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TopoDiffusionNet: A Topology-aware Diffusion Model
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Proposes TopoDiffusionNet, which fine-tunes an ADM diffusion model to generate masks with an exact Betti number by adding a persistent-homology-based loss on the predicted clean image at each timestep.
Problem
Diffusion models fail to generate images with a specified topology, i.e., a given number of connected components (0-dim) or holes/regions (1-dim) as measured by Betti numbers.
Approach
Use persistent homology on the predicted noiseless image at each denoising step to separate the top-c most persistent structures from noise, and add a loss that boosts the persistence of the top-c dots and suppresses the rest, enforcing exactly c structures.
Key results
- On Shapes (0-dim), TDN reaches 0.9478 accuracy versus 0.7500 for ADM-T and 0.6381 for Stable Diffusion.
- On COCO animals, TDN reaches 0.8557 accuracy versus 0.6685 for ADM-T and 0.6379 for Attention Refocusing.
- On 1-dim datasets, TDN improves accuracy from 0.5494 to 0.8318 on Google Maps and from 0.5357 to 0.7785 on CREMI over ADM-T.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the topology-based objective to ADM-T | ADM-T · Shapes | Accuracy: 0.7500 ± 0.1889 -> 0.9478 ± 0.0420 | Helped |
| Add the topology-based objective to ADM-T | ADM-T · COCO (Animals) | Accuracy: 0.6685 ± 0.1485 -> 0.8557 ± 0.0805 | Helped |
| Add the topology-based objective to ADM-T | ADM-T · Google Maps | Accuracy: 0.5494 ± 0.1386 -> 0.8318 ± 0.1159 | Helped |
| Add the topology-based objective to ADM-T | ADM-T · CREMI | Accuracy: 0.5357 ± 0.1879 -> 0.7785 ± 0.1901 | Helped |
| Add L_preserve to the topology loss (ablation) | TDN · Shapes | Accuracy: 0.9186 ± 0.1129 -> 0.9478 ± 0.0420 | Helped |
| Add L_denoise to the topology loss (ablation) | TDN · Shapes | Accuracy: 0.8926 ± 0.1821 -> 0.9478 ± 0.0420 | Helped |
| Set λ to 1e-3 instead of 0 (ablation) | TDN · Shapes | Accuracy: 0.7500 ± 0.1889 -> 0.9066 ± 0.0612 | Helped |
| Set λ to 1e-5 instead of 0 (ablation) | TDN · Shapes | Accuracy: 0.7500 ± 0.1889 -> 0.9478 ± 0.0420 | Helped |
| Set λ to 1e-7 instead of 0 (ablation) | TDN · Shapes | Accuracy: 0.7500 ± 0.1889 -> 0.9176 ± 0.0407 | Helped |
| Use Min-SNR (5) loss weighting instead of λ=0 (ablation) | TDN · Shapes | Accuracy: 0.7500 ± 0.1889 -> 0.9286 ± 0.0320 | Helped |
Surprising findings
- Conditioning on the topological constraint alone (ADM-T) is insufficient, and text-to-image models like Stable Diffusion and DALL·E 3 perform poorly at exact object counts.
- The denoise (suppression) loss contributes more than the preserve loss in ablations (0.9186 vs 0.8926 accuracy alone).
Limitations
- Evaluated only on simple 0-dim and 1-dim Betti constraints with up to ten structures.
- Generates masks only; textures are rendered by a separate ControlNet, so final image topology depends on that second stage.
- Computing persistent homology at every training timestep adds cost, and compute resources are not reported.
Experimental setup
Datasets: Shapes COCO CREMI Google Maps
Benchmarks: Shapes COCO (Animals) Google Maps CREMI
Metrics: Accuracy Precision F1 FID
Baselines: Stable Diffusion DALL·E 3 Attention Refocusing ADM-T
Builds on: ADM persistent homology ControlNet DDIM Attention Refocusing
Closest papers by meaning
- Hyperbolic Geometric Latent Diffusion Model for Graph Generation (ICML 2024 · similarity 0.88)
- Topograph: An Efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation (ICLR 2025 · similarity 0.88)
- Graph Generation with Diffusion Mixture (ICML 2024 · similarity 0.88)
- Tracing the Roots: Leveraging Temporal Dynamics in Diffusion Trajectories for Origin Attribution (NeurIPS 2025 · similarity 0.88)
- Hierarchical Koopman Diffusion: Fast Generation with Interpretable Diffusion Trajectory (NeurIPS 2025 · similarity 0.87)
- What Matters for Diffusion-Friendly Latent Manifold? Prior-Aligned Autoencoders for Latent Diffusion (NeurIPS 2026 · similarity 0.87)
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