# Invariance-Preserving Localized Activation Functions for Graph Neural Networks

arXiv: https://arxiv.org/abs/1903.12575 · HTML: https://mlsanity.com/paper/1903.12575/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2019.

## Built on by (most cited first, 14 of 14)

- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Transferability Properties of Graph Neural Networks](https://mlsanity.com/paper/2112.04629/) (2021)
- [Constrained Learning with Non-Convex Losses](https://mlsanity.com/paper/2103.05134/) (2021)
- [Understanding Structural Vulnerability in Graph Convolutional Networks](https://mlsanity.com/paper/2108.06280/) (2021)
- [Learning by Transference: Training Graph Neural Networks on Growing Graphs](https://mlsanity.com/paper/2106.03693/) (2021)
- [Unsupervised Learning of Sampling Distributions for Particle Filters](https://mlsanity.com/paper/2302.01174/) (2023)
- [What Functions Can Graph Neural Networks Generate?](https://mlsanity.com/paper/2202.08833/) (2022)
- [Training Graph Neural Networks on Growing Stochastic Graphs](https://mlsanity.com/paper/2210.15567/) (2022)
- [Covariance Density Neural Networks](https://mlsanity.com/paper/2505.11139/) (2025)
- [Deceptive Path Planning via Reinforcement Learning with Graph Neural Networks](https://mlsanity.com/paper/2402.06552/) (2024)
- [Spectral Convergence of Complexon Shift Operators](https://mlsanity.com/paper/2309.07169/) (2023)
- [Enhancing Cross-Market Recommendation System with Graph Isomorphism Networks: A Novel Approach to Personalized User Experience](https://mlsanity.com/paper/2409.07850/) (2024)
- [Sampling Transferable Graph Neural Networks with Limited Graph Information](https://mlsanity.com/paper/2410.16593/) (2024)
- [Graph Semi-Supervised Learning for Point Classification on Data Manifolds](https://mlsanity.com/paper/2506.12197/) (2025)

## Abstract (arXiv)

Graph signals are signals with an irregular structure that can be described by a graph. Graph neural networks (GNNs) are information processing architectures tailored to these graph signals and made of stacked layers that compose graph convolutional filters with nonlinear activation functions. Graph convolutions endow GNNs with invariance to permutations of the graph nodes' labels. In this paper, we consider the design of trainable nonlinear activation functions that take into consideration the structure of the graph. This is accomplished by using graph median filters and graph max filters, which mimic linear graph convolutions and are shown to retain the permutation invariance of GNNs. We also discuss modifications to the backpropagation algorithm necessary to train local activation functions. The advantages of localized activation function architectures are demonstrated in four numerical experiments: source localization on synthetic graphs, authorship attribution of 19th century novels, movie recommender systems and scientific article classification. In all cases, localized activation functions are shown to improve model capacity.