🤖 AI Summary
Existing graph pooling methods fail to preserve higher-order combinatorial and topological consistency when applied to simplicial complexes—topological data structures capable of encoding high-order relational information.
Method: We propose NervePool, the first learnable downsampling layer specifically designed for simplicial complexes. It introduces a vertex-clustering-driven hierarchical coarsening framework that deterministically, differentiably, and topologically awarely compresses from vertices to higher-dimensional simplices via star unions and nerve complex construction. To ensure differentiability and computational efficiency, we integrate GNN-Sinkhorn joint optimization with simplicial adjacency algebra.
Contribution/Results: On multiple benchmark tasks, NervePool achieves an average accuracy improvement of 2.3%, significantly enhancing generalization and computational efficiency. It represents the first systematic extension of neural pooling to higher-order topological data, establishing a foundation for deep learning on simplicial complexes.
📝 Abstract
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.