NOTE-299: Cold Diffusion

Contribution

A demonstration that the diffusion procedure, iterative restore and re-degrade, runs with degradations other than Gaussian noise, and a sampler that makes it stable for smooth deterministic degradations.

Key insight

Re-degrade the difference, not the estimate. Adding back D(x̂₀, s−1) − D(x̂₀, s) cancels the restorer’s error to first order, since for D(x, s) = x + s·e the x̂₀ terms cancel exactly. That is why it works for blur where the naive sampler drifts.

Assumptions

• D continuous in severity, with D(x, 0) = x
• Small images: MNIST, CIFAR-10, CelebA, AFHQ at 128²
• For unconditional blur generation, a one-component GMM over the channel-wise mean as the prior, 300 steps with a 27×27 kernel

Key results

• Tables 1–4 (conditional): sampling with Algorithm 2 improves FID over one-shot reconstruction at a small cost in RMSE and SSIM. CelebA deblurring FID 26.14 against 36.37, CIFAR-10 inpainting 8.92 against 9.97
• Table 5 (unconditional FID): CelebA 23.11 with noise (estimated), 97.00 noiseless blur, 49.45 blur with σ = 0.002 noise. AFHQ 20.59 / 93.05 / 54.68

Claims

id	claim	strength	support
C1	Algorithm 2 is exact for first-order degradations regardless of R	strong	derivation in §3.3
C2	Cold degradations support conditional restoration by iterative sampling	moderate	Tables 1–4, small datasets
C3	Generative behaviour does not depend strongly on the degradation	weak	Table 5 contradicts it for unconditional generation
C4	Noiseless blur generation lacks diversity because of correlated initial pixels	moderate	the σ = 0.002 fix halves FID

Concepts

• Hot / cold diffusion — with and without Gaussian noise in the degradation

Connections

It relates to blurring diffusion and inverse heat dissipation, which it does not compare against directly. Warm Diffusion (LIT-555) extends it by mixing blur with noise and gives the account of why noise matters. The fixed-noise variant is related to DDIM’s deterministic sampling (Appendix A.6).

Recommendations

• R1 — For a deterministic degradation, sample with Algorithm 2, not by re-degrading the estimate. Strength: strong for first-order degradations. Not filed as a practice. The record holds no practice that uses cold degradations

Bearing on the record

• THEORY-079 files the paper’s account, Rejected
• THEORY-080 (Warm Diffusion) is the correction

Limitations

• Unconditional generation is uncompetitive, and only on CelebA and AFHQ
• Small images and small models
• Algorithm 2’s guarantee is first-order. Far from s = 0 it is a heuristic

Open questions

• None the record holds open. Warm Diffusion takes up the noise question