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.
The notebook complex_examples.ipynb walks through a series of examples, using NervePool to pool various simplicial compelxes, for different choices of input vertex cover.
Each simplicial complex is an object defined by complex.py, which also defines the collection of functions that perform the pooling on a complex.
The content of this repository is released under the terms of the MIT license. S. McGuire, E. Munch, and M. Hirn. NervePool: A Simplicial Pooling Layer, 2023.
@article{mcguire2023nervepool,
title = {NervePool: A Simplicial Pooling Layer},
author = {McGuire, Sarah and Munch, Elizabeth and Hirn, Matthew},
year = {2023},
archiveprefix = {arXiv},
eprint = {2305.06315},
url = {https://arxiv.org/abs/2305.06315},
}