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.

介绍了一种名为广义单纯姿态注意力神经网络（GSANs）的新型神经架构，用于处理在单纯复合体上定义的数据，通过掩码自注意力层。利用拓扑信号处理原理，设计了一系列自注意机制，能够处理在不同单纯阶数上定义的数据组件，如节点、边、三角形等。证明了 GSANs 具有置换等变性和单纯意识。在几个任务（归纳和传导）中应用到轨迹预测、缺失数据补全、图分类和单纯预测中，与其他方法相比取得了有利的效果。

广义简单注意力神经网络

Generalized Simplicial Attention Neural Networks

We study data-driven representations for three-dimensional triangle meshes,
which are one of the prevalent objects used to represent 3D geometry. Recent
works have developed models that exploit the intrinsic geometry of manifolds
and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants,
which learn from the local metric tensor via the Laplacian operator. Despite
offering excellent sample complexity and built-in invariances, intrinsic
geometry alone is invariant to isometric deformations, making it unsuitable for
many applications. To overcome this limitation, we propose several upgrades to
GNNs to leverage extrinsic differential geometry properties of
three-dimensional surfaces, increasing its modeling power.
In particular, we propose to exploit the Dirac operator, whose spectrum
detects principal curvature directions --- this is in stark contrast with the
classical Laplace operator, which directly measures mean curvature. We coin the
resulting models \emph{Surface Networks (SN)}. We prove that these models
define shape representations that are stable to deformation and to
discretization, and we demonstrate the efficiency and versatility of SNs on two
challenging tasks: temporal prediction of mesh deformations under non-linear
dynamics and generative models using a variational autoencoder framework with
encoders/decoders given by SNs.

本研究提出了一种名为 Surface Networks（SN）的模型，该模型通过利用三维曲面的外部微分几何属性，如 Dirac 算子的频谱来定义稳定的形状表示，并证明其对变形和离散化具有稳定性，在两项挑战性任务上展示了 SN 的高效性和多功能性，包括使用 SN 构建的变分自编码器进行时间预测。