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.

从不变性观点研究深度神经网络的几何和容量度量之间的关系，引入了具有期望不变性的 Fisher-Rao 范数作为新的容量概念，并发现了其分析特征和规范比较不等式，证明了其作为多种基于范数的复杂度度量的伞兵角色，讨论了引入新的度量方式对泛化误差的影响，使用 CIFAR-10 数据集的大量数值实验支持了理论分析的发现，研究的分析基于多层整流器网络局部导数的关键结构引理。