We introduce a large class of manifold neural networks (MNNs) which we call Manifold Filter-Combine Networks. This class includes as special cases, the MNNs considered in previous work by Wang, Ruiz, and Ribeiro, the manifold scattering transform (a wavelet-based model of neural networks), and other interesting examples not previously considered in the literature such as the manifold equivalent of Kipf and Welling's graph convolutional network. 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 MNN architectures and graph constructions), our rate of convergence does not explicitly 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.

翻译：我们提出了一类广泛的流形神经网络，称为流形滤波-组合网络。该类网络特例包括Wang、Ruiz和Ribeiro先前工作中考虑的流形神经网络、流形散射变换（一种基于小波的神经网络模型），以及文献中尚未讨论的其他有趣示例，例如Kipf和Welling图卷积网络的流形等价物。随后，我们考虑一种基于构建数据驱动图的方法，用于在缺乏流形全局知识而仅能获取有限采样点的情况下实现此类网络。我们提供了充分条件，保证当采样点数量趋于无穷时，网络能够可证明地收敛到其连续极限。与先前工作（侧重于特定流形神经网络架构和图构建）不同，我们的收敛速度不显式依赖于所用滤波器的数量。此外，该收敛速度与网络深度呈线性关系，而非先前获得的指数关系。