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Variational Control for Guidance in Diffusion Models
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Proposes Diffusion Trajectory Matching (DTM), a variational-control framework for training-free guidance of pretrained diffusion models, instantiated as NDTM, which optimizes a per-step control signal against a terminal cost and is evaluated on linear, non-linear, and blind inverse problems plus style-guided Stable Diffusion generation.
Problem
Guiding pretrained diffusion models to satisfy external terminal constraints (inverse problems, style guidance) without retraining, while staying close to the unguided sample distribution.
Approach
Treats guidance as a stochastic optimal control problem with KL transient cost and terminal cost; at each DDIM step, optimizes a non-linear control inserted into the denoiser input (x_t + gamma*u_t) via a DDIM-based upper bound, then samples from the guided posterior, generalizing classifier guidance and RB-Modulation as special cases.
Key results
- On noisy non-linear deblur (ImageNet), NDTM reaches LPIPS 0.163 and FID 34.31 versus 0.249/47.60 for the next best RB-Modulation.
- On 4x super-resolution (ImageNet), NDTM achieves FID 28.75, best among compared baselines.
- On style guidance with Stable Diffusion 1.4, NDTM gets CLIP score 31.34 and style score 475.62, beating MPGD and FreeDOM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NDTM instead of RB-Modulation | FFHQ pretrained diffusion model · Non-linear deblurring, FFHQ (256 × 256) | LPIPS: 0.064 -> 0.046 | Helped |
| Use NDTM instead of RB-Modulation | OpenAI unconditional pretrained diffusion model · Non-linear deblurring, ImageNet (256 × 256) | LPIPS: 0.249 -> 0.163 | Helped |
| Use NDTM instead of DPS | OpenAI unconditional pretrained diffusion model · Super-resolution (4x), ImageNet (256 × 256) | LPIPS: 0.195 -> 0.158 | Helped |
| Use NDTM instead of DPS | FFHQ pretrained diffusion model · Random inpainting (90%), FFHQ (256 × 256) | LPIPS: 0.058 -> 0.059 | Hurt |
| Use NDTM instead of DPS | OpenAI unconditional pretrained diffusion model · Random inpainting (90%), ImageNet (256 × 256) | LPIPS: 0.152 -> 0.149 | Helped |
| Use NDTM instead of DMPlug | FFHQ pretrained diffusion model · Blind image deblurring, Gaussian blur | LPIPS: 0.147 -> 0.103 | Helped |
| Use NDTM instead of DMPlug | FFHQ pretrained diffusion model · Blind image deblurring, Motion blur | LPIPS: 0.118 -> 0.086 | Helped |
| Use NDTM instead of FreeDOM | Stable Diffusion 1.4 · Style guidance | CLIP score: 30.86 -> 31.34 | Helped |
| Use NDTM instead of MPGD | Stable Diffusion 1.4 · Style guidance | Style Score: 498.85 -> 475.62 | Helped |
| Use NDTM instead of RB-Modulation | OpenAI unconditional pretrained diffusion model · Non-linear deblurring, ImageNet (256 × 256) | PSNR: 22.11 -> 24.41 | Helped |
| Increase terminal cost weight w_T (ablation) | NDTM · Non-linear deblurring, ImageNet | PSNR | Helped |
| Increase terminal cost weight w_T toward infinity (ablation) | NDTM · Non-linear deblurring, ImageNet | LPIPS | Hurt |
Surprising findings
- DPS is reported as highly sensitive to step size and unstable on non-linear deblurring.
- C-ΠGDM failed to produce plausible random inpainting results despite tuning.
Limitations
- Requires an inner optimization loop per diffusion step, increasing sampling cost.
- Restricted to differentiable terminal costs.
- Only one instantiation (DDIM-based, additive control) explored; alternative aggregation functions and weighting schedules left open.
Experimental setup
Datasets: FFHQ ImageNet WikiArt PartiPrompt
Benchmarks: FFHQ 256x256 inverse problems ImageNet 256x256 inverse problems WikiArt/PartiPrompt style guidance
Metrics: LPIPS FID KID PSNR SSIM CLIP score Style score
Baselines: DPS DDRM RED-diff MPGD C-ΠGDM RB-Modulation DMPlug FreeDOM
Builds on: DDIM Diffusion Posterior Sampling Path Integral Control RB-Modulation
Closest papers by meaning
- Hierarchical Variational Policies for Reward-Guided Diffusion (NeurIPS 2026 · similarity 0.90)
- Gradient Guidance for Diffusion Models: An Optimization Perspective (NeurIPS 2024 · similarity 0.90)
- Theoretical insights for diffusion guidance: A case study for Gaussian mixture models (ICML 2024 · similarity 0.90)
- A Variational Perspective on Solving Inverse Problems with Diffusion Models (ICLR 2024 · similarity 0.89)
- Tortoise and Hare Guidance: Accelerating Diffusion Model Inference with Multirate Integration (NeurIPS 2025 · similarity 0.89)
- Plug-and-Play Diffusion Distillation (CVPR 2024 · similarity 0.88)
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