Neural denoising diffusion models

4.1 Generic conditioning

Rozet and Louppe (2023b) summarises:

With score-based generative models, we can generate samples from the unconditional distribution $p(x(0))\approx p(x)$. To solve inverse problems, however, we need to sample from the posterior distribution $p(x\mid y)$. This could be accomplished by training a conditional score network $s_{\varphi }(x(t),t\mid y)$ to approximate the posterior score ${\mathrm{\nabla }}_{x(t)}\log p(x(t)\mid y)$ and plugging it into the reverse SDE (4). However, this would require data pairs $(x,y)$ during training and one would need to retrain a new score network each time the observation process $p(y\mid x)$ changes. Instead, many have observed (Y. Song, Sohl-Dickstein, et al. 2022; Adam et al. 2022; Chung et al. 2023; Kawar, Vaksman, and Elad 2021; Y. Song, Shen, et al. 2022) that the posterior score can be decomposed into two terms thanks to Bayes’ rule $${\mathrm{\nabla }}_{x(t)}\log p(x(t)\mid y)={\mathrm{\nabla }}_{x(t)}\log p(x(t))+{\mathrm{\nabla }}_{x(t)}\log p(p\mid x(t)).$$

Since the prior score ${\mathrm{\nabla }}_{x(t)}\log p(x(t))$ can be approximated with a single score network, the remaining task is to estimate the likelihood score ${\mathrm{\nabla }}_{x(t)}\log p(y\mid x(t))$. Assuming a differentiable measurement function $\mathcal{A}$ and a Gaussian observation process $p(y\mid x)=\mathcal{N}(y\mid \mathcal{A}(x),{\mathrm{\Sigma }}_y)$, Chung et al. (2023) proposes the approximation $$p(y\mid x(t))=\int p(y\mid x)p(x\mid x(t))\mathrm{d}x\approx \mathcal{N}(y\mid \mathcal{A}(\hat{x}(x(t))),{\mathrm{\Sigma }}_y)$$ where the mean $\hat{x}(x(t))={\mathbb{E}}_{p(x\mid x(t))}[x]$ is given by Tweedie’s formula (Efron 2011; Kim and Ye 2021) $$\begin{array}{rl}{\mathbb{E}}_{p(x\mid x(t))}[x]& =\frac{x(t)+\sigma (t{)}^2{\mathrm{\nabla }}_{x(t)}\log p(x(t))}{\mu (t)}\\ & =\frac{x(t)+\sigma (t{)}^2{s}_{\varphi }(x(t),t)}{\mu (t)}.\end{array}$$

As the log-likelihood of a multivariate Gaussian is known analytically and $s_{\varphi }(x(t),t)$ is differentiable, we can compute the likelihood score ${\mathrm{\nabla }}_{x(t)}\log p(y\mid x(t))$ with this approximation in zero-shot, that is, without training any other network than $s_{\varphi }(x(t),t)$.

4.2 Inpainting

If we want coherence with part of an existing image, we call that inpainting (Ajay et al. 2023; Grechka, Couairon, and Cord 2024; A. Liu, Niepert, and Broeck 2023; Lugmayr et al. 2022; Sharrock et al. 2022; Wu et al. 2023; Zhang et al. 2023).

4.3 Super-resolution

Coherence, with a sparse regular subset (Zamir et al. 2021; Choi et al. 2021).

4.4 Reconstruction/inversion

Perturbed and partial observations (Choi et al. 2021; Kawar et al. 2022; Nair, Mei, and Patel 2023; Peng et al. 2024; Xie and Li 2022; Zhao et al. 2023; Y. Song, Shen, et al. 2022; Zamir et al. 2021; Chung et al. 2023; Sui et al. 2024).

5.1 Generic

• Diffusion Autoencoders: Toward a Meaningful and Decodable Representation (Preechakul et al. 2022)

6.1 Language

See language models for using diffusion models in NLP.

6.2 Solutions satisfying physical constraints

See PDE diffusion models for using diffusion models to generate PDE solutions.

6.3 PD manifolds

6.4 Proteins

Baker Lab (Torres et al. 2022; Watson et al. 2022)

• A diffusion model for protein design
• RF Diffusion now free and open source