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

TL;DR

This paper demonstrates that Spherical CNNs are theoretically stable to general structure perturbations modeled as diffeomorphisms, providing guarantees for their robustness and generalization in 3D shape analysis tasks.

Contribution

The paper extends the understanding of Spherical CNNs by proving their stability to arbitrary structure perturbations using a novel rotation distance measure.

Findings

Spherical CNNs are stable to diffeomorphism perturbations proportional to the perturbation size.

Theoretical guarantees support empirical observations of robustness and generalization.

Rotation equivariance combined with stability ensures reliable performance under perturbations.

Abstract

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.