  Recently mean field theory has been successfully used to analyze properties
of wide, random neural networks. It gave rise to a prescriptive theory for
initializing feed-forward neural networks with orthogonal weights, which
ensures that both the forward propagated activations and the backpropagated
gradients are near $\ell_2$ isometries and as a consequence training is orders
of magnitude faster. Despite strong empirical performance, the mechanisms by
which critical initializations confer an advantage in the optimization of deep
neural networks are poorly understood. Here we show a novel connection between
the maximum curvature of the optimization landscape (gradient smoothness) as
measured by the Fisher information matrix (FIM) and the spectral radius of the
input-output Jacobian, which partially explains why more isometric networks can
train much faster. Furthermore, given that orthogonal weights are necessary to
ensure that gradient norms are approximately preserved at initialization, we
experimentally investigate the benefits of maintaining orthogonality throughout
training, from which we conclude that manifold optimization of weights performs
well regardless of the smoothness of the gradients. Moreover, motivated by
experimental results we show that a low condition number of the FIM is not
predictive of faster learning.
