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Diffusion Models With Learned Adaptive Noise
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Proposes MuLAN, a learned, input-conditioned multivariate (per-pixel) noise schedule for diffusion models, trained by optimizing the ELBO with auxiliary latents, achieving state-of-the-art likelihood on CIFAR-10 and ImageNet-32.
Problem
Diffusion models' noise (forward) process is usually fixed or scalar, and prior work claimed the ELBO is invariant to the noise schedule; the paper aims to improve log-likelihood estimation by learning the forward process.
Approach
Treat the learned forward diffusion as a variational posterior with a per-pixel polynomial noise schedule conditioned on auxiliary latents (inferred from the image), showing the ELBO is not invariant to multivariate/adaptive schedules; the auxiliary latent lets the reverse model use noise parameterization.
Key results
- MuLAN reaches CIFAR-10 BPD 2.55 (vs VDM 2.65), setting a new state of the art without data augmentation.
- On CIFAR-10, MuLAN matches VDM's 2.65 BPD in 2M steps versus VDM's 10M steps.
- On ImageNet-32, MuLAN reaches 3.67 BPD in the main table (VDM 3.72), with VLB 3.71 at 2M steps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace VDM's noise process with MuLAN | VDM · CIFAR-10 test | BPD: ≤2.65 -> 2.55 ± 10^{-3} | Helped |
| Replace VDM's noise process with MuLAN | VDM · ImageNet-32 test | BPD: ≤3.72 -> 3.67 ± 10^{-3} | Helped |
| Replace VDM's noise process with MuLAN | VDM · CIFAR-10 | steps to VLB 2.65: 10M -> 2M | Helped |
| Remove conditioning on an auxiliary latent, leaving the multivariate schedule conditioned only on time (ablation) | MuLAN · CIFAR-10 | likelihood | Hurt |
| Replace multivariate noise with a scalar schedule based on latent z (ablation) | MuLAN · CIFAR-10 | likelihood | Hurt |
| Use an input-conditioned scalar schedule instead of VDM's scalar schedule (ablation) | MuLAN · CIFAR-10 | likelihood | No clear effect |
| Use a linear schedule and remove auxiliary latents (ablation) | MuLAN · CIFAR-10 | likelihood | Hurt |
| Replace the polynomial schedule with a monotonic neural network schedule (ablation) | MuLAN · CIFAR-10 | likelihood | Hurt |
| Replace the polynomial schedule with a linear schedule (ablation) | MuLAN · CIFAR-10 | likelihood | Hurt |
| Replace the learned multivariate schedule with a scalar schedule (ablation) | trained MuLAN denoising model · CIFAR-10 | likelihood: MuLAN likelihood -> 2.65 | Hurt |
| Replace the learned multivariate schedule with a linear schedule (ablation) | trained MuLAN denoising model · CIFAR-10 | likelihood: MuLAN likelihood -> 2.65 | Hurt |
| Use the continuous-time objective as the number of diffusion steps tends to infinity (ablation) | MuLAN · CIFAR-10 | performance | Helped |
Surprising findings
- Replacing the noise schedule of a trained VDM with a scalar or linear schedule gives the same likelihood (2.65), while the learned multivariate schedule shows no clear human-interpretable patterns.
Limitations
- Ablations run only on CIFAR-10 with reduced batch size and 2.5M steps due to training cost
- FID is not the focus and is worse than some methods (e.g., 17.62 vs strong FID baselines not compared)
- Learned noise schedules show no interpretable patterns
Experimental setup
Datasets: CIFAR-10 ImageNet-32
Benchmarks: CIFAR-10 ImageNet-32
Metrics: bits per dimension FID NFE
Baselines: VDM PixelCNN Image Transformer DDPM ScoreFlow Flow Matching Reflected Diffusion Models DNF NDM DiffEnc
Builds on: Variational Diffusion Models Diffusion Normalizing Flows Neural Diffusion Models DiffEnc
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- Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data (ICML 2024 · similarity 0.89)
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