Prerequisites & Notation

Before You Begin

This chapter builds on the denoising and PnP frameworks of Chapter 21 and the iterative algorithms of Chapter 17. Familiarity with the following is assumed.

Score function and denoising score matching (Chapter 21)(Review ch21)
Self-check: Can you state the relationship between the score $\nabla_\mathbf{x}\log p_\sigma(\mathbf{x})$ and the MMSE denoiser?
OAMP/VAMP iterative reconstruction (Chapter 17)(Review ch17)
Self-check: Can you write the OAMP iteration and explain the role of the denoiser module?
Linear inverse problems: $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ (Chapter 12)(Review ch12)
Self-check: Can you define the forward model, the measurement residual, and the pseudoinverse?
Basic probability: Bayes rule, Gaussian conditioning
Self-check: Can you state Bayes rule and compute the posterior for a linear Gaussian model?

Notation for This Chapter

Symbols introduced or emphasised in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\nabla_\mathbf{x}\log p(\mathbf{x})$	Score function of the distribution $p(\mathbf{x})$	s01
$\mathbf{s}_\theta(\mathbf{x}_t, t)$	Learned score network at diffusion time $t$	s01
$\bar{\alpha}_t$	Cumulative product of noise schedule: $\bar{\alpha}_t = \prod_{s=1}^t (1-\beta_s)$	s01
$\hat{\mathbf{x}}_0(\mathbf{x}_t)$	Tweedie denoised estimate: $\mathbb{E}[\mathbf{x}_0 \mid \mathbf{x}_t]$	s01
$\zeta$	Guidance scale controlling measurement consistency strength in DPS	s02
$T$	Total number of diffusion steps	s01
$\text{NFE}$	Number of neural-network function evaluations per reconstruction	s04

← Ch 21 Score-Based Diffusion Models Recap