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Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling
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Theoretically and empirically explains why diffusion-model features peak at intermediate noise levels via a noisy mixture-of-low-rank-Gaussians model, and links the unimodal pattern to generalization versus memorization.
Problem
Diffusion models yield useful self-supervised representations whose linear-probing quality follows a unimodal curve over noise level, but the cause of this unimodal representation dynamics is poorly understood.
Approach
Models the data as a noisy mixture of low-rank Gaussians and analyzes a shallow, U-Net-like denoiser with softmax group weighting; derives an SNR metric whose approximation is unimodal because denoising strength and class confidence trade off across noise levels. Empirically, unimodality disappears when the model memorizes training data.
Key results
- Theory shows SNR of the optimal DAE is approximately unimodal in noise level, driven by the interplay of denoising rate and class confidence rate.
- Unimodal representation dynamics vanish as training data shrinks or training proceeds into memorization; peak probing accuracy negatively correlates with FID on CIFAR10/CIFAR100 with 2^12 samples.
- Ensembling features across timesteps improves DiT on MiniImageNet under 80% label noise from 47.3% to 56.3% test accuracy.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Ensemble features from multiple timesteps | DiT · MiniImageNet Test, 80% label noise | accuracy: 47.3 -> 56.3 | Helped |
| Ensemble features from multiple timesteps | EDM · CIFAR100 transfer, 80% label noise | accuracy: 35.6 -> 43.9 | Helped |
| Ensemble features from multiple timesteps | DiT · DTD transfer, 80% label noise | accuracy: 27.0 -> 30.6 | Helped |
| Ensemble features from multiple timesteps | DiT · Flowers102 transfer, 80% label noise | accuracy: 13.3 -> 14.7 | Helped |
| Add dropout (0.5) (ablation) | Vanilla DAE on CIFAR10 · CIFAR10 test | accuracy: $32.3$ -> $35.3$ | Helped |
| Add dropout (0.9) (ablation) | Vanilla DAE on CIFAR10 · CIFAR10 test | accuracy: $32.3$ -> $36.4$ | Helped |
| Add dropout (0.95) (ablation) | Vanilla DAE on CIFAR10 · CIFAR10 test | accuracy: $32.3$ -> $38.1$ | Helped |
| Add EDM-based preconditioning (ablation) | Vanilla DAE on CIFAR10 · CIFAR10 test | accuracy: $32.3$ -> $49.2$ | Helped |
| Train at noise levels σ={0.002, 0.012, 0.102} (ablation) | Vanilla DAE on CIFAR10 · CIFAR10 test | accuracy: $32.3$ -> $58.6$ | Helped |
| Allow class subspaces to overlap (θ = 30°) (ablation) | Parameterized DAE on MoLRG · MoLRG, σ_t = 0.296 | SNR: 58.27 -> 38.54 | Hurt |
| Vary class subspace ranks (d₀ = 10, d₁ = 2) (ablation) | Parameterized DAE on MoLRG · MoLRG, σ_t = 0.585 | SNR: Class 0: 215.31; Class 1: 1.80 | Mixed |
| Use non-uniform class mixing weights (π₀ = 0.8, π₁ = 0.2) (ablation) | Parameterized DAE on MoLRG · MoLRG, σ_t = 0.296 | SNR: Class 0: 134.76; Class 1: 23.91 | Mixed |
Surprising findings
- Individual DAEs trained on a single noise level collapse into near-identity mappings at low noise, while multi-noise-level diffusion training avoids this.
- Unimodal dynamics disappear when class clusters are well separated, with accuracy and SNR peaking at t=0.
Limitations
- Theory relies on simplified assumptions (orthogonal subspaces, equal ranks, uniform mixing, shallow U-Net-like parameterization).
- Theorem gives an approximation of SNR rather than the exact quantity, with a constant C_t assumed to have minimal impact on shape.
Experimental setup
Datasets: CIFAR10 CIFAR100 TinyImageNet Oxford-IIIT Pet Flowers-102 ImageNet-1K MiniImageNet DTD synthetic noisy MoLRG
Benchmarks: CIFAR10 CIFAR100 TinyImageNet MiniImageNet CIFAR100 transfer DTD Flowers102
Metrics: linear probing accuracy SNR FID mIoU test accuracy
Baselines: MAE EDM DiT
Builds on: DDPM EDM REPA
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