  Deep neural networks (DNNs) exhibit a surprising structure in their final
layer known as neural collapse (NC), and a growing body of works has currently
investigated the propagation of neural collapse to earlier layers of DNNs -- a
phenomenon called deep neural collapse (DNC). However, existing theoretical
results are restricted to special cases: linear models, only two layers or
binary classification. In contrast, we focus on non-linear models of arbitrary
depth in multi-class classification and reveal a surprising qualitative shift.
As soon as we go beyond two layers or two classes, DNC stops being optimal for
the deep unconstrained features model (DUFM) -- the standard theoretical
framework for the analysis of collapse. The main culprit is a low-rank bias of
multi-layer regularization schemes: this bias leads to optimal solutions of
even lower rank than the neural collapse. We support our theoretical findings
with experiments on both DUFM and real data, which show the emergence of the
low-rank structure in the solution found by gradient descent.
