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On the Trajectory Regularity of ODE-based Diffusion Sampling
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Identifies a linear-nonlinear-linear "boomerang" regularity in ODE-based diffusion sampling trajectories, explains it via an implicit denoising trajectory under time-varying-bandwidth KDE, and uses dynamic programming (GITS) to pick time schedules that improve few-NFE image generation.
Problem
ODE-based diffusion sampling with few function evaluations suffers from truncation error; the shape structure of sampling trajectories is empirically observed but unexplained, and time schedules are hand-crafted.
Approach
Shows the sampling trajectory is a convex combination driven by an implicit denoising trajectory; under Gaussian KDE with time-varying bandwidth the denoising output has a closed form resembling annealed mean shift, explaining the boomerang shape. A dynamic-programming search over a fine time grid, using Euler-vs-reference local truncation costs from a few warmup samples, yields a time schedule th
Key results
- On CIFAR-10 at 5 NFE, iPNDM + GITS reaches FID 8.38 versus 13.59 for iPNDM with polynomial schedule.
- On ImageNet 64x64 at 5 NFE, iPNDM + GITS achieves FID 10.79 versus 18.99 for iPNDM.
- On Stable Diffusion v1.4 at 10 NFE, DPM-Solver++(2M) + GITS improves FID from 17.16 to 15.53.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add GITS time scheduling to iPNDM | iPNDM · CIFAR-10 32×32, NFE 5 | FID: 13.59 -> 8.38 | Helped |
| Add GITS time scheduling to iPNDM | iPNDM · FFHQ 64×64, NFE 5 | FID: 17.17 -> 11.22 | Helped |
| Add GITS time scheduling to iPNDM | iPNDM · ImageNet 64×64, NFE 5 | FID: 18.99 -> 10.79 | Helped |
| Add GITS time scheduling to iPNDM | iPNDM · LSUN Bedroom 256×256, NFE 5 | FID: 26.65 -> 15.85 | Helped |
| Add GITS time scheduling to DPM-Solver++(2M) | DPM-Solver++(2M) · MS-COCO validation captions, Stable Diffusion v1.4, NFE 10 | FID: 17.16 -> 15.53 | Helped |
| Use GITS instead of the polynomial schedule | iPNDM · CIFAR-10, NFE 10 | FID: 2.77 -> 2.49 | Helped |
| Use GITS instead of the polynomial schedule | DDIM · CIFAR-10, NFE 10 | FID: 15.69 -> 10.09 | Helped |
Surprising findings
- Trajectory deviation from the endpoint line is tiny (about 0.0034 of endpoint distance), so trajectories are nearly straight despite the boomerang shape.
- Dynamic programming schedules searched with the Euler solver transfer effectively to high-order solvers such as iPNDM.
Limitations
- Theoretical analysis relies on an idealized KDE/optimal denoiser on training samples, which is only approximately valid for trained networks.
- Time schedule search requires a fine-grained ground-truth trajectory and a warmup sample set, adding cost for large datasets (10-15 minutes for LSUN and LAION).
Experimental setup
Datasets: CIFAR-10 FFHQ ImageNet 64x64 LSUN Bedroom MS-COCO validation
Benchmarks: CIFAR-10 FFHQ 64x64 ImageNet 64x64 LSUN Bedroom 256x256 MS-COCO 2014 (Stable Diffusion FID)
Metrics: FID running time
Baselines: DDIM DPM-Solver-2 DPM-Solver++(3M) DEIS-tAB3 UniPC AMED-Solver AMED-Plugin iPNDM AYS
Builds on: EDM DDIM iPNDM DPM-Solver annealed mean shift
Closest papers by meaning
- Tracing the Roots: Leveraging Temporal Dynamics in Diffusion Trajectories for Origin Attribution (NeurIPS 2025 · similarity 0.90)
- Diffusion Bridge Implicit Models (ICLR 2025 · similarity 0.90)
- Adaptive Stochastic Coefficients for Accelerating Diffusion Sampling (NeurIPS 2025 · similarity 0.90)
- Generative Modeling with Phase Stochastic Bridge (ICLR 2024 · similarity 0.90)
- Denoising Diffusion Bridge Models (ICLR 2024 · similarity 0.89)
- On Accelerating Diffusion-Based Sampling Processes via Improved Integration Approximation (ICLR 2024 · similarity 0.89)
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