Stability of Spherical Convolutional Neural Networks to Rotation Diffeomorphisms
Zhan Gao, Fernando Gama, Alejandro Ribeiro · 2021 29th European Signal Processing Conference (EUSIPCO) · 2021
Spherical convolutional neural networks (Spherical CNNs) learn nonlinear representations from spherical signals. These are mathematical models for data arising in 3-D objects and have found applications in computer vision, light detection and ranging (LIDAR), and planning among others. The Spherical CNN comprises a cascade of layers, each with a series of spherical convolutions (spherical filters) followed by a pointwise nonlinearity. This paper investigates the impact that structure perturbations in spherical signals have on Spherical CNN outputs. We consider general perturbations as rotation diffeomorphisms in the spherical surface, and show that Spherical CNNs with Lipschitz filters are stable to such perturbations. In particular, we establish that the output difference of Spherical CNN induced by the diffeomorphism perturbation is bounded by the perturbation size. This result also shows the role of the nonlinearity and the architecture width and depth, and indicates how Spherical CNNs exploit the rotational structure of spherical signals to gain superior performance. We corroborate theoretical findings in 3-D object classification, and observe stable performance to rotation diffeomorphisms.