Parallel Transport Convolution: A New Tool for Convolutional Neural Networks on Manifolds

TL;DR

This paper introduces Parallel Transport Convolution (PTC), a novel convolution method on manifolds that preserves directionality, supports compact filters, and is robust to deformations, enabling deep learning on curved domains.

Contribution

The paper develops PTC, a new convolution operation on Riemannian manifolds that maintains key properties of Euclidean convolutions, facilitating neural networks on curved spaces.

Findings

PTC preserves directionality on manifolds.

PTC supports compactly supported filters.

PTC is robust to manifold deformations.

Abstract

Convolution has been playing a prominent role in various applications in science and engineering for many years. It is the most important operation in convolutional neural networks. There has been a recent growth of interests of research in generalizing convolutions on curved domains such as manifolds and graphs. However, existing approaches cannot preserve all the desirable properties of Euclidean convolutions, namely compactly supported filters, directionality, transferability across different manifolds. In this paper we develop a new generalization of the convolution operation, referred to as parallel transport convolution (PTC), on Riemannian manifolds and their discrete counterparts. PTC is designed based on the parallel transportation which is able to translate information along a manifold and to intrinsically preserve directionality. PTC allows for the construction of compactly supported filters and is also robust to manifold deformations. This enables us to preform wavelet-like operations and to define deep convolutional neural networks on curved domains.