We study the discrete-to-continuum consistency of the training of shallow graph convolutional neural networks (GCNNs) on proximity graphs of sampled point clouds under a manifold assumption. Graph convolution is defined spectrally via the graph Laplacian, whose low-frequency spectrum approximates that of the Laplace-Beltrami operator of the underlying smooth manifold, and shallow GCNNs of possibly infinite width are linear functionals on the space of measures on the parameter space. From this functional-analytic perspective, graph signals are seen as spatial discretizations of functions on the manifold, which leads to a natural notion of training data consistent across graph resolutions. To enable convergence results, the continuum parameter space is chosen as a weakly compact product of unit balls, with Sobolev regularity imposed on the output weight and bias, but not on the convolutional parameter. The corresponding discrete parameter spaces inherit the corresponding spectral decay, and are additionally restricted by a frequency cutoff adapted to the informative spectral window of the graph Laplacians. Under these assumptions, we prove $Γ$-convergence of regularized empirical risk minimization functionals and corresponding convergence of their global minimizers, in the sense of weak convergence of the parameter measures and uniform convergence of the functions over compact sets. This provides a formalization of mesh and sample independence for the training of such networks.

翻译：我们在流形假设下，研究了浅层图卷积神经网络（GCNN）在采样点云邻近图上的训练过程的离散-连续一致性。图卷积通过图拉普拉斯算子谱定义，其低频谱近似于底层光滑流形的拉普拉斯-贝尔特拉米算子谱，而可能具有无限宽度的浅层GCNN是参数空间上测度空间的线性泛函。从这一泛函分析视角来看，图信号被视为流形上函数的空间离散化，这自然引出了一个在图分辨率间保持一致性的训练数据概念。为了获得收敛性结果，连续参数空间被选为单位球的弱紧乘积，并在输出权重和偏置上施加Sobolev正则性，但不对卷积参数施加。相应的离散参数空间继承了对应的谱衰减特性，并额外受到一个频率截断的限制，该截断适应于图拉普拉斯算子的信息谱窗。在这些假设下，我们证明了正则化经验风险最小化泛函的$Γ$-收敛性，以及其全局极小值点的相应收敛性，其收敛意义在于参数测度的弱收敛和在紧集上函数的一致收敛。这为此类网络的训练提供了网格和样本独立性的形式化表述。