  Neural collapse (NC) is a simple and symmetric phenomenon for deep neural
networks (DNNs) at the terminal phase of training, where the last-layer
features collapse to their class means and form a simplex equiangular tight
frame aligning with the classifier vectors. However, the relationship of the
last-layer features to the data and intermediate layers during training remains
unexplored. To this end, we characterize the geometry of intermediate layers of
ResNet and propose a novel conjecture, progressive feedforward collapse (PFC),
claiming the degree of collapse increases during the forward propagation of
DNNs. We derive a transparent model for the well-trained ResNet according to
that ResNet with weight decay approximates the geodesic curve in Wasserstein
space at the terminal phase. The metrics of PFC indeed monotonically decrease
across depth on various datasets. We propose a new surrogate model, multilayer
unconstrained feature model (MUFM), connecting intermediate layers by an
optimal transport regularizer. The optimal solution of MUFM is inconsistent
with NC but is more concentrated relative to the input data. Overall, this
study extends NC to PFC to model the collapse phenomenon of intermediate layers
and its dependence on the input data, shedding light on the theoretical
understanding of ResNet in classification problems.
