This paper studies the relationship between a graph neural network (GNN) and a manifold neural network (MNN) when the graph is constructed from a set of points sampled from the manifold, thus encoding geometric information. We consider convolutional MNNs and GNNs where the manifold and the graph convolutions are respectively defined in terms of the Laplace-Beltrami operator and the graph Laplacian. Using the appropriate kernels, we analyze both dense and moderately sparse graphs. We prove non-asymptotic error bounds showing that convolutional filters and neural networks on these graphs converge to convolutional filters and neural networks on the continuous manifold. As a byproduct of this analysis, we observe an important trade-off between the discriminability of graph filters and their ability to approximate the desired behavior of manifold filters. We then discuss how this trade-off is ameliorated in neural networks due to the frequency mixing property of nonlinearities. We further derive a transferability corollary for geometric graphs sampled from the same manifold. We validate our results numerically on a navigation control problem and a point cloud classification task.

中文翻译：

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几何图过滤器和神经网络：限制属性和可辨别性权衡

本文研究了图神经网络（GNN）和流形神经网络（MNN）之间的关系，当图是由流形采样的一组点构造而成，从而编码几何信息时。我们考虑卷积 MNN 和 GNN，其中流形和图卷积分别根据 Laplace-Beltrami 算子和图拉普拉斯算子定义。使用适当的内核，我们分析密集图和适度稀疏图。我们证明了非渐近误差界，表明这些图上的卷积滤波器和神经网络收敛到连续流形上的卷积滤波器和神经网络。作为此分析的副产品，我们观察到图过滤器的可区分性与其近似流形过滤器所需行为的能力之间的重要权衡。然后，我们讨论如何由于非线性的混频特性而在神经网络中改善这种权衡。我们进一步推导出从同一流形采样的几何图的可转移性推论。我们在导航控制问题和点云分类任务上以数值方式验证了我们的结果。