参考文献：
什么是扩散模型?
Generative Modeling by Estimating Gradients of the Data Distribution
Understanding Diffusion Models: A Unified Perspective
给定来自感兴趣分布的观察到的样本x x x p ( x ) p(x) p ( x )
基于似然的模型 ：通过（近似）最大似然直接学习分布的概率密度（或质量）函数。典型的基于似然的模型包括autoregressive models、flow-based models、energy-based models（EBM）和variational auto-encoders（VAE）。隐式生成模型 ：其中概率分布由其采样过程的模型隐式表示。最突出的例子是生成对抗网络（GAN），其中通过用生成器转换随机高斯向量来合成来自数据分布的新样本。
VAE（Variational Auto-Encoder）地基本思想是首先采样 z ∼ p ( z ) z\sim p(z) z ∼ p ( z ) z z z x ∼ p θ ( x ∣ z ) x\sim p_\theta(x|z) x ∼ p θ ( x ∣ z ) p θ ( x ∣ z ) p_\theta(x|z) p θ ( x ∣ z ) z z z x x x z z z p ( z ) = N ( 0 , I ) p(z)=\mathcal{N}(0,I) p ( z ) = N ( 0 , I ) p θ ( x ∣ z ) p_\theta(x|z) p θ ( x ∣ z ) x x x p ( x ) p(x) p ( x )
在数学上，我们可以把我们观察到的潜在变量和数据想象成一个联合分布 p ( x , z ) p(x, z) p ( x , z ) p ( x ) p(x) p ( x )
将潜在变量 z z z
p ( x ) = ∫ p ( x , z ) d z p(x) =\int p(x, z) dz
p ( x ) = ∫ p ( x , z ) d z
使用链式规则
p ( x ) = p ( x , z ) p ( z ∣ x ) p(x)=\frac {p(x,z)}{p(z| x)}
p ( x ) = p ( z ∣ x ) p ( x , z )
训练VAE要基于最大似然 log p ( x ) \log p(x) log p ( x ) p ( x ) p(x) p ( x ) z z z p ( z ∣ x ) p(z| x) p ( z ∣ x ) q ϕ ( z ∣ x ) q_\phi(z | x) q ϕ ( z ∣ x )
log p ( x ) = log ∫ p ( x , z ) d z = log ∫ p ( x , z ) q ϕ ( z ∣ x ) q ϕ ( z ∣ x ) d z = log E q ϕ ( z ∣ x ) [ p ( x , z ) q ϕ ( z ∣ x ) ] ≥ E q ϕ ( z ∣ x ) [ log p ( x , z ) q ϕ ( z ∣ x ) ] = E q ϕ ( z ∣ x ) [ log p θ ( x ∣ z ) p ( z ) q ϕ ( z ∣ x ) ] = E q ϕ ( z ∣ x ) [ log p θ ( x ∣ z ) ] + E q ϕ ( z ∣ x ) [ log p ( z ) q ϕ ( z ∣ x ) ] = E q ϕ ( z ∣ x ) [ log p θ ( x ∣ z ) ] ⏟ reconstruction term − D K L ( q ϕ ( z ∣ x ) ∥ p ( z ) ) ⏟ prior matching term \begin{aligned}
\log p(x) & =\log \int p(x, z) d z \\
& =\log \int \frac{p(x, z) q_\phi(z | x)}{q_\phi(z | x)} d z \\
& =\log \mathbb{E}_{q_\phi(z | x)}\left[\frac{p(x, z)}{q_\phi(z | x)}\right] \\
& \geq \mathbb{E}_{q_\phi(z | x)}\left[\log \frac{p(x, z)}{q_\phi(z | x)}\right] \\
& =\mathbb{E}_{q_\phi(z | x)}\left[\log \frac{p_{\theta}(x | z) p(z)}{q_\phi(z | x)}\right] \\
& =\mathbb{E}_{q_\phi(z | x)}\left[\log p_\theta(x | z)\right]+\mathbb{E}_{q_\phi(z | x)}\left[\log \frac{p(z)}{q_\phi(z | x)}\right] \\
& =\underbrace{\mathbb{E}_{q_\phi(z | x)}\left[\log p_\theta(x | z)\right]}_{\text {reconstruction term }}-\underbrace{D_{\mathrm{KL}}\left(q_\phi(z | x) \| p(z)\right)}_{\text {prior matching term }}
\end{aligned}
log p ( x ) = log ∫ p ( x , z ) d z = log ∫ q ϕ ( z ∣ x ) p ( x , z ) q ϕ ( z ∣ x ) d z = log E q ϕ ( z ∣ x ) [ q ϕ ( z ∣ x ) p ( x , z ) ] ≥ E q ϕ ( z ∣ x ) [ log q ϕ ( z ∣ x ) p ( x , z ) ] = E q ϕ ( z ∣ x ) [ log q ϕ ( z ∣ x ) p θ ( x ∣ z ) p ( z ) ] = E q ϕ ( z ∣ x ) [ log p θ ( x ∣ z ) ] + E q ϕ ( z ∣ x ) [ log q ϕ ( z ∣ x ) p ( z ) ] = reconstruction term E q ϕ ( z ∣ x ) [ log p θ ( x ∣ z ) ] − prior matching term D K L ( q ϕ ( z ∣ x ) ∥ p ( z ) )
