The aim of this work is to introduce Generalized Simplicial Attention Neural Networks (GSANs), i.e., novel neural architectures designed to process data defined on simplicial complexes using masked self-attentional layers. Hinging on topological signal processing principles, we devise a series of self-attention schemes capable of processing data components defined at different simplicial orders, such as nodes, edges, triangles, and beyond. These schemes learn how to weight the neighborhoods of the given topological domain in a task-oriented fashion, leveraging the interplay among simplices of different orders through the Dirac operator and its Dirac decomposition. We also theoretically establish that GSANs are permutation equivariant and simplicial-aware. Finally, we illustrate how our approach compares favorably with other methods when applied to several (inductive and transductive) tasks such as trajectory prediction, missing data imputation, graph classification, and simplex prediction.

翻译：本文旨在介绍广义单纯形注意力神经网络（GSANs），即一类基于掩码自注意力层、专为处理定义在单纯复形上的数据而设计的新型神经架构。基于拓扑信号处理原理，我们设计了一系列自注意力机制，能够处理不同单纯形阶（如节点、边、三角形及更高阶）上定义的数据成分。这些机制通过狄拉克算子及其狄拉克分解，以任务导向的方式学习如何对给定拓扑域的邻域进行加权，从而利用不同阶单纯形之间的相互作用。我们还在理论上证明了GSANs具有置换等变性和单纯形感知性。最后，我们通过轨迹预测、缺失数据填补、图分类及单纯形预测等多项（归纳式和直推式）任务，展示了该方法相较于其他方法的优越性能。