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An analytic theory of creativity in convolutional diffusion models
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Derives analytic local score (LS) and equivariant local score (ELS) machines from locality and translational equivariance constraints on the optimal score, and shows they predict trained convolutional diffusion models' outputs case-by-case after calibrating one time-dependent locality scale.
Problem
Score-matching diffusion models with ideal score can only memorize training data, yet practical convolutional diffusion models generate highly original images; the theory-experiment gap about the mechanism of creativity is unexplained.
Approach
Two inductive biases of CNNs, locality and translational equivariance, are imposed on the MMSE score estimate; the resulting closed-form ELS machine generates locally consistent patch mosaics from training patches at any location, and with a calibrated time-dependent patch scale it reproduces trained ResNet/UNet outputs sample by sample.
Key results
- The ELS machine with calibrated locality scale achieves median r^2 of 0.94 (ResNet, MNIST), 0.90 (ResNet, CIFAR10) and 0.96 (ResNet, CelebA32x32) against trained ResNet outputs.
- On CIFAR10, the ELS machine predicts a pretrained self-attention UNet's outputs with median r^2 about 0.77, versus about 0.48 for the IS machine baseline.
- The UNet on CelebA is better predicted by the fully-local LS machine (median r^2 about 0.90) than by the ELS machine.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the ELS theory instead of the IS machine to predict pretrained UNet+SA outputs | CIFAR10 pretrained UNet+SA · CIFAR10, 100 sample images | median r^2: $r^2 \sim 0.48$ -> $r^2 \sim 0.77$ | Helped |
| Use the LS machine rather than the ELS machine | UNet on CelebA32x32 · CelebA32x32 | median r^2: $r^2 \sim 0.90$ | Helped |
| Add self-attention to the UNet architecture | CIFAR10 diffusion model · CIFAR10 | median r^2 against ELS predictions | Mixed |
| Use excess locality at late stages of the reverse flow | Diffusion models on FashionMNIST · FashionMNIST | spatial consistency | Hurt |
| Use circular padding rather than zero padding | ResNet diffusion model on MNIST · MNIST | localized digit-like outputs | Hurt |
Surprising findings
- Late-time locality of about 5 pixels in trained models explains characteristic spatial inconsistencies (extra limbs, wrong numbers of fingers) in generated images.
- On CelebA the UNet is better explained by a fully equivariance-broken LS machine than by the boundary-broken ELS machine, since the UNet can use absolute position via its global receptive field.
Limitations
- Theory is restricted to convolution-only diffusion models; self-attention UNets are only partially predicted (r^2 about 0.77).
- Evaluated only on small datasets (MNIST, FashionMNIST, CIFAR10, CelebA32x32) because latent-space and attention models were excluded.
Experimental setup
Datasets: MNIST FashionMNIST CIFAR10 CelebA
Benchmarks: MNIST FashionMNIST CIFAR10 CelebA
Metrics: r^2 between theory and experiment outputs
Baselines: ideal score machine IS machine local score (LS) machine
Builds on: DDIM score matching diffusion
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