提出流形滤波-组合网络，实现点云上神经网络的收敛性分析。
Convergence of Manifold Filter-Combine Networks
- 用稀疏图近似流形，构建滤波-组合架构
- 证明数据点增多时算法趋于连续极限
- 为流形神经网络提供理论支撑，适合几何深度学习研究者
为更好地理解流形神经网络（MNNs），我们提出了流形滤波-组合网络（MFCNs）。该框架类比图神经网络（GNNs）中流行的聚合-组合范式，自然引出多种可解释为经典GNNs流形类比的MNN族。我们进一步提出一种在高维点云上实现MFCNs的方法，依赖于用稀疏图近似流形。我们证明了该方法具有一致性：当数据点数量趋于无穷时，其收敛到一个连续极限。
原文摘要 · Abstract (English)
In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). The 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 the manifold analog of various popular GNNs. We then propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating the manifold by a sparse graph. We 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.
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