For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and to minimize overfitting. We define a pooling layer, NervePool, for data structured as simplicial complexes, which are generalizations of graphs that include higher-dimensional simplices beyond vertices and edges; this structure allows for greater flexibility in modeling higher-order relationships. The proposed simplicial coarsening scheme is built upon partitions of vertices, which allow us to generate hierarchical representations of simplicial complexes, collapsing information in a learned fashion. NervePool builds on the learned vertex cluster assignments and extends to coarsening of higher dimensional simplices in a deterministic fashion. While in practice, the pooling operations are computed via a series of matrix operations, the topological motivation is a set-theoretic construction based on unions of stars of simplices and the nerve complex

翻译：针对图结构数据的深度学习问题，池化层对下采样、降低计算成本以及减少过拟合至关重要。我们定义了一个名为 NervePool 的池化层，适用于以单纯复形结构组织的数据——这类数据是图的泛化形式，除顶点和边外还包含更高维度的单形体；该结构为建模高阶关系提供了更强的灵活性。所提出的单纯形粗化方案基于顶点划分构建，能够生成单纯复形的层级表示，并以学习方式压缩信息。NervePool 基于学习到的顶点簇分配，并以确定性方式扩展至更高维单形体的粗化。尽管实际应用中池化操作通过一系列矩阵运算实现，其拓扑动机源于基于单形体星形并集与神经复形的集合论构造。