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Graphcode: Learning from multiparameter persistent homology using graph neural networks
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Introduces graphcodes, an embedded-graph summary of two-parameter persistent homology computed via batched matrix reduction, and classifies them with a GAT-based GNN, outperforming vectorization baselines on synthetic shape, point-process, and orbit datasets.
Problem
Multi-parameter persistent homology gives complete invariants that are hard to compute and must be vectorized partially for machine learning, losing topological information.
Approach
Represent a bifiltered complex as a layered graph (graphcode) by connecting barcode bases of consecutive one-parameter slices via the inclusion-induced map, computed in cubic time by a single out-of-order matrix reduction, and feed this graph directly to a graph neural network with per-slice max-pooling.
Key results
- On the synthetic shape dataset, graphcodes reach 86.9% accuracy vs. 74.9% for the next best (GRIL) and 64.1% for MP-I, while computing in 95 s.
- On Orbit100k, graphcodes reach 92.3% accuracy vs. 89.2% for PersLay; the 100000 point clouds took 27 minutes to compute graphcodes.
- On the graph TUDatasets, graphcodes are competitive but not best: e.g., PROTEINS 73.6% vs. 74.6% for MP-HSM-C, and IMDB-BINARY 65.4% vs. 74.8%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add edges connecting consecutive graphcode layers (ablation) | Graphcode GNN · Point cloud shape dataset | accuracy: 82.8$\pm$1.9 -> 86.9$\pm$1.4 | Helped |
| add edges connecting consecutive graphcode layers (ablation) | Graphcode GNN · Processes | accuracy: 83.1$\pm$3.7 -> 83.4$\pm$2.5 | No clear effect |
| add edges connecting consecutive graphcode layers (ablation) | Graphcode GNN · Orbit5k | accuracy: 88.4$\pm$1.5 -> 88.5$\pm$1.1 | No clear effect |
| add edges connecting consecutive graphcode layers (ablation) | Graphcode GNN · Orbit100k | accuracy: 91.5$\pm$0.3 -> 92.3$\pm$0.3 | Helped |
| augment node attributes with multiplicative and additive persistence (ablation) | Graphcode GNN · TUDatasets graph classification | accuracy | Helped |
| use primary parameter 1 for slicing (ablation) | Graphcode GNN · TUDatasets graph classification | accuracy | Helped |
| replace separate pooling by slice with global pooling (ablation) | Graphcode GNN · Graphcode classification | accuracy | Hurt |
Surprising findings
- On the graph TUDatasets, graphcodes underperform several baselines (e.g., IMDB-BINARY 65.4% vs. 74.8% for MP-HSM-C), which the authors attribute partly to small training sets.
- Removing graphcode edges still gives better accuracy than related approaches on the shape dataset (82.8% vs. 74.9% for GRIL), but edges add a further gain.
Limitations
- Graphcodes depend on the chosen barcode bases and are not topological invariants or uniquely defined.
- Graph classification results on TUDatasets are not consistently superior to baselines; the authors question whether these datasets have a strong topological signal.
- GNN training is slower than CNN-based training on vectorizations (about 9 minutes versus about 1 minute).
Experimental setup
Datasets: PROTEINS DHFR COX2 MUTAG IMDB-BINARY Orbit5k Orbit100k Processes (synthetic point processes) Synthetic shape dataset (annuli and disks)
Benchmarks: PROTEINS DHFR COX2 MUTAG IMDB-BINARY Shape dataset Processes Orbit5k Orbit100k
Metrics: test accuracy computation time
Baselines: MP-I MP-K MP-L GRIL MP-HSM-C P-I PSS-K PWG-K SW-K PF-K PersLay
Builds on: PersLay multi-parameter persistence images graph attention networks Delaunay bifiltration MPFree minimal presentations
Closest papers by meaning
- Differentiability and Optimization of Multiparameter Persistent Homology (ICML 2024 · similarity 0.90)
- Positional Encoding meets Persistent Homology on Graphs (ICML 2025 · similarity 0.87)
- Boosting Graph Pooling with Persistent Homology (NeurIPS 2024 · similarity 0.87)
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.87)
- Persistent Homology for High-dimensional Data Based on Spectral Methods (NeurIPS 2024 · similarity 0.87)
- On topological descriptors for graph products (NeurIPS 2025 · similarity 0.87)
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