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Diffusion Models as Cartoonists: The Curious Case of High Density Regions
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Derives augmented SDEs that track log-density along stochastic diffusion sampling, a mode-tracking ODE for the denoising mode, and a high-density sampler that yields higher-likelihood (often cartoon-like or blurry) images on CIFAR-10, FFHQ-256, Churches-256 and ImageNet-64.
Problem
Stochastic diffusion samplers cannot track their own sample likelihood, and it is unknown what images lie in the highest-density regions of diffusion models.
Approach
Augmented reverse and forward SDEs that evolve log-density alongside the sample at no extra cost beyond the score, a mode-tracking ODE that follows the denoising mode, and a general instantaneous change-of-variables lemma used to compute density along the high-density ODE; the resulting sampler biases generation toward high-likelihood regions.
Key results
- Density estimate r_0 correlates with log p_0^ODE at 0.996 and 0.999 on two CIFAR-10 models.
- High-density sampler yields samples with higher likelihood than regular samples for all 512 CIFAR-10 comparisons across noise levels (0% of regular samples exceed the high-density sample).
- On FFHQ-256, log-likelihood correlates 97% with PNG file size (amount of information in image).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the high-density sampler instead of regular denoising sampling | Diffusion models · CIFAR-10 | denoising likelihood: 0 | Helped |
| Use the high-density sampler instead of regular sampling | Diffusion models · Different models and threshold values | sample likelihood | Helped |
| Blur FFHQ-256 test images | FFHQ-256 diffusion model · FFHQ-256 test images | likelihood | Helped |
| Increase blur strength | FFHQ-256 images · FFHQ-256 test images | likelihood | Helped |
| Use a sample-quality-optimized model instead of a maximum-likelihood-trained model | CIFAR-10 diffusion model · CIFAR-10 | correlation between r₀ and ODE likelihood: 0.996 -> 0.999 | Helped |
| Use a sample-quality-optimized model instead of a maximum-likelihood-trained model | CIFAR-10 diffusion model · CIFAR-10 | upper bound on SDE-ODE KL divergence | Helped |
| Use a sample-quality-optimized model instead of a maximum-likelihood-trained model | CIFAR-10 diffusion model · CIFAR-10 | likelihood-estimation bias bound | Helped |
| Use high-density sampling at higher noise levels | Diffusion models · FFHQ-256, Churches-256, and ImageNet-64 | sample likelihood | Helped |
| Use high-density sampling at lower noise levels | Diffusion models · FFHQ-256, Churches-256, and ImageNet-64 | sample likelihood | Helped |
| Replace image content with lower-information blurry or cartoon-like samples | Diffusion models · FFHQ-256, Churches-256, and ImageNet-64 | sample likelihood | Helped |
Surprising findings
- Highest-likelihood samples are cartoon-like drawings or blurry images, though the training datasets contain no cartoons.
- Blurring an image always increases its model likelihood, in proportion to blur strength.
Limitations
- Mode-tracking ODE requires Hessian inversion and third-order derivatives, too expensive in high dimensions, so a Gaussian-motivated approximation (HD-ODE) is used without guarantees for non-Gaussian data.
- Exact likelihood estimation for linear-drift SDEs is only possible if the marginals are Gaussian; the variance of the log-density estimate is not theoretically guaranteed.
Experimental setup
Datasets: CIFAR-10 FFHQ-256 Churches-256 ImageNet-64
Benchmarks: CIFAR-10 FFHQ-256 Churches-256 ImageNet-64
Metrics: log-likelihood (bits per dim / NLL) correlation between density estimates PNG file size correlation
Baselines: regular stochastic (reverse SDE) sampler probability flow ODE sampler guided diffusion
Builds on: score-based SDE probability flow ODE Itô's lemma ELBO for diffusion models EDM
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