Riemannian Curvature of Deep Neural Networks

Abstract

We analyze deep neural networks using the theory of Riemannian geometry and curvature. The objective is to gain insight into how Riemannian geometry can characterize and predict the trained behavior of neural networks. We define a method for calculating Riemann and Ricci curvature tensors, and Ricci scalar curvature values for a trained neural net, in such a way that the output classifier softmax values are related to the input transformations, through the curvature equations. We also measure these curvature tensors experimentally for different networks which are pretrained with stochastic gradient descent and offer a way of visualizing and understanding the measurements to gain insight into the effect curvature has on behavior the neural networks locally, and possibly predict their behavior for different transformations of the test data. We also analyze the effect of variation in depth of the neural networks as well as how it behaves for different choices of data set.