Convolutional Neural Networks on Manifolds: From Graphs and Back
This work addresses the challenge of processing non-Euclidean data like point clouds for applications in 3D modeling and sensor networks, offering a novel theoretical framework that bridges discrete graphs and continuous manifolds.
The paper tackles the problem of learning from geometric data on manifolds by proposing manifold neural networks (MNNs) that generalize graph neural networks to continuous domains, and demonstrates their performance on point-cloud datasets.
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.