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.

我们介绍了一类大型流形神经网络（MNNs），称为 Manifold Filter-Combine Networks。我们研究了此类神经网络的实现方法，并提供收敛条件以及收敛速度的改进。