论文阅读笔记：Denoising Diffusion Implicit Models

1、参考来源

论文《Denoising Diffusion Implicit Models》
来源：ICLR2021
https://iclr.cc/virtual/2021/poster/2804
论文链接：https://arxiv.org/abs/2010.02502
代码链接：https://github.com/ermongroup/ddim

2、符号表示的不同

在论文DDPM《Denoising Diffusion Implicit Models》当中，前向传播过程的$q(x_{t-1}|x_t,x_0)\sim N(x_{t-1};{\tilde{\mu }}_t(x_t,x_0),{\sigma }_t)$。并且${\tilde{\mu }}_t(x_t,x_0)和{\sigma }_t$分别如公式（1）所示。
$$\begin{array}{c}\begin{array}{rl}{\sigma }_t& =\sqrt{\frac{{\beta }_t\cdot (1-\overline{{\alpha }_{t-1}})}{(1-\overline{{\alpha }_t})}}\\ {\tilde{\mu }}_t(x_t,x_0)& =\frac{\sqrt{{\alpha }_t}\cdot (1-\overline{{\alpha }_{t-1}})}{1-\overline{{\alpha }_t}}\cdot x_t+\frac{{\beta }_t\cdot \sqrt{\overline{{\alpha }_{t-1}}}}{1-\overline{{\alpha }_t}}\cdot x_0\end{array}\end{array}$$
在DDIM《Denoising Diffusion Implicit Models》中对符号进行了重新定义。具体来说使用${\alpha }_t$替换掉了${\bar{\alpha }}_t$，而在DDPM当中
$$\begin{array}{c}\begin{array}{r}{\bar{\alpha }}_t=\prod _0^t{\alpha }_i\end{array}\end{array}$$
因此，在DDIM中会发生一些变化，例如${\beta }_t$的改变如公式（3）所示。
$$\begin{array}{c}\begin{array}{rl}{\beta }_t& =1-{\alpha }_t(DDPM)\\ & =1-\frac{{\alpha }_t}{{\alpha }_{t-1}}(DDIM)\end{array}\end{array}$$
前向加噪过程中的$q(x_{t-1}|x_t,x_0)$分布的方差和均值分别如公式（4）和（5）所示。
$$\begin{array}{c}\begin{array}{rl}{\sigma }_t^2& =\frac{1-{\bar{\alpha }}_{t-1}}{1-\overline{{\alpha }_t}}\cdot {\beta }_t(DDPM)\\ & =\frac{1-{\alpha }_{t-1}}{1-{\alpha }_t}\cdot (1-\frac{{\alpha }_t}{{\alpha }_{t-1}})(DDIM)\end{array}\end{array}$$
$$\begin{array}{c}\begin{array}{rl}{\tilde{\mu }}_t(x_t,x_0)& =\frac{\sqrt{{\alpha }_t}\cdot (1-{\bar{\alpha }}_{t-1})}{1-\overline{{\alpha }_t}}\cdot x_t+\frac{{\beta }_t\cdot \sqrt{{\bar{\alpha }}_{t-1}}}{1-\overline{{\alpha }_t}}\cdot x_0（DDPM）\\ & =\frac{\sqrt{{\alpha }_t}\cdot (1-{\alpha }_{t-1})}{\sqrt{{\alpha }_{t-1}}\cdot (1-{\alpha }_t)}\cdot x_t+(1-\frac{{\alpha }_t}{{\alpha }_{t-1}})\cdot \frac{\sqrt{{\alpha }_{t-1}}}{1-{\alpha }_t}\cdot x_0(DDIM)\\ & =\sqrt{\frac{{\alpha }_t\cdot (1-{\alpha }_{t-1}{)}^2}{{\alpha }_{t-1}\cdot (1-{\alpha }_t{)}^2}}\cdot x_t+\frac{{\alpha }_{t-1}-{\alpha }_t}{{\alpha }_{t-1}}\cdot \frac{\sqrt{{\alpha }_{t-1}}}{1-{\alpha }_t}\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}}{1-{\alpha }_t}\cdot \frac{{\alpha }_t-{\alpha }_t\cdot {\alpha }_{t-1}}{{\alpha }_{t-1}-{\alpha }_{t-1}\cdot {\alpha }_t}}\cdot x_t+\frac{{\alpha }_{t-1}-{\alpha }_t}{\sqrt{{\alpha }_{t-1}}\cdot (1-{\alpha }_t)}\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}}{1-{\alpha }_t}\cdot \frac{{\alpha }_t+{\alpha }_{t-1}-{\alpha }_{t-1}-{\alpha }_t\cdot {\alpha }_{t-1}}{{\alpha }_{t-1}-{\alpha }_{t-1}\cdot {\alpha }_t}}\cdot x_t+\frac{{\alpha }_{t-1}-{\alpha }_t\cdot {\alpha }_{t-1}+{\alpha }_t\cdot {\alpha }_{t-1}-{\alpha }_t}{\sqrt{{\alpha }_{t-1}}\cdot (1-{\alpha }_t)}\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}}{1-{\alpha }_t}\cdot (1+\frac{{\alpha }_t-{\alpha }_{t-1}}{{\alpha }_{t-1}-{\alpha }_{t-1}\cdot {\alpha }_t})}\cdot x_t+\frac{{\alpha }_{t-1}\cdot (1-{\alpha }_t)-{\alpha }_t\cdot (1-{\alpha }_{t-1})}{\sqrt{{\alpha }_{t-1}}\cdot (1-{\alpha }_t)}\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}}{1-{\alpha }_t}\cdot (1-\frac{{\alpha }_{t-1}-{\alpha }_t}{{\alpha }_{t-1}-{\alpha }_{t-1}\cdot {\alpha }_t})}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{{\alpha }_t\cdot (1-{\alpha }_{t-1})}{\sqrt{{\alpha }_{t-1}}\cdot (1-{\alpha }_t)}]\cdot x_0\\ & =\sqrt{\frac{1}{1-{\alpha }_t}\cdot (1-{\alpha }_{t-1}-\frac{({\alpha }_{t-1}-{\alpha }_t)\cdot (1-{\alpha }_{t-1})}{{\alpha }_{t-1}-{\alpha }_{t-1}\cdot {\alpha }_t})}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{{\alpha }_t^2\cdot (1-{\alpha }_{t-1}{)}^2}}{\sqrt{{\alpha }_{t-1}}\cdot (1-{\alpha }_t)}]\cdot x_0\\ & =\sqrt{\frac{1}{1-{\alpha }_t}\cdot (1-{\alpha }_{t-1}-\underset{={\sigma }_t^2}{\underbrace{\frac{({\alpha }_{t-1}-{\alpha }_t)\cdot (1-{\alpha }_{t-1})}{{\alpha }_{t-1}\cdot (1-{\alpha }_t)}}})}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{{\alpha }_t\cdot (1-{\alpha }_{t-1})\cdot ({\alpha }_t-{\alpha }_t\cdot {\alpha }_{t-1})}}{\sqrt{{\alpha }_{t-1}}\cdot (1-{\alpha }_t)}]\cdot x_0\\ & =\sqrt{\frac{1}{1-{\alpha }_t}\cdot (1-{\alpha }_{t-1}-{\sigma }_t^2)}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{{\alpha }_t\cdot (1-{\alpha }_{t-1})\cdot ({\alpha }_t+{\alpha }_{t-1}-{\alpha }_{t-1}-{\alpha }_t\cdot {\alpha }_{t-1})}}{\sqrt{{\alpha }_{t-1}\cdot (1-{\alpha }_t)}\cdot (\sqrt{1-{\alpha }_t})}]\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{1-{\alpha }_{t-1}}}{\sqrt{1-{\alpha }_t}}\cdot \frac{\sqrt{{\alpha }_t\cdot ({\alpha }_t-{\alpha }_{t-1}+{\alpha }_{t-1}\cdot (1-{\alpha }_t))}}{\sqrt{{\alpha }_{t-1}\cdot (1-{\alpha }_t)}}]\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{1-{\alpha }_{t-1}}}{\sqrt{1-{\alpha }_t}}\cdot \sqrt{\frac{{\alpha }_t\cdot ({\alpha }_t-{\alpha }_{t-1}+{\alpha }_{t-1}\cdot (1-{\alpha }_t))}{{\alpha }_{t-1}\cdot (1-{\alpha }_t)}}]\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{1-{\alpha }_{t-1}}}{\sqrt{1-{\alpha }_t}}\cdot \sqrt{{\alpha }_t\cdot (1+\frac{{\alpha }_t-{\alpha }_{t-1}}{{\alpha }_{t-1}\cdot (1-{\alpha }_t)}})]\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{1}{\sqrt{1-{\alpha }_t}}\cdot \sqrt{(1-{\alpha }_{t-1})\cdot {\alpha }_t\cdot (1-\frac{{\alpha }_{t-1}-{\alpha }_t}{{\alpha }_{t-1}\cdot (1-{\alpha }_t)}})]\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{1}{\sqrt{1-{\alpha }_t}}\cdot \sqrt{{\alpha }_t\cdot (1-{\alpha }_{t-1}-\underset{{\sigma }_t^2}{\underbrace{\frac{({\alpha }_{t-1}-{\alpha }_t)\cdot (1-{\alpha }_{t-1})}{{\alpha }_{t-1}\cdot (1-{\alpha }_t)}}}})]\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{1}{\sqrt{1-{\alpha }_t}}\cdot \sqrt{{\alpha }_t\cdot (1-{\alpha }_{t-1}-{\sigma }_t^2})]\cdot x_0\\ & =\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{{\alpha }_t\cdot (1-{\alpha }_{t-1}-{\sigma }_t^2})}{\sqrt{1-{\alpha }_t}}]\cdot x_0\end{array}\end{array}$$

因此，前向传播过程中的$q(x_{t-1}|x_t,x_0)\sim N(x_{t-1};\sqrt{\frac{1-{\alpha }_{t-1}-{\sigma }_t^2}{1-{\alpha }_t}}\cdot x_t+[\sqrt{{\alpha }_{t-1}}-\frac{\sqrt{{\alpha }_t\cdot (1-{\alpha }_{t-1}-{\sigma }_t^2})}{\sqrt{1-{\alpha }_t}}]\cdot x_0,{\sigma }_t^{2}I)$