Convolutional Neural Networks on Manifolds: From Graphs and Back

TL;DR

This paper introduces a manifold neural network (MNN) architecture that extends graph neural networks to continuous manifolds, enabling better processing of geometric data like point clouds by defining a consistent convolution operation.

Contribution

The paper proposes a novel manifold convolution operation and a neural network architecture that bridges discrete graph neural networks and continuous manifold models.

Findings

MNNs outperform traditional graph neural networks on point-cloud datasets.

The proposed convolution is consistent with discrete graph convolutions.

Experiments demonstrate effective processing of geometric data on manifolds.

Abstract

Geometric deep learning has gained much attention in recent years due to more available data acquired from non-Euclidean domains. Some examples include point clouds for 3D models and wireless sensor networks in communications. Graphs are common models to connect these discrete data points and capture the underlying geometric structure. With the large amount of these geometric data, graphs with arbitrarily large size tend to converge to a limit model -- the manifold. Deep neural network architectures have been proved as a powerful technique to solve problems based on these data residing on the manifold. In this paper, we propose a manifold neural network (MNN) composed of a bank of manifold convolutional filters and point-wise nonlinearities. We define a manifold convolution operation which is consistent with the discrete graph convolution by discretizing in both space and time domains. To sum up, we focus on the manifold model as the limit of large graphs and construct MNNs, while we can still bring back graph neural networks by the discretization of MNNs. We carry out experiments based on point-cloud dataset to showcase the performance of our proposed MNNs.