  Numerous researchers recently applied empirical spectral analysis to the
study of modern deep learning classifiers. We identify and discuss an important
formal class/cross-class structure and show how it lies at the origin of the
many visually striking features observed in deepnet spectra, some of which were
reported in recent articles, others are unveiled here for the first time. These
include spectral outliers, "spikes", and small but distinct continuous
distributions, "bumps", often seen beyond the edge of a "main bulk".
  The significance of the cross-class structure is illustrated in three ways:
(i) we prove the ratio of outliers to bulk in the spectrum of the Fisher
information matrix is predictive of misclassification, in the context of
multinomial logistic regression; (ii) we demonstrate how, gradually with depth,
a network is able to separate class-distinctive information from class
variability, all while orthogonalizing the class-distinctive information; and
(iii) we propose a correction to KFAC, a well-known second-order optimization
algorithm for training deepnets.
