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Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers
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Gives the first non-asymptotic KL bound for score-mismatched DDPM samplers, shows the mismatch induces an asymptotic bias, and derives a bias-optimal zero-shot linear-model sampler (BO-DDNM) that coincides with DDNM when there is no measurement noise.
Problem
Theoretical guarantees for diffusion samplers when the sampling (training-based) scores mismatch the target distribution's scores, notably zero-shot conditional sampling for linear measurement models (e.g., super-resolution, inpainting), where existing theory covers only plug-and-play or extra-train
Approach
Treats score mismatch as a tilting factor between target and sampling reverse-step distributions, bounding the KL divergence with an asymptotic bias term proportional to accumulated Fisher-type mismatch; minimizing that bias over the range-space correction yields a bias-optimal zero-shot sampler (BO-DDNM).
Key results
- Score mismatch yields an asymptotic distributional bias in KL bounded by the accumulated mismatch, with explicit dimension dependence for targets with finite sixth moment.
- For bounded-support targets, BO-DDNM gives bias scaling delta^-2 versus delta^-4 for vanilla DDNM under measurement noise.
- Numerical studies on Gaussian and Gaussian mixture targets show BO-DDNM achieves lower bias than DDNM and DDNM+, especially at large measurement noise.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BO-DDNM instead of DDNM under positive measurement noise | Zero-shot conditional sampling with linear models · Bounded-support targets | asymptotic bias bound: δ^{-2} vs. δ^{-4} | Helped |
| Use BO-DDNM instead of DDNM and DDNM$^+$ under measurement noise | Zero-shot conditional sampling · Gaussian and Gaussian mixture targets | distributional bias | Helped |
| Increase measurement noise variance | BO-DDNM · Gaussian mixture targets | asymptotic bias | Hurt |
| Increase the averaged distance between the conditioning and component means | BO-DDNM · Gaussian mixture targets | asymptotic bias | Hurt |
| Increase correlation between the conditioned and unconditioned coordinates | BO-DDNM · Gaussian mixture targets | asymptotic bias | Hurt |
| Introduce score mismatch during sampling | Score-mismatched DDPM · General target distributions with finite second moments | distributional bias | Hurt |
| Set measurement noise to zero | BO-DDNM · Linear conditional models | sampler identity | No clear effect |
Limitations
- Assumes finite sixth moment and bounded-support or Gaussian mixture targets for explicit dimension dependence
- Requires higher accuracy score estimation (epsilon = O~(T^-2)) under mismatch
- Analysis restricted to first-order score information; higher-order derivative samplers left open
Experimental setup
Metrics: KL divergence (theoretical bound and numerical bias)
Baselines: DDNM DDNM+ CCDF
Builds on: DDPM DDNM Come-Closer-Diffuse-Faster tilting factor analysis for DDPM convergence
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