We propose new scattering networks for signals measured on simplicial complexes, which we call \emph{Multiscale Hodge Scattering Networks} (MHSNs). Our construction is based on multiscale basis dictionaries on simplicial complexes, i.e., the $\kappa$-GHWT and $\kappa$-HGLET, which we recently developed for simplices of dimension $\kappa \in \mathbb{N}$ in a given simplicial complex by generalizing the node-based Generalized Haar-Walsh Transform (GHWT) and Hierarchical Graph Laplacian Eigen Transform (HGLET). The $\kappa$-GHWT and the $\kappa$-HGLET both form redundant sets (i.e., dictionaries) of multiscale basis vectors and the corresponding expansion coefficients of a given signal. Our MHSNs use a layered structure analogous to a convolutional neural network (CNN) to cascade the moments of the modulus of the dictionary coefficients. The resulting features are invariant to reordering of the simplices (i.e., node permutation of the underlying graphs). Importantly, the use of multiscale basis dictionaries in our MHSNs admits a natural pooling operation that is akin to local pooling in CNNs, and which may be performed either locally or per-scale. These pooling operations are harder to define in both traditional scattering networks based on Morlet wavelets, and geometric scattering networks based on Diffusion Wavelets. As a result, we are able to extract a rich set of descriptive yet robust features that can be used along with very simple machine learning methods (i.e., logistic regression or support vector machines) to achieve high-accuracy classification systems with far fewer parameters to train than most modern graph neural networks. Finally, we demonstrate the usefulness of our MHSNs in three distinct types of problems: signal classification, domain (i.e., graph/simplex) classification, and molecular dynamics prediction.

翻译：我们提出了针对单纯复形上测量信号的新型散射网络，称为**多尺度霍奇散射网络（MHSNs）**。该构造基于我们在单纯复形上建立的多尺度基字典，即通过将基于节点的广义哈尔-沃尔什变换（GHWT）和分层图拉普拉斯特征变换（HGLET）推广到给定单纯复形中维度为$\kappa \in \mathbb{N}$的单纯形，近期开发的$\kappa$-GHWT和$\kappa$-HGLET。$\kappa$-GHWT和$\kappa$-HGLET均构成冗余集（即字典），包含多尺度基向量及其对应信号的展开系数。我们的MHSNs采用类似于卷积神经网络（CNN）的分层结构，级联字典系数模的矩。所得特征对单纯形的重排（即底层图的节点置换）具有不变性。重要的是，MHSNs中使用多尺度基字典允许自然的池化操作（类似CNN中的局部池化），且该操作可在局部或每尺度层面执行。传统基于莫莱小波的散射网络和基于扩散小波的几何散射网络均难以定义此类池化操作。因此，我们能够提取一组丰富且鲁棒的描述性特征，可与极简单的机器学习方法（如逻辑回归或支持向量机）结合，构建高精度分类系统，且其训练参数远少于大多数现代图神经网络。最后，我们在三类不同问题中展示了MHSNs的有效性：信号分类、域（即图/单纯形）分类以及分子动力学预测。