Research map / Image generation / Fast diffusion sampling
Hierarchical Koopman Diffusion: Fast Generation with Interpretable Diffusion Trajectory
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes Hierarchical Koopman Diffusion (HKD), a one-step generator that lifts diffusion dynamics into multi-scale Koopman subspaces with closed-form linear evolution, enabling interpretable spectral analysis and intermediate-state editing.
Problem
One-step diffusion generators are fast but discard the intermediate trajectory, so they lose interpretability and fine-grained control that diffusion dynamics provide.
Approach
Encode noisy images with a U-Net into multi-scale latents, evolve each spatial location's latent by a learned block-diagonal linear operator exp(A(t-s)) (Koopman dynamics) to the clean time, and decode in one step; supervise intermediate-time reconstructions with a trajectory consistency loss.
Key results
- Ablation on CIFAR-10: one-step baseline without Koopman dynamics gives FID 5.72; adding Koopman evolution, trajectory consistency loss and hierarchical design further improves it.
- Training takes 2-3 days on 8×V100 GPUs, claimed comparable to state-of-the-art consistency methods that need about a week on 8 GPUs.
- Spectral masking shows low-range Koopman modes capture global structure, mid-range modes shape and pose, high-range modes local details.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add Koopman evolution at the skip and bottleneck paths (ablation) | One-step baseline without Koopman dynamics · CIFAR-10 | FID | Helped |
| add trajectory consistency loss (ablation) | HKD · CIFAR-10 | FID | Helped |
| use hierarchical Koopman dynamics (ablation) | HKD · CIFAR-10 | FID | Helped |
Surprising findings
- Retraining consistency distillation reportedly degrades to FID 10.53, indicating instability of consistency-based methods.
Limitations
- Limited to standard-resolution datasets (CIFAR-10, FFHQ 64x64); high-resolution generation underexplored.
- Theoretical equivalence and error bounds rely on strong structural assumptions (disentangled levels, isotropy, spectrum near 1).
Experimental setup
Datasets: CIFAR-10 FFHQ
Benchmarks: CIFAR-10 FFHQ 64x64
Metrics: FID-50k
Baselines: DDPM Score SDE DDIM EDM KD PD CD (LPIPS) DMD 2-Rectified flow++ iCT iCT-deep ECM
Builds on: EDM Probability flow ODE Koopman operator theory LPIPS
Closest papers by meaning
- Tracing the Roots: Leveraging Temporal Dynamics in Diffusion Trajectories for Origin Attribution (NeurIPS 2025 · similarity 0.90)
- One-Step Offline Distillation of Diffusion-based Models via Koopman Modeling (NeurIPS 2025 · similarity 0.90)
- DiffuseHigh: Training-Free Progressive High-Resolution Image Synthesis Through Structure Guidance (AAAI 2025 · similarity 0.89)
- On the Trajectory Regularity of ODE-based Diffusion Sampling (ICML 2024 · similarity 0.89)
- Spectral Progressive Diffusion for Efficient Image and Video Generation (NeurIPS 2026 · similarity 0.88)
- Nested Diffusion Models Using Hierarchical Latent Priors (CVPR 2025 · similarity 0.88)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.