Unifying Low Dimensional Observations in Deep Learning Through the Deep
  Linear Unconstrained Feature Model
Connall Garrod Jonathan P. Keating
Modern deep neural networks have achieved high performance across various
tasks. Recently, researchers have noted occurrences of low-dimensional
structure in the weights, Hessian's, gradients, and feature vectors of these
networks, spanning different datasets and architectures when trained to
convergence. In this analysis, we theoretically demonstrate these observations
arising, and show how they can be unified within a generalized unconstrained
feature model that can be considered analytically. Specifically, we consider a
previously described structure called Neural Collapse, and its multi-layer
counterpart, Deep Neural Collapse, which emerges when the network approaches
global optima. This phenomenon explains the other observed low-dimensional
behaviours on a layer-wise level, such as the bulk and outlier structure seen
in Hessian spectra, and the alignment of gradient descent with the outlier
eigenspace of the Hessian. Empirical results in both the deep linear
unconstrained feature model and its non-linear equivalent support these
predicted observations.