Research map / Probabilistic and scientific modeling / Variational and generative sampling
Towards Hierarchical Rectified Flow
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Proposes hierarchical rectified flow, which couples multiple ODEs (location, velocity, acceleration, ...) to model the full multimodal velocity distribution so integration paths can intersect and become straighter, reducing NFEs for generation.
Problem
Classic rectified flow regresses only the expected velocity field under an MSE objective, which averages over multimodal ground-truth velocities and yields curved integration paths that require many function evaluations.
Approach
Apply the rectified flow objective recursively in velocity space (learning acceleration), and generally in deeper hierarchy (jerk, etc.), so the random velocity field is modeled in its entirety; sampling uses nested Euler integrations of coupled ODEs, yielding a random differential equation whose trajectories can intersect and be straighter.
Key results
- On 1D 5-Gaussian target, HRF models achieve lower 1-Wasserstein distance than RF at the same NFE, with HRF2 using 74,497 parameters versus 297,089 for RF.
- On MNIST and CIFAR-10, HRF2 achieves better FID than the RF baseline at the same total NFEs, with on-par results on ImageNet-32.
- Density estimation bpd improves over RF: CIFAR-10 2.975 vs 2.980 and ImageNet-32 3.397 vs 3.416.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use HRF2 instead of RF | Density estimation · 1D 1N→2N | NLL (BPD): 0.275 -> 0.261 | Helped |
| Use HRF2 instead of RF | Density estimation · 2D 8N→moon | NLL (BPD): 2.119 -> 2.113 | Helped |
| Use HRF2 instead of RF | Density estimation · CIFAR-10 | NLL (BPD): 2.980 -> 2.975 | Helped |
| Use HRF2 instead of RF | Density estimation · ImageNet-32 | NLL (BPD): 3.416 -> 3.397 | Helped |
| Use sampling steps (2,50) rather than (1,100) (ablation) | HRF2 at fixed NFE=100 · 2D N→6N | 2-SWD: 0.070 -> 0.037 | Helped |
| Use sampling steps (20,5) rather than (1,100) (ablation) | HRF2 at fixed NFE=100 · 1D N→5N | 1-WD: 0.031 -> 0.017 | Helped |
| Use sampling steps (50,2) rather than (2,50) (ablation) | HRF2 at fixed NFE=100 · 1D 2N→2N | 1-WD: 0.011 -> 0.075 | Hurt |
Surprising findings
- Fewer parameters (about one quarter of the baseline) were used for the low-dimensional HRF2 model while still outperforming RF.
Limitations
- Sampling uses only a basic Euler scheme and a basic grid search over integration schedules; other solvers were not explored.
- Gains over the rectified flow baseline on image datasets are slight (on-par on ImageNet-32).
Experimental setup
Datasets: MNIST CIFAR-10 ImageNet-32 synthetic 1D Gaussian mixtures synthetic 2D Gaussian mixtures and moons
Benchmarks: MNIST CIFAR-10 ImageNet-32 synthetic 1D and 2D Gaussian mixture and moons
Metrics: FID 1-Wasserstein distance sliced 2-Wasserstein distance bits per dimension NFE
Baselines: Rectified Flow (RF)
Builds on: rectified flow flow matching U-Net diffusion architecture
Closest papers by meaning
- Improving the Training of Rectified Flows (NeurIPS 2024 · similarity 0.89)
- Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow (ICLR 2025 · similarity 0.88)
- Variational Rectified Flow Matching (ICML 2025 · similarity 0.87)
- Text-to-Image Rectified Flow as Plug-and-Play Priors (ICLR 2025 · similarity 0.87)
- SCoT: Unifying Consistency Models and Rectified Flows via Straight-Consistent Trajectories (NeurIPS 2025 · similarity 0.86)
- On the Relation between Rectified Flows and Optimal Transport (NeurIPS 2025 · similarity 0.86)
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