# Which Graph Shift Operator? A Spectral Answer to an Empirical Question

arXiv: https://arxiv.org/abs/2602.06557 · HTML: https://mlsanity.com/paper/2602.06557/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 30 references that are in the map.
- First posted on arXiv 6 Feb 2026.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [GPT-GNN: Generative Pre-Training of Graph Neural Networks](https://mlsanity.com/paper/2006.15437/) (2020)
- [Rademacher Complexity for Adversarially Robust Generalization](https://mlsanity.com/paper/1810.11914/) (2018)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Spatio-Spectral Graph Neural Networks](https://mlsanity.com/paper/2405.19121/) (2024)
- [Learning Parametrised Graph Shift Operators](https://mlsanity.com/paper/2101.10050/) (2021)
- [SPADE: A Spectral Method for Black-Box Adversarial Robustness Evaluation](https://mlsanity.com/paper/2102.03716/) (2021)
- [Bounding the Expected Robustness of Graph Neural Networks Subject to Node Feature Attacks](https://mlsanity.com/paper/2404.17947/) (2024)
- [How Fair is Your Diffusion Recommender Model?](https://mlsanity.com/paper/2409.04339/) (2024)
- [Flexible Diffusion Scopes with Parameterized Laplacian for Heterophilic Graph Learning](https://mlsanity.com/paper/2409.09888/) (2024)
- [ADMP-GNN: Adaptive Depth Message Passing GNN](https://mlsanity.com/paper/2509.01170/) (2025)
- [Centrality Graph Shift Operators for Graph Neural Networks](https://mlsanity.com/paper/2411.04655/) (2024)

## Built on by (most cited first, 1 of 1)

- [Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature](https://mlsanity.com/paper/2607.22381/) (2026)

## Abstract (arXiv)

Graph Neural Networks (GNNs) have established themselves as the leading models for learning on graph-structured data, generally categorized into spatial and spectral approaches. Central to these architectures is the Graph Shift Operator (GSO), a matrix representation of the graph structure used to filter node signals. However, selecting the optimal GSO, whether fixed or learnable, remains largely empirical. In this paper, we introduce a novel alignment gain metric that quantifies the geometric distortion between the input signal and label subspaces. Crucially, our theoretical analysis connects this alignment directly to generalization bounds via a spectral proxy for the Lipschitz constant. This yields a principled, computation-efficient criterion to rank and select the optimal GSO for any prediction task prior to training, eliminating the need for extensive search.