The paper defines and studies manifold (M) convolutional filters and neural networks (NNs). \emph{Manifold} filters and MNNs are defined in terms of the Laplace-Beltrami operator exponential and are such that \emph{graph} (G) filters and neural networks (NNs) are recovered as discrete approximations when the manifold is sampled. These filters admit a spectral representation which is a generalization of both the spectral representation of graph filters and the frequency response of standard convolutional filters in continuous time. The main technical contribution of the paper is to analyze the stability of manifold filters and MNNs to smooth deformations of the manifold. This analysis generalizes known stability properties of graph filters and GNNs and it is also a generalization of known stability properties of standard convolutional filters and neural networks in continuous time. The most important observation that follows from this analysis is that manifold filters, same as graph filters and standard continuous time filters, have difficulty discriminating high frequency components in the presence of deformations. This is a challenge that can be ameliorated with the use of manifold, graph, or continuous time neural networks. The most important practical consequence of this analysis is to shed light on the behavior of graph filters and GNNs in large scale graphs.

翻译：本文定义并研究了流形(M)卷积滤波器与流形神经网络(MNN)。流形滤波器与MNN基于拉普拉斯-贝尔特拉米算子的指数形式定义，当流形被采样时，图(G)滤波器与图神经网络(GNN)可作为离散近似得以恢复。这些滤波器具有谱表示，该表示既是图谱表示与连续时间标准卷积滤波器频率响应的泛化。本文的主要技术贡献在于分析流形滤波器与MNN对流形光滑形变的稳定性。该分析既推广了图滤波器与GNN的已知稳定性性质，也推广了连续时间标准卷积滤波器与神经网络的已知稳定性性质。该分析最重要的发现是：流形滤波器与图滤波器及标准连续时间滤波器相同，在形变存在时难以区分高频成分。这一挑战可通过使用流形、图或连续时间神经网络得到缓解。该分析最实际的意义在于揭示了大规模图中图滤波器与GNN的行为特性。