# Lecture 10
## Diffusion Models
Learning to generate by denoising.
2 processes, forward diffusion process that gradually add noise to input and reverse denoising process that learn to generate data by denoising.
**Forward**
$$q(x\_t | x\_{t-1}) = N (x\_t; \sqrt{1- \beta\_t}x\_{t-1}, \beta\_tI)$$ -> $$q(x\_{1:T}|x\_0) = \prod\_{t=1}^T q(x\_t|x\_{t-1})$$
**Diffusion Kernel**
define $$\hat{\alpha}\_t = \prod{s=1}^t (1-\beta\_s)$$ -> $$q(x\_t|x\_0) = N(x\_t; \sqrt{\hat{\alpha}\_t}x\_0, (1-\hat{\alpha}\_t)I))$$
For sampling: $$x\_t = \sqrt{\hat{\alpha}\_t}x\_0 + \sqrt{(1-\hat{\alpha}\_t}\epsilon$$ where $$\epsilon \sim N(0, I)$$
$$\beta\_t$$ values schedule is designed such that $$\hat{\alpha}\_t \rightarrow 0$$ and $$q(x\_T|x\_0) \approx N(x\_T; 0, I)$$
Forward distribution
### Denoising
Generation:
Sample $$x\_T \sim N(x\_T ; 0,I)$$
Iteratively sample $$x\_{t-1} \sim q(x\_{t-1}|x\_t)$$
In general, $$q(x\_{t-1}|x\_t) \propto q(x\_{t-1})q(x\_t|x\_{t-1})$$ is intractable
We can approximate $$q(x\_{t-1}|x\_t)$$ use normal distribution if $$\beta\_t$$ is small each forward diffusion step.
Reverse
Learning
Learning
Parameterization
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