  Spherical convolutional neural networks (Spherical CNNs) learn nonlinear
representations from 3D data by exploiting the data structure and have shown
promising performance in shape analysis, object classification, and planning
among others. This paper investigates the properties that Spherical CNNs
exhibit as they pertain to the rotational structure inherent in spherical
signals. We build upon the rotation equivariance of spherical convolutions to
show that Spherical CNNs are stable to general structure perturbations. In
particular, we model arbitrary structure perturbations as diffeomorphism
perturbations, and define the rotation distance that measures how far from
rotations these perturbations are. We prove that the output change of a
Spherical CNN induced by the diffeomorphism perturbation is bounded
proportionally by the perturbation size under the rotation distance. This
stability property coupled with the rotation equivariance provide theoretical
guarantees that underpin the practical observations that Spherical CNNs exploit
the rotational structure, maintain performance under structure perturbations
that are close to rotations, and offer good generalization and faster learning.
