearest-class-mean decision
rule. Recent works demonstrated that deep nets trained with mean squared error
(MSE) loss perform comparably to those trained with CE. As a preliminary, we
empirically establish that NC emerges in such MSE-trained deep nets as well
through experiments on three canonical networks and five benchmark datasets. We
provide, in a Google Colab notebook, PyTorch code for reproducing MSE-NC and
CE-NC: at
https://colab.research.google.com/github/neuralcollapse/neuralcollapse/blob/main/neuralcollapse.ipynb.
The analytically-tractable MSE loss offers more mathematical opportunities than
the hard-to-analyze CE loss, inspiring us to leverage MSE loss towards the
theoretical investigation of NC. We develop three main contributions: (I) We
show a new decomposition of the MSE loss into (A) terms directly interpretable
through the lens of NC and which assume the last-layer classifier is exactly
the least-squares classifier; and (B) a term capturing the deviation from this
least-squares classifier. (II) We exhibit experiments on canonical datasets and
networks demonstrating that term-(B) is negligible during training. This
motivates us to introduce a new theoretical construct: the central path, where
the linear classifier stays MSE-optimal for feature activations throughout the
dynamics. (III) By studying renormalized gradient flow along the central path,
we derive exact dynamics that predict NC.
