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Probability Density Geodesics in Image Diffusion Latent Space
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Shows that geodesics in Stable Diffusion latent space, with a metric inversely proportional to probability density, can be computed via BVP/IVP solvers, and applies them to training-free image interpolation and extrapolation.
Problem
Studying the structure of a learned image data space by computing probability-density-weighted shortest paths, and using them for image sequence interpolation and extrapolation without training.
Approach
Defines a Riemannian metric whose norm is inversely proportional to the probability density estimated by a diffusion model's score, so geodesics prefer high-density regions; solves boundary and initial value problems in Stable Diffusion latent space using noise-free score distillation, a sphere-projected control-point BVP solver, and an RK4 IVP solver.
Key results
- On Animals and Humans, the w/o-opt great-circle path achieves FID 24.71 and PPL 0.874, with the optimized geodesic at FID 33.87 and PPL 0.960, versus fine-tuning baselines IMPUS (FID 25.75) and DiffMorpher (FID 32.89).
- Optimized geodesic gives the lowest PDV on Animals and Humans (0.016), better than all listed methods, while TOPIQ is slightly below fine-tuning methods (0.607 vs 0.686 for DiffMorpher).
- Video analysis on 40 CLEVR clips shows the average geodesic gradient norm of original videos (1.86e-5) is close to the optimized path (1.63e-5) and lower than perturbed or smoothed paths (about 1.9-2.2e-5).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Optimize the great-circle initialization into a geodesic path (ablation) | Ours image interpolation · Animals and Humans | PDV: 0.032 -> 0.016 | Helped |
| Optimize the great-circle initialization into a geodesic path (ablation) | Ours image interpolation · MorphBench | PDV: 0.035 -> 0.022 | Helped |
| Optimize the great-circle initialization into a geodesic path (ablation) | Ours image interpolation · Web | PDV: 0.053 -> 0.026 | Helped |
| Optimize the great-circle initialization into a geodesic path (ablation) | Ours image interpolation · Animals and Humans | FID: 24.71 -> 33.87 | Hurt |
| Optimize the great-circle initialization into a geodesic path (ablation) | Ours image interpolation · MorphBench | FID: 37.85 -> 46.51 | Hurt |
| Optimize the great-circle initialization into a geodesic path (ablation) | Ours image interpolation · Web | FID: 112.81 -> 134.68 | Hurt |
| Use text inversion, positive prompts, and negative prompts (ablation) | BVP solver · validation dataset | FID: 70.56 -> 63.28 | Helped |
| Use text inversion, positive prompts, and negative prompts (ablation) | BVP solver · validation dataset | PPL: 1.06 -> 1.000 | Helped |
| Use text inversion, positive prompts, and negative prompts (ablation) | BVP solver · validation dataset | TOPIQ: 0.518 -> 0.553 | Helped |
| Use text inversion, positive prompts, and negative prompts (ablation) | BVP solver · validation dataset | PDV: 0.017 -> 0.017 | No clear effect |
| Use geodesic interpolation instead of the corresponding strongest baseline | Ours image interpolation · Animals and Humans | PDV: 0.018 -> 0.016 | Helped |
| Use geodesic interpolation instead of the corresponding strongest baseline | Ours image interpolation · MorphBench | FID: 36.33 -> 46.51 | Hurt |
Surprising findings
- The training-free geodesic method is roughly on par with fine-tuning-based methods such as IMPUS and DiffMorpher, despite not fine-tuning the diffusion model.
- The unoptimized great-circle initialization often scores better than the optimized geodesic on FID, PPL and TOPIQ, so optimization does not uniformly improve interpolation metrics.
Limitations
- Struggles with large camera motions or domain gaps; great-circle initialization may trap the optimizer in local optima.
- Score distillation gradient is an approximation that may not align well with the true log-probability gradient.
- Image extrapolation is unreliable because a good initial velocity is hard to determine.
Experimental setup
Datasets: MorphBench AFHQ CelebA-HQ CLEVR (rendered 40 videos) Web image pairs
Benchmarks: MorphBench Animals and Humans Web image pairs CLEVR videos
Metrics: FID PPL PDV TOPIQ geodesic gradient norm
Baselines: NoiseDiffusion AID IMPUS DiffMorpher SmoothDiffusion Ours w/o opt. (great circle)
Builds on: DDIM inversion score distillation sampling noise-free score distillation classifier-free guidance DiffMorpher
Closest papers by meaning
- Simpler Diffusion: 1.5 FID on ImageNet512 with Pixel-space Diffusion (CVPR 2025 · similarity 0.88)
- Smooth Diffusion: Crafting Smooth Latent Spaces in Diffusion Models (CVPR 2024 · similarity 0.88)
- Isometric Representation Learning for Disentangled Latent Space of Diffusion Models (ICML 2024 · similarity 0.87)
- Understanding the Generalization of Pretrained Diffusion Models on Out-of-Distribution Data (AAAI 2024 · similarity 0.87)
- Exploiting Diffusion Prior for Generalizable Dense Prediction (CVPR 2024 · similarity 0.86)
- Diffusion Models Without Attention (CVPR 2024 · similarity 0.86)
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