NervePool: A Simplicial Pooling Layer

TL;DR

NervePool introduces a novel pooling layer for simplicial complexes, enabling hierarchical data reduction and modeling higher-order relationships in graph-structured data.

Contribution

It presents a new simplicial pooling method that extends graph pooling to higher-dimensional complexes with a topologically motivated coarsening scheme.

Findings

Enables hierarchical representations of simplicial complexes.

Supports coarsening of higher-dimensional simplices.

Built on a topologically motivated set-theoretic construction.

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.