# NervePool: A Simplicial Pooling Layer

arXiv: https://arxiv.org/abs/2305.06315 · HTML: https://mlsanity.com/paper/2305.06315/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 36 references that are in the map.
- First posted on arXiv 10 May 2023.

## Builds on (most cited first, 23 of 23)

- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [Understanding Attention and Generalization in Graph Neural Networks](https://mlsanity.com/paper/1905.02850/) (2019)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Towards Sparse Hierarchical Graph Classifiers](https://mlsanity.com/paper/1811.01287/) (2018)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Understanding Pooling in Graph Neural Networks](https://mlsanity.com/paper/2110.05292/) (2021)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (2021)
- [Signal Processing on Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Deep Graph Mapper: Seeing Graphs through the Neural Lens](https://mlsanity.com/paper/2002.03864/) (2020)
- [Towards Interpretable Sparse Graph Representation Learning with Laplacian Pooling](https://mlsanity.com/paper/1905.11577/) (2019)
- [Dist2Cycle: A Simplicial Neural Network for Homology Localization](https://mlsanity.com/paper/2110.15182/) (2021)
- [A Non-Negative Factorization approach to node pooling in Graph Convolutional Neural Networks](https://mlsanity.com/paper/1909.03287/) (2019)
- [Clique pooling for graph classification](https://mlsanity.com/paper/1904.00374/) (2019)
- [Pooling Strategies for Simplicial Convolutional Networks](https://mlsanity.com/paper/2210.05490/) (2022)

## Abstract (arXiv)

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.