  The implicit bias induced by the training of neural networks has become a
topic of rigorous study. In the limit of gradient flow and gradient descent
with appropriate step size, it has been shown that when one trains a deep
linear network with logistic or exponential loss on linearly separable data,
the weights converge to rank-1 matrices. In this paper, we extend this
theoretical result to the last few linear layers of the much wider class of
nonlinear ReLU-activated feedforward networks containing fully-connected layers
and skip connections. Similar to the linear case, the proof relies on specific
local training invariances, sometimes referred to as alignment, which we show
to hold for submatrices where neurons are stably-activated in all training
examples, and it reflects empirical results in the literature. We also show
this is not true in general for the full matrix of ReLU fully-connected layers.
Our proof relies on a specific decomposition of the network into a multilinear
function and another ReLU network whose weights are constant under a certain
parameter directional convergence.
