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

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.

arXiv comment10 pages

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Cited by in corpus (8)

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• Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs
• Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds
• Graph Convolution with Low-rank Learnable Local Filters
• HodgeNet: Learning Spectral Geometry on Triangle Meshes
• Unsupervised Geometric Disentanglement for Surfaces via CFAN-VAE
• NPTC-net: Narrow-Band Parallel Transport Convolutional Neural Network on Point Clouds