# Graphon Pooling for Reducing Dimensionality of Signals and Convolutional Operators on Graphs

arXiv: https://arxiv.org/abs/2212.08171 · HTML: https://mlsanity.com/paper/2212.08171/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 15 Dec 2022.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015)
- [Graphon Signal Processing](https://mlsanity.com/paper/2003.05030/) (2020)
- [Learning Graphons via Structured Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2012.05644/) (2020)
- [Algebraic Neural Networks: Stability to Deformations](https://mlsanity.com/paper/2009.01433/) (2020)
- [Graphon Filters: Graph Signal Processing in the Limit](https://mlsanity.com/paper/2003.02099/) (2020)
- [Graphon Pooling in Graph Neural Networks](https://mlsanity.com/paper/2003.01795/) (2020)
- [Stability of Algebraic Neural Networks to Small Perturbations](https://mlsanity.com/paper/2010.11544/) (2020)
- [Sampling and Recovery of Graph Signals](https://mlsanity.com/paper/1712.09310/) (2017)
- [Blue-Noise Sampling on Graphs](https://mlsanity.com/paper/1811.12542/) (2018)

## Built on by (most cited first, 8 of 8)

- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Approximately Equivariant Graph Networks](https://mlsanity.com/paper/2308.10436/) (2023)
- [Sampling and Uniqueness Sets in Graphon Signal Processing](https://mlsanity.com/paper/2401.06279/) (2024)
- [Generalization of Graph Neural Networks is Robust to Model Mismatch](https://mlsanity.com/paper/2408.13878/) (2024)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)
- [RKHS Representation of Algebraic Convolutional Filters with Integral Operators](https://mlsanity.com/paper/2602.19094/) (2026)
- [Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations](https://mlsanity.com/paper/2310.03879/) (2023)
- [Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions](https://mlsanity.com/paper/2609.36106/) (2026)

## Abstract (arXiv)

In this paper we propose a pooling approach for convolutional information processing on graphs relying on the theory of graphons and limits of dense graph sequences. We present three methods that exploit the induced graphon representation of graphs and graph signals on partitions of [0, 1]2 in the graphon space. As a result we derive low dimensional representations of the convolutional operators, while a dimensionality reduction of the signals is achieved by simple local interpolation of functions in L2([0, 1]). We prove that those low dimensional representations constitute a convergent sequence of graphs and graph signals, respectively. The methods proposed and the theoretical guarantees that we provide show that the reduced graphs and signals inherit spectral-structural properties of the original quantities. We evaluate our approach with a set of numerical experiments performed on graph neural networks (GNNs) that rely on graphon pooling. We observe that graphon pooling performs significantly better than other approaches proposed in the literature when dimensionality reduction ratios between layers are large. We also observe that when graphon pooling is used we have, in general, less overfitting and lower computational cost.