Spherical Convolutional Neural Networks: Stability to Perturbations in SO(3)

Spherical signals are useful mathematical models for data arising in many 3-D applications such as LIDAR images, panorama cameras, and optical scanners. Successful processing of spherical signals entails architectures capable of exploiting their inherent data structure. In particular, spherical convolutional neural networks (Spherical CNNs) have shown promising performance in shape analysis and object recognition. In this paper, we focus on analyzing the properties that Spherical CNNs exhibit as they pertain to the rotational structure present in spherical signals. More specifically, we prove that they are equivariant to rotations and stable to rotation diffeomorphisms. These two properties illustrate how Spherical CNNs exploit the rotational structure of spherical signals, thus offering good generalization and faster learning. We corroborate these properties through controlled numerical experiments.