Manifold Filter-Combine Networks
This work provides a theoretical foundation for manifold neural networks, which could benefit researchers in machine learning and data analysis dealing with high-dimensional point cloud data, though it appears incremental as it adapts existing graph neural network concepts to manifolds.
The authors tackled the challenge of understanding manifold neural networks by introducing Manifold Filter-Combine Networks (MFCNs), a framework analogous to graph neural networks, and demonstrated its effectiveness on real-world and synthetic datasets with proven convergence to a continuum limit.
In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity, and we numerically demonstrate its effectiveness on real-world and synthetic data sets.