NervePool: A Simplicial Pooling Layer
Sarah McGuire Elizabeth Munch Matthew Hirn
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