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Assessing the quality of denoising diffusion models in Wasserstein distance: noisy score and optimal bounds
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Shows empirically that DDPMs tolerate constant-variance noise in score evaluations and proves W2 bounds for DDPM with randomized noisy score estimators that achieve the sqrt(D)/eps rate matching the Gaussian case.
Problem
Quantifying how noisy score evaluations affect DDPM sampling and the number of score queries needed for Wasserstein-2 accuracy; prior bounds scaled poorly (D^4/eps^2, D/eps^2) and did not match the Gaussian optimal rate.
Approach
Analyze DDPM with randomized noisy score estimators in W2 by applying the recursive argument to the squared Wasserstein distance, exploiting smoothness of Gaussian-smoothed score; the discretization error is linear in the maximal step size and noise enters only via a term vanishing as the number of queries grows.
Key results
- On CelebA-HQ FID rises only moderately up to noise sigma about 1, while CIFAR-10 stays robust up to sigma about 2 under constant-variance score noise.
- The W2 discretization error term depends linearly on the maximal step size h_max, improving on h_max^{1/12} (Chen et al.), h_max^{1/4} and h_max^{1/2} from prior work.
- For strongly log-concave targets, W2 error scales as sqrt(LD)(log L log K / K + eps_b + sqrt(log L log K / K) eps_v), matching the Gaussian lower bound sqrt(D) h_max.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add constant-variance noise to score evaluations | DDPM · CIFAR-10 and CelebA-HQ | visual quality | No clear effect |
| increase score-noise scale | DDPM · CIFAR-10 and CelebA-HQ | visual quality: σ ∈ {0.25, 0.5, 1, 2, 3, 4} | Hurt |
| increase score-noise scale | DDPM · CelebA-HQ | FID: σ ≈ 1 | Hurt |
| increase score-noise scale | DDPM · CIFAR-10 | FID: σ ≈ 2 | Hurt |
| change the score-noise distribution among centered Uniform, Gaussian, Laplace, and Student's-t | DDPM · CIFAR-10 and CelebA-HQ | FID | No clear effect |
Surprising findings
- Adding constant-variance noise to every score evaluation has negligible visual impact on generated samples.
- Sharp quality degradation occurs beyond a certain noise threshold, not predicted by the theory.
Limitations
- Bounds include exponential factors in parameters such as support diameter for semi-log-concave targets.
- Score error assumed uniform over x and time grid, stronger than prior L2 assumptions.
- Empirical study is small-scale and does not explain the sharp degradation at high noise.
Experimental setup
Datasets: CIFAR-10 CelebA-HQ LSUN-Church
Benchmarks: CIFAR-10 CelebA-HQ LSUN-Church
Metrics: FID Wasserstein-2 bound
Baselines: Chen et al. 2023 DDPM bound Benton et al. bound Strasman et al. 2025
Builds on: DDPM Pinsker-Girsanov analysis recursive Langevin analysis
Closest papers by meaning
- O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions (ICLR 2025 · similarity 0.94)
- Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models (NeurIPS 2024 · similarity 0.92)
- Interpreting and Improving Diffusion Models from an Optimization Perspective (ICML 2024 · similarity 0.92)
- Denoising Diffusion Bridge Models (ICLR 2024 · similarity 0.91)
- Efficient Denoising Diffusion via Probabilistic Masking (ICML 2024 · similarity 0.91)
- On Inference Stability for Diffusion Models (AAAI 2024 · similarity 0.91)
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