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Diffeomorphic interpolation for efficient persistence-based topological optimization
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Turns sparse persistence-based topological gradients into smooth, invertible diffeomorphic vector fields via RKHS (Gaussian kernel) interpolation, enabling subsampled topological optimization on large point clouds and black-box autoencoder latent-space regularization.
Problem
Topological optimization of point clouds and latent spaces with persistence-based losses is hampered by extremely sparse gradients (only few points move per step) and poor scalability of Vietoris-Rips persistence computation, and existing approaches cannot be reused for new data or invertibly sample
Approach
Interpolate the sparse vanilla topological gradient with a Gaussian-kernel diffeomorphism (minimum-RKHS-norm vector field) defined on all of R^d, prove it remains a descent direction and bound its Lipschitz constant, combine it with subsampling so the whole point cloud is updated, and learn a fixed invertible flow for black-box AE latent spaces.
Key results
- On a 200-point noisy circle, diffeomorphic interpolation converges faster than vanilla gradients and is faster in wall-clock time than oineus big-step gradients.
- On the Stanford Bunny (n=35,947) with subsampling s=100, vanilla subsampled gradient barely changes the cloud in 1,000 epochs while diffeomorphic updates decrease the augmentation loss.
- For VAE latent spaces on COIL, angle correlation improves from 0.56 to 0.6 (Duck), 0.17 to 0.74 (Pig) and 0.85 to 0.93 (Vase) for vanilla vs diffeo.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use diffeomorphic gradient descent instead of vanilla gradients | Black-box VAE latent spaces · COIL Duck | angle correlation: 0.56 -> 0.6 | Helped |
| Use diffeomorphic gradient descent instead of vanilla gradients | Black-box VAE latent spaces · COIL Cat | angle correlation: 0.79 -> 0.83 | Helped |
| Use diffeomorphic gradient descent instead of vanilla gradients | Black-box VAE latent spaces · COIL Pig | angle correlation: 0.17 -> 0.74 | Helped |
| Use diffeomorphic gradient descent instead of vanilla gradients | Black-box VAE latent spaces · COIL Vase | angle correlation: 0.85 -> 0.93 | Helped |
| Use diffeomorphic gradient descent instead of vanilla gradients | Black-box VAE latent spaces · COIL Teapot | angle correlation: 0.32 -> 0.39 | Helped |
| Apply diffeomorphic interpolation to gradients computed on subsamples | Topological optimization with subsampling · Stanford Bunny point cloud | topological augmentation loss | Helped |
| Use diffeomorphic interpolation rather than vanilla gradients | Topological optimization · Noisy unit-circle point cloud | loss over iterations | Helped |
| Combine big-step gradients with diffeomorphic interpolation | Topological optimization · Noisy unit-circle point cloud | loss over iterations | Helped |
| Use diffeomorphic interpolation rather than vanilla gradients | Topological optimization · Noisy unit-circle point cloud | time to convergence | Helped |
| Apply diffeomorphic interpolation to gradients computed on subsamples | Topological optimization with subsampling · 2000-point cloud | topological loss | Helped |
Surprising findings
- Vanilla subsampled topological gradients barely change the point cloud even after 1,000 epochs on the Stanford Bunny.
Limitations
- Initial latent spaces with nontrivial topology (zero or two loops) cannot be unfolded by diffeomorphisms, so the optimized latent spaces may not match the target topology.
- Diffeomorphic step is about 10x slower per iteration than a vanilla step on the bunny (20 s vs 3 s), and the vector field evaluation scales with n.
- Stability between diffeomorphic interpolations on a dataset and its subsamples is left as future theoretical work.
Experimental setup
Datasets: COIL Stanford Bunny synthetic noisy circle point cloud
Benchmarks: COIL Stanford Bunny synthetic noisy circle
Metrics: topological loss vs iterations running time angle correlation with ground truth
Baselines: vanilla topological gradient big-step gradient (oineus)
Builds on: vanilla topological gradient (Gradient-based topological optimization) big-step gradient (oineus) Gudhi critical set method
Closest papers by meaning
- Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.88)
- Towards Scalable Topological Regularizers (ICLR 2025 · similarity 0.87)
- TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information (NeurIPS 2026 · similarity 0.87)
- Differentiable Mapper for Topological Optimization of Data Representation (ICML 2024 · similarity 0.86)
- TopoDiffusionNet: A Topology-aware Diffusion Model (ICLR 2025 · similarity 0.85)
- Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms (NeurIPS 2024 · similarity 0.85)
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