# Graphcode: Learning from multiparameter persistent homology using graph neural networks

arXiv: https://arxiv.org/abs/2405.14302 · HTML: https://mlsanity.com/paper/2405.14302/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 37 references that are in the map.
- First posted on arXiv 23 May 2024.

## Builds on (most cited first, 18 of 18)

- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Hierarchical Graph Pooling with Structure Learning](https://mlsanity.com/paper/1911.05954/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [Topological analysis of the connectome of digital reconstructions of neural microcircuits](https://mlsanity.com/paper/1601.01580/) (2016)
- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [A Kernel for Multi-Parameter Persistent Homology](https://mlsanity.com/paper/1809.10231/) (2018)
- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [Keeping it sparse: Computing Persistent Homology revisited](https://mlsanity.com/paper/2211.09075/) (2022)
- [Delaunay Bifiltrations of Functions on Point Clouds](https://mlsanity.com/paper/2310.15902/) (2023)
- [Combinatorial presentation of multidimensional persistent homology](https://mlsanity.com/paper/1409.7936/) (2014)

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

- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data](https://mlsanity.com/paper/2502.16049/) (2025)
- [TMetaNet: Topological Meta-Learning Framework for Dynamic Link Prediction](https://mlsanity.com/paper/2506.00453/) (2025)
- [Kernel Foundry: A Diagnosis-driven Evolutionary Kernel Optimizer with Multi-Experts](https://mlsanity.com/paper/2605.30359/) (2026)
- [Heat Field Signatures: From Point Clouds to Smooth Geometry](https://mlsanity.com/paper/2609.07975/) (2026)
- [COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules](https://mlsanity.com/paper/2609.22012/) (2026)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)
- [Exact Topological Compliance: Generating Persistence-Equivalent Graphs](https://mlsanity.com/paper/2610.18856/) (2026)

## Abstract (arXiv)

We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale parameters. Such multi-parameter topological summaries are usually based on complicated theoretical foundations and difficult to compute; in contrast, graphcodes yield an informative and interpretable summary and can be computed as efficient as one-parameter summaries. Moreover, a graphcode is simply an embedded graph and can therefore be readily integrated in machine learning pipelines using graph neural networks. We describe such a pipeline and demonstrate that graphcodes achieve better classification accuracy than state-of-the-art approaches on various datasets.