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Kuramoto Orientation Diffusion Models
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Proposes a score-based diffusion model whose forward process is stochastic Kuramoto synchronization on periodic phase variables (global or local coupling plus a reference phase), trained with local score matching on wrapped Gaussian transitions.
Problem
Standard isotropic Euclidean diffusion models handle orientation-rich images (fingerprints, textures, directional fields) poorly because angular data lives on periodic domains and angular discontinuities cause artifacts.
Approach
Replace the linear Gaussian forward SDE with nonlinear stochastic Kuramoto dynamics that synchronize phase variables toward a global reference phase (forward = synchronization, reverse = desynchronization), with wrapped Gaussian transition kernels, local score matching, and sinusoidal periodicity-aware score networks.
Key results
- On Brodatz textures, the 100-step globally coupled model reaches FID 20.26 vs 38.33 for SGM at 100 steps, and the locally coupled model gets 14.19 at 1000 steps vs 20.37 for SGM.
- On SOCOFing at 100 steps, FID drops from 104.92 (SGM) to 67.49 (locally coupled Kuramoto).
- On CIFAR-10 at 1000 steps, SGM (FID 3.17) beats the Kuramoto models (10.79-11.58), so the method is not better on general natural images at long schedules.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Kuramoto synchronization diffusion instead of SGM | Image generation · SOCOFing, Brodatz, and ground terrain | FID: SOCOFing: 104.92 -> 67.49 (100 steps); Brodatz: 38.33 -> 18.47 (100 steps); ground terrain: 114.90 -> 92.86 (100 steps) | Helped |
| Use Kuramoto synchronization diffusion instead of SGM at 100 steps | CIFAR-10 generation · CIFAR-10 | FID: 38.04 -> 28.17 | Helped |
| Use Kuramoto synchronization diffusion instead of SGM at 300 steps | CIFAR-10 generation · CIFAR-10 | FID: 25.76; 25.83; 24.86 | Mixed |
| Use Kuramoto synchronization diffusion instead of SGM at 1000 steps | CIFAR-10 generation · CIFAR-10 | FID: 3.17; 11.58; 10.79 | Hurt |
| Add local coupling rather than global coupling | Kuramoto Orientation Diffusion · SOCOFing, Brodatz, ground terrain, and CIFAR-10 | FID: SOCOFing 100 steps: 74.41 -> 67.49; Brodatz 100 steps: 20.26 -> 18.47; ground terrain 100 steps: 101.65 -> 92.86; CIFAR- | Helped |
| Remove Kuramoto coupling (set K(t)=0) (ablation) | Kuramoto Orientation Diffusion · Brodatz | FID: 20.26 -> 33.76 (100 steps); 18.51 -> 20.54 (300 steps); 15.42 -> 19.01 (1000 steps) | Hurt |
Surprising findings
- On CIFAR-10 the advantage reverses at 1000 steps: SGM reaches FID 3.17 while the Kuramoto variants are at 10.79 to 11.58.
- Removing the nonlinear Kuramoto coupling (reference-only drift) raises 100-step Brodatz FID from 20.26 to 33.76, so the coupling term is the main source of gains.
Limitations
- Training requires simulating the forward Markov chain, costing O(T) per training step.
- Gains are reported on orientation-dense data; the structured bias may limit expressiveness on natural images with many steps (CIFAR-10 at 1000 steps).
- Results are single-run FID without error bars.
Experimental setup
Datasets: CIFAR-10 SOCOFing fingerprint Brodatz textures ground terrain dataset Earth/climate event datasets on sphere (volcano, earthquake, flood, wildfire) Navier-Stokes velocity fields
Benchmarks: CIFAR-10 SOCOFing Brodatz ground terrain Earth/climate sphere datasets Navier-Stokes
Metrics: FID CMMD test NLL spectral slope difference Wasserstein distance MSE
Baselines: SGM (VP-SDE score-based model) Riemannian CNF Moser flow CNF matching Riemannian diffusion model Riemannian flow matching
Builds on: score-based SDE diffusion Kuramoto model denoising score matching variance-preserving SDE
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