NervePool: A Simplicial Pooling Layer
This provides a method for deep learning on simplicial complexes, addressing a domain-specific need in topological data analysis and graph-based models.
The paper tackles the problem of downsampling data on simplicial complexes, which generalize graphs to include higher-dimensional relationships, by introducing NervePool, a pooling layer that coarsens these structures hierarchically based on learned vertex clusters.
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.