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Understanding Diffusion Models by Feynman's Path Integral
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Reformulates score-based diffusion models via Feynman's path integral, links the interpolation parameter between SDE and probability flow ODE to Planck's constant, and applies a WKB-style first-order expansion to estimate NLL in 1-D Gaussian and 2-D synthetic experiments.
Problem
Explaining why stochastic and deterministic (probability flow ODE) sampling in score-based diffusion models differ in performance, and quantifying the effect of sampling noise on log-likelihood, which was previously not computable for stochastic sampling.
Approach
Write the diffusion path probability as a path integral with Onsager-Machlup action, treat the interpolating parameter h (SDE to probability flow ODE) as analogous to Planck's constant, and perform a perturbative (WKB-like) expansion in h to compute the NLL correction for stochastic sampling.
Key results
- On Swiss-roll, the NLL first-order correction is -0.31±0.21 (simple SDE) and -1.59±0.57 (cosine SDE) at tol 1e-3, indicating noise lowers NLL.
- On 25-Gaussian, the first-order correction is -3.64±0.49 (simple) and -17.57±5.56 (cosine) at tol 1e-3.
- In the 1-D Gaussian toy model, NLL and W2 decrease with h for imperfect scores, consistent with noise improving sample quality.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add sampling noise (increase h) with imperfect score estimation | 1D Gaussian diffusion model · 1D Gaussian toy model | NLL | Mixed |
| Add sampling noise at small h | Pretrained diffusion models · Swiss-roll and 25-Gaussian | 2-Wasserstein distance | Helped |
| Add sampling noise (first-order increase in h) | Simple-SDE diffusion model · Swiss-roll | NLL: -0.31$\pm$0.21; -0.44$\pm$0.38 | Helped |
| Add sampling noise (first-order increase in h) | Cosine-SDE diffusion model · Swiss-roll | NLL: -1.59$\pm$0.57; -3.27$\pm$1.11 | Helped |
| Add sampling noise (first-order increase in h) | Simple-SDE diffusion model · 25-Gaussian | NLL: -3.64$\pm$0.49; -3.61$\pm$0.64 | Helped |
| Add sampling noise (first-order increase in h) | Cosine-SDE diffusion model · 25-Gaussian | NLL: -17.57$\pm$5.56; -19.65$\pm$17.46 | Helped |
Surprising findings
- The noise-level dependence of NLL can be increasing rather than decreasing in h depending on score error, so injecting noise can degrade quality.
Limitations
- Experiments do not use real image data because the NLL evaluation via discrete differentials does not scale to high dimensions.
- Numerical error may be underestimated since the error estimate relies on a score model that does not exactly match the true score.
- Results limited to 1-D and 2-D synthetic data.
Experimental setup
Datasets: Swiss-roll synthetic 2-D 25-Gaussian synthetic 2-D 1-D Gaussian toy
Benchmarks: Swiss-roll 25-Gaussian 1-D Gaussian toy model
Metrics: negative log-likelihood 2-Wasserstein distance
Baselines: probability flow ODE
Builds on: score-based SDE probability flow ODE Onsager-Machlup function Girsanov's theorem DDPM
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