# On the Transferability of Spectral Graph Filters

arXiv: https://arxiv.org/abs/1901.10524 · HTML: https://mlsanity.com/paper/1901.10524/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2019.

## Built on by (most cited first, 28 of 28)

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- [Generalization Analysis of Message Passing Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2202.00645/) (2022)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [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)
- [Non-IID Transfer Learning on Graphs](https://mlsanity.com/paper/2212.08174/) (2022)
- [Towards Graph Foundation Models: Learning Generalities Across Graphs via Task-Trees](https://mlsanity.com/paper/2412.16441/) (2024)
- [Interpretable Stability Bounds for Spectral Graph Filters](https://mlsanity.com/paper/2102.09587/) (2021)
- [EvenNet: Ignoring Odd-Hop Neighbors Improves Robustness of Graph Neural Networks](https://mlsanity.com/paper/2205.13892/) (2022)
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- [Stability of Aggregation Graph Neural Networks](https://mlsanity.com/paper/2207.03678/) (2022)
- [On the Stability of Low Pass Graph Filter With a Large Number of Edge Rewires](https://mlsanity.com/paper/2110.07234/) (2021)
- [Stability of Graph Convolutional Neural Networks through the lens of small perturbation analysis](https://mlsanity.com/paper/2312.12934/) (2023)
- [Covariance Density Neural Networks](https://mlsanity.com/paper/2505.11139/) (2025)
- [Framelet Message Passing](https://mlsanity.com/paper/2302.14806/) (2023)
- [Simplified PCNet with Robustness](https://mlsanity.com/paper/2403.03676/) (2024)
- [Stable and Transferable Hyper-Graph Neural Networks](https://mlsanity.com/paper/2211.06513/) (2022)
- [Limitless stability for Graph Convolutional Networks](https://mlsanity.com/paper/2301.11443/) (2023)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)
- [Graph Convolutional Neural Networks Sensitivity under Probabilistic Error Model](https://mlsanity.com/paper/2203.07831/) (2022)
- [Graph Neural Networks Are Not Continuous Across Graph Resolutions](https://mlsanity.com/paper/2605.31315/) (2026)
- [Generalized Graphon Process: Convergence of Graph Frequencies in Stretched Cut Distance](https://mlsanity.com/paper/2309.05260/) (2023)
- [A Review of Graph-Powered Data Quality Applications for IoT Monitoring Sensor Networks](https://mlsanity.com/paper/2410.21006/) (2024)
- [On the Stability of Graph Convolutional Neural Networks: A Probabilistic Perspective](https://mlsanity.com/paper/2506.01213/) (2025)
- [A Fully Spectral Neuro-Symbolic Reasoning Architecture with Graph Signal Processing as the Computational Backbone](https://mlsanity.com/paper/2508.14923/) (2025)
- [Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs](https://mlsanity.com/paper/2602.10031/) (2026)
- [A Topological Characterization of Graph Neural Networks via Stochastic Block Model Embeddings on the n-Sphere](https://mlsanity.com/paper/2606.07598/) (2026)

## Abstract (arXiv)

This paper focuses on spectral filters on graphs, namely filters defined as elementwise multiplication in the frequency domain of a graph. In many graph signal processing settings, it is important to transfer a filter from one graph to another. One example is in graph convolutional neural networks (ConvNets), where the dataset consists of signals defined on many different graphs, and the learned filters should generalize to signals on new graphs, not present in the training set. A necessary condition for transferability (the ability to transfer filters) is stability. Namely, given a graph filter, if we add a small perturbation to the graph, then the filter on the perturbed graph is a small perturbation of the original filter. It is a common misconception that spectral filters are not stable, and this paper aims at debunking this mistake. We introduce a space of filters, called the Cayley smoothness space, that contains the filters of state-of-the-art spectral filtering methods, and whose filters can approximate any generic spectral filter. For filters in this space, the perturbation in the filter is bounded by a constant times the perturbation in the graph, and filters in the Cayley smoothness space are thus termed linearly stable. By combining stability with the known property of equivariance, we prove that graph spectral filters are transferable.