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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 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, and we conclude that manifold optimization of weights performs well regardless of the smoothness of the gradients. Moreover, we observe a surprising yet robust behavior of highly isometric initializations --- even though such networks have a lower FIM condition number \emph{at initialization}, and therefore by analogy to convex functions should be easier to optimize, experimentally they prove to be much harder to train with stochastic gradient descent. We conjecture the FIM condition number plays a non-trivial role in the optimization. 

Understanding the nature of representation in neural networks is a goal shared by neuroscience and machine learning. It is therefore exciting that both fields converge not only on shared questions but also on similar approaches. A pressing question in these areas is understanding how the structure of the representation used by neural networks affects both their generalization, and robustness to perturbations. In this work, we investigate the latter by juxtaposing experimental results regarding the covariance spectrum of neural representations in the mouse V1 (Stringer et al) with artificial neural networks. We use adversarial robustness to probe Stringer et al's theory regarding the causal role of a 1/n covariance spectrum. We empirically investigate the benefits such a neural code confers in neural networks, and illuminate its role in multi-layer architectures. Our results show that imposing the experimentally observed structure on artificial neural networks makes them more robust to adversarial attacks. Moreover, our findings complement the existing theory relating wide neural networks to kernel methods, by showing the role of intermediate representations. 

Exploding and vanishing gradient are both major problems often faced when an artificial neural network is trained with gradient descent. Inspired by the ubiquity and robustness of nonlinear oscillations in biological neural systems, we investigate the properties of their artificial counterpart, the stable limit cycle neural networks. Using a continuous time dynamical system interpretation of neural networks and backpropagation, we show that stable limit cycle neural networks have non-exploding gradients, and at least one effective non-vanishing gradient dimension. We conjecture that limit cycles can support the learning of long temporal dependence in both biological and artificial neural networks.