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How Much is a Noisy Image Worth? Data Scaling Laws for Ambient Diffusion.
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Trains over 80 diffusion models on noisy and clean mixtures (CIFAR-10, CelebA-HQ, ImageNet) and shows a small clean fraction plus mostly noisy data nearly matches clean-data performance, backed by heterogeneous-variance GMM sample complexity bounds.
Problem
Training diffusion models from corrupted (noisy) data underperforms clean-data training at realistic sample sizes; how much is a noisy sample worth relative to a clean one?
Approach
Train diffusion models on a mixture of a small clean set and a large noisy set using Ambient Denoising Score Matching for noisy samples and standard DSM for clean ones, without consistency loss; complement with minimax rates for Gaussian mixtures with heterogeneous variances that quantify the exponential price of noisy samples.
Key results
- Training on 90% noisy (sigma=0.2) plus 10% clean ImageNet yields FID 1.68, versus 1.41 for 100% clean, 10.57 for 10% clean only, and 5.32 for 100% noisy with consistency.
- On CIFAR-10, 1% clean plus 99% noisy at sigma=0.2 gives FID 3.53, versus 60.73 for 100% noisy with truncated-free full sampling.
- Estimated noisy-sample pricing: roughly 1.5 to 1.75 noisy samples at sigma=0.2 are needed to replace one clean sample.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Mix 90% noisy and 10% clean data at σ=0.2 instead of using only noisy data | Diffusion model training · CIFAR-10 | FID: 11.93 \scriptstyle{\pm 0.09} -> 2.81 \scriptstyle{\pm 0.02} | Helped |
| Mix 90% noisy and 10% clean data at σ=0.2 instead of using only noisy data | Diffusion model training · CelebA-HQ | FID: 12.97 \scriptstyle{\pm 0.11} -> 2.75 \scriptstyle{\pm 0.02} | Helped |
| Mix 90% noisy and 10% clean data at σ=0.2 instead of using only noisy data | Diffusion model training · ImageNet | FID: 5.32 \scriptstyle{\pm 0.08} -> 1.68 \scriptstyle{\pm 0.02} | Helped |
| Add truncated sampling instead of full sampling (ablation) | Noisy-only diffusion training · CIFAR-10, σ=0.2 | FID: 60.73 -> 12.12 | Helped |
| Add consistency training instead of using full sampling without consistency (ablation) | Noisy-only diffusion training · CIFAR-10, σ=0.2 | FID: 60.73 -> 11.93 | Helped |
| Add consistency training instead of using full sampling without consistency (ablation) | Noisy-only diffusion training · ImageNet, σ=0.2 | FID: 18.08 -> 5.32 | Helped |
| Train for 500K more steps (ablation) | ImageNet mixture model (90% noisy, 10% clean) · ImageNet | FID: 1.68 -> 1.63 | Helped |
| Increase sampling steps for t ≤ t_n (ablation) | ImageNet mixture model (90% noisy, 10% clean) · ImageNet | FID: 1.63 -> 1.60 | Helped |
| Add consistency fine-tuning (ablation) | ImageNet mixture model (90% noisy, 10% clean) · ImageNet | FID: 1.60 -> 1.58 | Helped |
| Add weight decay (ablation) | ImageNet mixture model (90% noisy, 10% clean) · ImageNet | FID: 1.58 -> 1.55 | Helped |
| Use only 1% clean data with 99% noisy data at σ=0.2 | CIFAR-10 diffusion training · CIFAR-10 | FID: 60.73 -> 3.53 | Helped |
| Increase corruption to σ=0.4 while keeping 90% noisy and 10% clean data (ablation) | CIFAR-10 diffusion training · CIFAR-10 | FID: 2.81 -> 3.56 | Hurt |
Surprising findings
- Adding only 1% clean data to noisy data at sigma=0.2 dramatically reduces FID (from 60.73 to 3.53 on CIFAR-10).
Limitations
- Only additive Gaussian noise corruption studied, with two noise levels.
- Experiments limited to pixel-space models up to ImageNet at 64x64 resolution.
- The Gaussian mixture theory may not capture real data distributions.
Experimental setup
Datasets: CIFAR-10 CelebA-HQ ImageNet
Benchmarks: CIFAR-10 CelebA-HQ ImageNet
Metrics: FID
Baselines: Ambient Diffusion Consistent Diffusion Meets Tweedie Truncated sampling Clean-data-only training
Builds on: Ambient Diffusion Consistent Diffusion Meets Tweedie EDM Noisier2Noise
Closest papers by meaning
- Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data (ICML 2024 · similarity 0.89)
- Ambient Diffusion Omni: Training Good Models with Bad Data (NeurIPS 2025 · similarity 0.88)
- Slight Corruption in Pre-training Data Makes Better Diffusion Models (NeurIPS 2024 · similarity 0.88)
- Your Diffusion Model is Secretly a Noise Classifier and Benefits from Contrastive Training (NeurIPS 2024 · similarity 0.87)
- Test-Time Scaling of Diffusion Models via Noise Trajectory Search (NeurIPS 2025 · similarity 0.86)
- Stochastic Forward–Backward Deconvolution: Training Diffusion Models with Finite Noisy Datasets (ICML 2025 · similarity 0.86)
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