  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.
