We introduce a class of manifold neural networks (MNNs) that we call Manifold Filter-Combine Networks (MFCNs), that aims to further our understanding of MNNs, analogous to how the aggregate-combine framework helps with the understanding of graph neural networks (GNNs). This class includes a wide variety of subclasses that can be thought of as the manifold analog of various popular GNNs. We then consider a method, based on building a data-driven graph, for implementing such networks when one does not have global knowledge of the manifold, but merely has access to finitely many sample points. We provide sufficient conditions for the network to provably converge to its continuum limit as the number of sample points tends to infinity. Unlike previous work (which focused on specific graph constructions), our rate of convergence does not directly depend on the number of filters used. Moreover, it exhibits linear dependence on the depth of the network rather than the exponential dependence obtained previously. Additionally, we provide several examples of interesting subclasses of MFCNs and of the rates of convergence that are obtained under specific graph constructions.

翻译：我们引入一类称为流形滤波-组合网络（MFCNs）的流形神经网络（MNNs），旨在深化对MNNs的理解，类似于聚合-组合框架对图神经网络（GNNs）理解的促进作用。该类网络包含多种子类，可视为各类流行GNNs的流形对应物。随后，我们考虑一种基于数据驱动图构建的方法，用于在缺乏流形全局知识、仅具有有限采样点的情况下实现此类网络。我们给出了当采样点数量趋于无穷时，网络可证明收敛至其连续极限的充分条件。与先前研究（聚焦于特定图构建）不同，我们的收敛速率不直接依赖于所用滤波器的数量，且呈现对网络深度的线性依赖，而非先前获得的指数依赖。此外，我们提供了MFCNs若干有趣子类及特定图构建下所得收敛速率的实例。