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Differentiability and Optimization of Multiparameter Persistent Homology
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Develops a differentiability and stochastic subgradient convergence framework for multiparameter persistent homology descriptors that are semilinearly determined on grids, and applies it to Hilbert decomposition signed measures and multiparameter persistence landscapes with experiments showing gains over one-parameter topological optimization.
Problem
Real-valued barcode-based losses have a well-understood optimization theory, but no general framework exists for differentiating and optimizing losses built from multiparameter persistent homology descriptors, which have no complete discrete invariant.
Approach
Stratify the space of n-filtrations into cells on which the filtration order is fixed, show descriptors that are semilinearly determined on grids are semilinear with explicit Clarke subdifferentials, and combine with o-minimal definability to get convergence of stochastic subgradient descent for multiparameter topological losses.
Key results
- On graph classification, GCN-MPTOP reaches 78.4% on IMDB-B versus 75.2% for GCN-OPTOP and 73.2% for GCN-NOTOP.
- GraphDenseNet-MPTOP achieves 92.6% on MUTAG, versus 86.2% for GraphDenseNet-OPTOP and 88.8% for GraphDenseNet-NOTOP.
- Multiparameter topological regularization preserves circle topology in noisy autoencoder latent spaces where one-parameter regularization fails.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace no topological descriptor with a multiparameter descriptor | GCN · ENZYMES | accuracy: 30.3$\pm$8.1 -> 39.0$\pm$10.1 | Helped |
| Replace one-parameter descriptors with a multiparameter descriptor | GraphDenseNet · ENZYMES | accuracy: 47.3$\pm$12.3 -> 48.0$\pm$11.4 | No clear effect |
| Replace one-parameter descriptors with a multiparameter descriptor | GCN · IMDB-B | accuracy: 75.2$\pm$5.6 -> 78.4$\pm$5.1 | Helped |
| Replace one-parameter descriptors with a multiparameter descriptor | GIN · IMDB-B | accuracy: 75.0$\pm$2.7 -> 71.3$\pm$5.1 | Hurt |
| Replace no topological descriptor with a multiparameter descriptor | GraphResNet · IMDB-B | accuracy: 75.3$\pm$5.3 -> 69.4$\pm$5.8 | Hurt |
| Replace no topological descriptor with a multiparameter descriptor | GraphDenseNet · IMDB-B | accuracy: 50.3$\pm$5.9 -> 52.2$\pm$7.7 | No clear effect |
| Replace one-parameter descriptors with a multiparameter descriptor | GCN · IMDB-M | accuracy: 51.2$\pm$4.4 -> 51.1$\pm$3.5 | No clear effect |
| Replace one-parameter descriptors with a multiparameter descriptor | GIN · IMDB-M | accuracy: 47.5$\pm$5.0 -> 48.5$\pm$4.2 | No clear effect |
| Replace no topological descriptor with a multiparameter descriptor | GraphResNet · IMDB-M | accuracy: 49.4$\pm$4.3 -> 50.1$\pm$4.4 | No clear effect |
| Replace no topological descriptor with a multiparameter descriptor | GraphDenseNet · IMDB-M | accuracy: 33.1$\pm$2.7 -> 34.1$\pm$3.1 | No clear effect |
| Replace no topological descriptor with a multiparameter descriptor | GCN · MUTAG | accuracy: 87.2$\pm$5.6 -> 85.1$\pm$7.7 | Hurt |
| Replace one-parameter descriptors with a multiparameter descriptor | GIN · MUTAG | accuracy: 88.3$\pm$8.9 -> 87.2$\pm$6.1 | No clear effect |
Surprising findings
- Multiparameter gains are not uniform: GIN-MPTOP underperforms GIN-OPTOP on IMDB-B (71.3 vs 75.0) and GraphResNet-MPTOP underperforms GraphResNet-NOTOP on IMDB-B.
- No topological regularization behaves better than one-parameter regularization when there are many noise points in the autoencoder experiment.
Limitations
- Multiparameter persistent homology is generally computationally demanding.
- The framework may not cover future descriptors that are not semilinearly determined on grids.
- Graph classification results show mixed gains; the paper does not aim for state-of-the-art.
Experimental setup
Datasets: ENZYMES IMDB-B IMDB-M MUTAG two interlaced circles in R^9 with background noise (synthetic) uniform unit-square point clouds (synthetic)
Benchmarks: ENZYMES IMDB-B IMDB-M MUTAG
Metrics: classification accuracy optimal transport distance
Baselines: one-parameter VR barcode optimization (OPTOP) no topological regularization (NOTOP) GCN GIN GraphResNet GraphDenseNet
Builds on: barcode optimization framework (one-parameter) Hilbert decomposition signed measure multiparameter persistence landscape Clarke subdifferential o-minimal structures
Closest papers by meaning
- Graphcode: Learning from multiparameter persistent homology using graph neural networks (NeurIPS 2024 · similarity 0.90)
- On topological descriptors for graph products (NeurIPS 2025 · similarity 0.87)
- Persistent Homology for High-dimensional Data Based on Spectral Methods (NeurIPS 2024 · similarity 0.86)
- Positional Encoding meets Persistent Homology on Graphs (ICML 2025 · similarity 0.86)
- The Flood Complex: Large-Scale Persistent Homology on Millions of Points (NeurIPS 2025 · similarity 0.85)
- Graph Persistence goes Spectral (NeurIPS 2025 · similarity 0.85)
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