diffusion models笔记

ELBO of VDM

Understanding^[1] 中讲 variational diffusion models（VDM）的 evidence lower bound（ELBO）推导时，(53) 式有一个容易引起误会的记号： … = E q ( x 1 : T ∣ x 0 ) [ log ⁡ p ( x T ) p θ ( x 0 ∣ x 1 ) q ( x 1 ∣ x 0 ) + log ⁡ ∏ t = 2 T p θ ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) q ( x t ∣ x 0 ) q ( x t − 1 ∣ x 0 ) ] ( 53 ) = E q ( x 1 : T ∣ x 0 ) [ log ⁡ p ( x T ) p θ ( x 0 ∣ x 1 ) q ( x 1 ∣ x 0 ) + log ⁡ q ( x 1 ∣ x 0 ) q ( x T ∣ x 0 ) + log ⁡ ∏ t = 2 T p θ ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] ( 54 ) \begin{aligned} \dots &= \mathbb{E}_{q\left(\boldsymbol{x}_{1: T} \mid \boldsymbol{x}_0\right)}\left[\log \frac{p\left(\boldsymbol{x}_T\right) p_{\boldsymbol{\theta}}\left(\boldsymbol{x}_0 \mid \boldsymbol{x}_1\right)}{q\left(\boldsymbol{x}_1 \mid \boldsymbol{x}_0\right)}+\log \prod_{t=2}^T \frac{p_{\boldsymbol{\theta}}\left(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t\right)}{\frac{q\left(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t, \boldsymbol{x}_0\right) \cancel{q\left(\boldsymbol{x}_t \mid \boldsymbol{x}_0\right)}}{\cancel{q\left(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_0\right)}} }\right] & (53) \\ &= \mathbb{E}_{q\left(\boldsymbol{x}_{1: T} \mid \boldsymbol{x}_0\right)}\left[\log \frac{p\left(\boldsymbol{x}_T\right) p_{\boldsymbol{\theta}}\left(\boldsymbol{x}_0 \mid \boldsymbol{x}_1\right)}{\cancel{q\left(\boldsymbol{x}_1 \mid \boldsymbol{x}_0\right)}}+\log \frac{\cancel{q\left(\boldsymbol{x}_1 \mid \boldsymbol{x}_0\right)}}{q\left(\boldsymbol{x}_T \mid \boldsymbol{x}_0\right)}+\log \prod_{t=2}^T \frac{p_{\boldsymbol{\theta}}\left(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t\right)}{q\left(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t, \boldsymbol{x}_0\right)}\right] &(54) \end{aligned} …​=Eq(x1:T​∣x0​)​ ​logq(x1​∣x0​)p(xT​)pθ​(x0​∣x1​)​+logt=2∏T​q(xt−1​∣x0​) ​q(xt−1​∣xt​,x0​)q(xt​∣x0​) ​​pθ​(xt−1​∣xt​)​ ​=Eq(x1:T​∣x0​)​[logq(x1​∣x0​) ​p(xT​)pθ​(x0​∣x1​)​+logq(xT​∣x0​)q(x1​∣x0​) ​​+logt=2∏T​q(xt−1​∣xt​,x0​)pθ​(xt−1​∣xt​)​]​(53)(54)​

其中 (53) 式第二项看起来像是消项，但显然两者并不相等！而明显 (54) 中第二项显然非零，却不知从何冒出来。听 [2] 讲到「拆开」、看 [3] 的评论问答才知道：这个推导是将 (53) 式「划掉」的部分拿出来单独放在一项 $\log$ 中，而这一项内部可以消项，消剩的就是 (54) 式中的第二项，即：$$\begin{array}{rl}\text{(53)第二项}& =\log \prod _{t=2}^T\frac{p_{\theta }\left(x_{t-1}\mid x_t\right)}{\frac{q\left(x_{t-1}\mid x_t,x_0\right)q\left(x_t\mid x_0\right)}{q\left(x_{t-1}\mid x_0\right)}}\\ & =\log \prod _{t=2}^T\left[\frac{p_{\theta }\left(x_{t-1}\mid x_t\right)}{q\left(x_{t-1}\mid x_t,x_0\right)}\cdot \frac{q\left(x_{t-1}\mid x_0\right)}{q\left(x_t\mid x_0\right)}\right]\\ & =\underset{\text{(54) 第三项}}{\log \prod _{t=2}^T\frac{p_{\theta }\left(x_{t-1}\mid x_t\right)}{q\left(x_{t-1}\mid x_t,x_0\right)}}+\log \prod _{t=2}^T\frac{q\left(x_{t-1}\mid x_0\right)}{q\left(x_t\mid x_0\right)}\\ & =\log \left[\frac{q\left(x_1\mid x_0\right)}{q\left(x_2\mid x_0\right)}\times \frac{q\left(x_2\mid x_0\right)}{q\left(x_3\mid x_0\right)}\times \cdots \times \frac{q\left(x_{T-1}\mid x_0\right)}{q\left(x_T\mid x_0\right)}\right]+\text{(54) 第三项}\\ & =\underset{\text{(54) 第二项}}{\log \frac{q\left(x_1\mid x_0\right)}{q\left(x_T\mid x_0\right)}}+\text{(54) 第三项}\end{array}$$

DDIM

DDIM^[7] 在其第 2 节 background 回顾 DDPM^[5] 时，(3) 式 $q(x_t|x_{t-1})$ 的形式与 Understanding^[1] 的 (31) 式、DDPM 的 (2) 式都不同，但对照其 (3) 式下面的公式（无标号那条） 与 Understanding 的 (70)、DDPM 的 (4) 可知：DDIM 中的 ${\alpha }_t$ 其实对应 Understanding / DDPM 中的 ${\bar{\alpha }}_t$。其 appendix C.2 也有明确讲到这点。

References

1. Understanding Diffusion Models: A Unified Perspective - arXiv, blog（abbr: Understanding）
2. Diffusion model (3) 概述：VDM相关公式推导
3. [論文導讀] Understanding Diffusion Models: A Unified Perspective 擴散模型的數學原理
4. (ICML 2015) Deep Unsupervised Learning using Nonequilibrium Thermodynamics
5. (NIPS 2020) Denoising Diffusion Probabilistic Models（DDPM）
6. (ICML 2021) Improved Denoising Diffusion Probabilistic Models（Improved DDPM）
7. (ICLR 2021) Denoising Diffusion Implicit Models（DDIM）