High-dimensional data arises in numerous applications, and the rapidly developing field of geometric deep learning seeks to develop neural network architectures to analyze such data in non-Euclidean domains, such as graphs and manifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced a method for constructing manifold neural networks using the spectral decomposition of the Laplace Beltrami operator. Moreover, in this work, the authors provide a numerical scheme for implementing such neural networks when the manifold is unknown and one only has access to finitely many sample points. The authors show that this scheme, which relies upon building a data-driven graph, converges to the continuum limit as the number of sample points tends to infinity. Here, we build upon this result by establishing a rate of convergence that depends on the intrinsic dimension of the manifold but is independent of the ambient dimension. We also discuss how the rate of convergence depends on the depth of the network and the number of filters used in each layer.

翻译：高维数据出现在众多应用中，快速发展的几何深度学习领域旨在开发神经网络架构，用于分析图与流形等非欧几里得域中的此类数据。Z. Wang、L. Ruiz 和 A. Ribeiro 近期的工作引入了一种利用拉普拉斯-贝尔特拉米算子谱分解构造流形神经网络的方法。此外，该工作还提供了一种当流形未知且仅能获取有限采样点时实现此类神经网络的数值方案。作者证明，这一基于数据驱动图构建的方案随着采样点数量趋于无穷大，将收敛至连续极限。在此，我们基于该结果进一步确立了依赖于流形固有维度但与环境维度无关的收敛速率。我们还讨论了收敛速率如何随网络深度及每层使用的滤波器数量变化。