Title: Fisher-Rao Metric, Geometry, and Complexity of Neural Networks,
Abstract: We study the relationship between geometry and capacity measures for deep
neural networks from an invariance viewpoint. We introduce a new notion of
capacity --- the Fisher-Rao norm --- that possesses desirable invariance
properties and is motivated by Information Geometry. We discover an analytical
characterization of the new capacity measure, through which we establish
norm-comparison inequalities and further show that the new measure serves as an
umbrella for several existing norm-based complexity measures. We discuss upper
bounds on the generalization error induced by the proposed measure. Extensive
numerical experiments on CIFAR-10 support our theoretical findings. Our
theoretical analysis rests on a key structural lemma about partial derivatives
of multi-layer rectifier networks.