# Spectro-Riemannian Graph Neural Networks

arXiv: https://arxiv.org/abs/2502.00401 · HTML: https://mlsanity.com/paper/2502.00401/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 58 references that are in the map.
- First posted on arXiv 1 Feb 2025.

## Builds on (most cited first, 30 of 31)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Inductive Representation Learning on Temporal Graphs](https://mlsanity.com/paper/2002.07962/) (2020)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Interpreting and Unifying Graph Neural Networks with An Optimization Framework](https://mlsanity.com/paper/2101.11859/) (2021)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Transformer for Graphs: An Overview from Architecture Perspective](https://mlsanity.com/paper/2202.08455/) (2022)
- [Hyperbolic Graph Attention Network](https://mlsanity.com/paper/1912.03046/) (2019)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphs](https://mlsanity.com/paper/2104.02228/) (2021)
- [Learning heat diffusion graphs](https://mlsanity.com/paper/1611.01456/) (2016)
- [Optimizing Generalized PageRank Methods for Seed-Expansion Community Detection](https://mlsanity.com/paper/1905.10881/) (2019)
- [A Self-supervised Mixed-curvature Graph Neural Network](https://mlsanity.com/paper/2112.05393/) (2021)
- [Pseudo-Riemannian Graph Convolutional Networks](https://mlsanity.com/paper/2106.03134/) (2021)
- [Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning](https://mlsanity.com/paper/2309.04082/) (2023)
- [Curvature-based Clustering on Graphs](https://mlsanity.com/paper/2307.10155/) (2023)
- [On properties of the Generalized Wasserstein distance](https://mlsanity.com/paper/1304.7014/) (2013)
- [Public Wisdom Matters! Discourse-Aware Hyperbolic Fourier Co-Attention for Social-Text Classification](https://mlsanity.com/paper/2209.13017/) (2022)
- [Block Models and Personalized PageRank](https://mlsanity.com/paper/1607.03483/) (2016)
- [A Comprehensive Benchmark on Spectral GNNs: The Impact on Efficiency, Memory, and Effectiveness](https://mlsanity.com/paper/2406.09675/) (2024)

## Built on by (most cited first, 6 of 6)

- [The Post-GCN Decade Revisited: Curvature-Stratified Evaluation of Relational Learning](https://mlsanity.com/paper/2606.06397/) (2026)
- [Learnable Kernel Density Estimation for Graphs and Its Application to Graph-Level Anomaly Detection](https://mlsanity.com/paper/2505.21285/) (2025)
- [Text Has Curvature](https://mlsanity.com/paper/2602.13418/) (2026)
- [Geometric Mixture-of-Experts with Curvature-Guided Adaptive Routing for Graph Representation Learning](https://mlsanity.com/paper/2603.22317/) (2026)
- [Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds](https://mlsanity.com/paper/2512.10407/) (2025)
- [Constant-Curvature Sliced Gromov-Wasserstein for Heterogeneous Cross-Curvature Alignment](https://mlsanity.com/paper/2610.07218/) (2026)

## Abstract (arXiv)

Can integrating spectral and curvature signals unlock new potential in graph representation learning? Non-Euclidean geometries, particularly Riemannian manifolds such as hyperbolic (negative curvature) and spherical (positive curvature), offer powerful inductive biases for embedding complex graph structures like scale-free, hierarchical, and cyclic patterns. Meanwhile, spectral filtering excels at processing signal variations across graphs, making it effective in homophilic and heterophilic settings. Leveraging both can significantly enhance the learned representations. To this end, we propose Spectro-Riemannian Graph Neural Networks (CUSP) - the first graph representation learning paradigm that unifies both CUrvature (geometric) and SPectral insights. CUSP is a mixed-curvature spectral GNN that learns spectral filters to optimize node embeddings in products of constant-curvature manifolds (hyperbolic, spherical, and Euclidean). Specifically, CUSP introduces three novel components: (a) Cusp Laplacian, an extension of the traditional graph Laplacian based on Ollivier-Ricci curvature, designed to capture the curvature signals better; (b) Cusp Filtering, which employs multiple Riemannian graph filters to obtain cues from various bands in the eigenspectrum; and (c) Cusp Pooling, a hierarchical attention mechanism combined with a curvature-based positional encoding to assess the relative importance of differently curved substructures in our graph. Empirical evaluation across eight homophilic and heterophilic datasets demonstrates the superiority of CUSP in node classification and link prediction tasks, with a gain of up to 5.3% over state-of-the-art models. The code is available at: https://github.com/amazon-science/cusp.