The manifold scattering transform is a deep feature extractor for data defined on a Riemannian manifold. It is one of the first examples of extending convolutional neural network-like operators to general manifolds. The initial work on this model focused primarily on its theoretical stability and invariance properties but did not provide methods for its numerical implementation except in the case of two-dimensional surfaces with predefined meshes. In this work, we present practical schemes, based on the theory of diffusion maps, for implementing the manifold scattering transform to datasets arising in naturalistic systems, such as single cell genetics, where the data is a high-dimensional point cloud modeled as lying on a low-dimensional manifold. We show that our methods are effective for signal classification and manifold classification tasks.

翻译：流形散射变换是一种针对定义在黎曼流形上的数据的深度特征提取器，它是将卷积神经网络类算子推广到一般流形的首批实例之一。该模型的初期工作主要关注其理论稳定性和不变性性质，但除预定义网格的二维曲面情形外，未提供数值实现方法。本文基于扩散映射理论，提出了实用的计算方案，用于将流形散射变换应用于自然系统（如单细胞遗传学）中产生的数据集，其中数据为建模于低维流形上的高维点云。我们证明了该方法在信号分类与流形分类任务中的有效性。