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Multidimensional Adaptive Coefficient for Inference Trajectory Optimization in Flow and Diffusion
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Proposes Multidimensional Adaptive Coefficient (MAC), a plug-in module that makes flow/diffusion interpolation coefficients per-dimension and trajectory-adaptive, trained with adversarial simulation-based feedback to optimize inference trajectories.
Problem
Flow and diffusion models fix unidimensional, predefined time coefficients and thus cannot adapt inference trajectories to final transport quality, unlike simulation-based methods.
Approach
Extend scalar interpolation coefficients to per-dimension multidimensional coefficients produced by a U-Net conditioned on the starting noise, parameterized by low-pass-filtered sinusoidal expansions, and optimize them (jointly with the vector field) via adversarial simulation-based feedback.
Key results
- EDM + adv theta,phi with 5 (+) NFE reaches FID 1.37 on CIFAR-10 conditional, the claimed state of the art in that table.
- SI with MAC and 10 (+) NFE reaches FID 4.14 on CIFAR-10 and 7.06 on ImageNet-32 after adversarial phi-optimization (~10x NFE efficiency).
- Adversarial training uses 1.5k kimg on CIFAR-10 versus 25.6k kimg for CTM + adv theta, roughly 17x fewer training images.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add MAC (γ) pre-training instead of α pre-training | SI · CIFAR-10, NFE 100 | FID: 4.75 -> 3.98 | Helped |
| Add MAC (γ) pre-training instead of α pre-training | FM · CIFAR-10, NFE 100 | FID: 4.52 -> 3.59 | Helped |
| Add MAC (γ) pre-training instead of α pre-training | DDPM · CIFAR-10, NFE 100 | FID: 6.64 -> 4.73 | Helped |
| Optimize the multidimensional coefficient φ | SIα · Gaussian to Moons, NFE 5 | W₂ distance: 0.882{ $\pm$0.035} -> 0.682{ $\pm$0.093} | Helped |
| Jointly optimize θ and φ instead of optimizing θ alone (ablation) | EDMγ · CIFAR-10 unconditional, 5 Euler steps | FID: 2.28 -> 1.81 | Helped |
| Jointly optimize θ and φ instead of optimizing θ alone (ablation) | EDMγ · CIFAR-10 conditional, 5 Euler steps | FID: 2.14 -> 1.42 | Helped |
| Use multidimensionality in φ rather than disabling it (ablation) | EDMγ + adversarial φ · CIFAR-10 unconditional, 5 Euler steps | FID: 33.55 -> 18.67 | Helped |
| Condition MAC on starting point x_T rather than on 1_d (ablation) | SIγ + adversarial φ · CIFAR-10, 10 NFE | FID: 7.84 -> 4.14 | Helped |
| Condition MAC on starting point x_T rather than on 1_d (ablation) | FMγ + adversarial φ · CIFAR-10, 10 NFE | FID: 9.20 -> 6.13 | Helped |
| Condition MAC on starting point x_T rather than on 1_d (ablation) | DDPMγ + adversarial φ · CIFAR-10, 10 NFE | FID: 26.09 -> 10.04 | Helped |
| Increase sinusoidal basis count M (ablation) | SIγ + adversarial φ · CIFAR-10, 10 NFE | FID: 6.89; 4.14; 4.42; 5.32; 6.11; 5.74 | Mixed |
| Set coefficient scale s to 0.0 instead of 0.005 (ablation) | SIγ + adversarial φ · CIFAR-10, 10 NFE, M 10 | FID: 4.79 -> 9.75 | Hurt |
Surprising findings
- Models pre-trained with multidimensional coefficients gamma achieve lower FID than standard alpha-trained models across frameworks and NFE, even without adversarial training.
- Optimized inference trajectories are clearly non-linear, and minibatch-OT-trained models still improve with MAC, suggesting straight trajectories are not optimal.
Limitations
- MAC is tied to a fixed NFE during trajectory optimization.
- Conditioning is only on the starting noise x_T rather than on the full trajectory.
- Training memory (VRAM) for simulation through the solver is a practical cost.
Experimental setup
Datasets: CIFAR-10 FFHQ-64 AFHQv2-64 ImageNet-64 ImageNet-32 2D synthetic (Gaussian, 8 Gaussians, Moons)
Benchmarks: CIFAR-10 FFHQ-64 AFHQ-64 ImageNet-64 ImageNet-32 2D synthetic transport
Metrics: FID FD_DINOv2 W2 distance
Baselines: CTM CD 2-Rectified Flow++ EDM StyleGAN-XL StyleGAN-Ada Score SDE ADM
Builds on: EDM Stochastic Interpolant Flow Matching CTM StyleGAN-XL
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