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Generalization in diffusion models arises from geometry-adaptive harmonic representations
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Shows that diffusion denoisers trained on large non-overlapping subsets converge to the same score function (generalization), and that their learned denoising acts as shrinkage in geometry-adaptive harmonic bases.
Problem
Whether diffusion-model denoisers trained on small image datasets memorize training data or learn a continuous density that generalizes, and what inductive biases of DNN denoisers enable this generalization in high dimensions.
Approach
Two UNet/BF-CNN denoisers trained on non-overlapping subsets of a dataset converge to nearly the same score function once the training set is large (~10^5 images), and their Jacobian eigenbases are geometry-adaptive harmonic bases; the denoiser's shrinkage in these bases is near-optimal for C^alpha images and suboptimal for manifold-supported data.
Key results
- Two denoisers trained on non-overlapping CelebA subsets of size 10^5 generate nearly identical samples and their train and test PSNR curves coincide, indicating vanishing model variance.
- UNet denoisers trained on 10^5 C^alpha images achieve PSNR slopes that closely match the theoretical optimal slopes alpha/(alpha+1).
- On the disk manifold, the DNN denoiser has PSNR slope below the optimal value of 1.0 at small noise, because it retains extra GAHB components beyond the 5-dimensional tangent space.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the training set from small subsets to 10^5 images | UNet denoiser · CelebA | train/test PSNR gap | Helped |
| Increase the training set to 10^5 images | UNet denoiser · CelebA | similarity between samples generated by models trained on disjoint subsets | Helped |
| Train on geometric C^α images, whose optimal basis is geometry-adaptive and harmonic | UNet denoiser · C^α images | PSNR slope | Helped |
| Use a DNN denoiser with geometry-adaptive harmonic components beyond the manifold tangent space | UNet denoiser · Translating and dilating disks | PSNR slope: below the optimal value of 1.0 | Hurt |
| Train on pixel-shuffled rather than ordinary faces | DNN denoiser · CelebA faces | PSNR slope | Hurt |
| Train on larger images | UNet denoiser · Generalization across image resolutions | training data required to reach the threshold PSNR gap | Hurt |
| Train on enough data for the model to leave the memorization regime | UNet denoiser · LSUN bedroom | train/test PSNR gap | Helped |
| Train on non-overlapping subsets of 20,000 images | BF-CNN denoiser · LSUN bedroom, 32×32 | similarity between generated samples | Helped |
| Train on non-overlapping subsets with sufficient data | DNN denoiser · CelebA and LSUN bedroom | model variance | Helped |
Surprising findings
- Shuffled-pixel faces are denoised substantially worse than ordinary faces, showing the inductive bias mismatches when locality is destroyed.
- Denoisers trained on a single image with rescaling retain oscillating harmonic components beyond the optimal ray projection.
Limitations
- Convergence at N=10^5 is not perfect; cosine similarity distribution is not a delta function at 1 and sampling trajectories bifurcate.
- The formal definition of the GAHB function class arising from DNN inductive biases is not provided.
- Optimal denoiser is unknown for photographic images, so model bias is hard to evaluate numerically.
Experimental setup
Datasets: CelebA LSUN bedroom CelebA HQ synthetic C^alpha images synthetic disk images
Benchmarks: CelebA LSUN bedroom C^alpha synthetic images
Metrics: PSNR MSE cosine similarity PSNR slope
Baselines: optimal denoiser (oracle) bandlet basis optimal thresholding
Builds on: Miyasawa relation bandlet bases BF-CNN DnCNN Stein's unbiased risk estimate
Closest papers by meaning
- On Inductive Biases That Enable Generalization in Diffusion Transformers (NeurIPS 2025 · similarity 0.92)
- Towards a Mechanistic Explanation of Diffusion Model Generalization (ICML 2025 · similarity 0.90)
- Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure (NeurIPS 2024 · similarity 0.90)
- An Analytical Theory of Spectral Bias in the Learning Dynamics of Diffusion Models (NeurIPS 2025 · similarity 0.90)
- When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Nets (ICML 2025 · similarity 0.90)
- Generalization through variance: how noise shapes inductive biases in diffusion models (ICLR 2025 · similarity 0.90)
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