训练VAE本质上就是最大化不等式的右边，这需要同时训练 p θ ( x ∣ z ) p_\theta(x|z) p θ ( x ∣ z ) q ϕ ( z ∣ x ) q_\phi(z|x) q ϕ ( z ∣ x ) q ϕ ( z ∣ x ) q_\phi(z|x) q ϕ ( z ∣ x ) q ϕ ( z ∣ x ) q_\phi(z|x) q ϕ ( z ∣ x ) log p ( x ) \log p(x) log p ( x )
相对于VAE，Flow-based Model和GAN只需要考虑生成器，用生成器把采样的高斯噪声 z z z p θ ( x ) p_\theta(x) p θ ( x )
Flow-based Model 要求模型是可逆函数
GAN 需要额外训练判别器，导致训练时间很长，浪费了一部分计算资源
为了克服这些问题，Diffusion Model横空出世。回想VAE，最大的难题是变分后验很难选择，这是因为我们首先定义了Decoder（条件分布 p θ ( x ∣ z ) p_\theta(x|z) p θ ( x ∣ z ) q ϕ ( z ∣ x ) q_\phi(z|x) q ϕ ( z ∣ x )
那么如何描述这个简单的过程呢？他的本质其实就是从一个数据分布转换成另一个分布。那么根据随机过程，如果我们可以构造适当的马尔科夫链，使得不管从什么分布出发，沿着马尔可夫链一直采样下去最终可以得到我们想要的平稳分布（stationary distribution），这也是马尔可夫链蒙特卡洛（MCMC）算法的核心。Diffusion的核心就是两方面：如何采样到高斯分布？（前向过程）如何拟合采样过程？（逆向过程）
扩散概率模型（简写为“扩散模型”）是一个参数化的马尔可夫链（parameterized Markov chain），通过变分推断（variational inference）进行训练，以生成在有限时间内匹配数据的样本。该链的转移是学习的，以便逆转扩散过程，即一个马尔可夫链，该链在采样的相反方向逐渐向数据中添加噪声，直到信号被破坏。当扩散过程由小量高斯噪声组成时，仅需要将采样链转移设置为条件高斯，从而允许使用特别简单的神经网络参数化。简单来说。就是diffusion model定义了一个简单的前向过程，不断地加噪来把真实数据映射到标准高斯；然后又定义一个逆向过程来去噪，并且逆向过程的每一步只需要是一个很简单的高斯分布。
Diffusion model最简单的理解方式是将其视为具有三个关键限制假设的马尔可夫层次变分自编码器（Hierarchical Variational Autoencoders）：
潜在变量维度与数据维度完全相等
在每个时间步骤中，潜在变量编码器的结构不是通过学习得到的，而是预定义为线性高斯模型，即它是围绕前一个时间步骤的输出中心化的高斯分布
潜在变量编码器的高斯参数随时间变化，以使得最终时间步骤T的潜变量分布为标准高斯分布
扩散模型是形式 p θ ( x 0 ) p_{\theta}(\mathbf{x}_0) p θ ( x 0 ) x 1 , … , x T \mathbf{x}_1,\dots,\mathbf{x}_T x 1 , … , x T x 0 ∼ q ( x 0 ) \mathbf{x}_0 \sim q(\mathbf{x}_0) x 0 ∼ q ( x 0 ) p θ ( x 0 : T ) p_{\theta}(\mathbf{x}_{0:T}) p θ ( x 0 : T ) p ( x T ) = N ( x T ; 0 , I ) p(\mathbf{x}_T)=\mathcal{N}(\mathbf{x}_T;0,I) p ( x T ) = N ( x T ; 0 , I )
p θ ( x 0 : T ) : = p ( x T ) ∏ t = 1 T p θ ( x t − 1 ∣ x t ) , p θ ( x t − 1 ∣ x t ) : = N ( x t − 1 ; μ θ ( x t , t ) , Σ θ ( x t , t ) ) p_{\theta}(\mathbf{x}_{0: T}):=p(\mathbf{x}_T) \prod_{t=1}^T p_{\theta}(\mathbf{x}_{t-1} | \mathbf{x}_t), \quad p_{\theta}(\mathbf{x}_{t-1} | \mathbf{x}_t):=\mathcal{N}(\mathbf{x}_{t-1} ; \mu_{\theta}(\mathbf{x}_t, t), \Sigma_{\theta}(\mathbf{x}_t, t))
p θ ( x 0 : T ) : = p ( x T ) t = 1 ∏ T p θ ( x t − 1 ∣ x t ) , p θ ( x t − 1 ∣ x t ) : = N ( x t − 1 ; μ θ ( x t , t ) , Σ θ ( x t , t ) )
根据第二个假设，我们知道编码器中每个潜变量的分布都是围绕其前一个层次潜变量的高斯分布。在每个时间步骤t中，编码器的结构不是通过学习得到的；它被固定为线性高斯模型，其中均值和标准差可以预先设置为超参数或学习为参数。扩散模型与其他类型的潜变量模型的区别在于，其近似后验编码器 q ( x 1 : T ∣ x 0 ) q(\mathbf{x}_{1: T} | \mathbf{x}_0) q ( x 1 : T ∣ x 0 ) β 1 , … , β T \beta_1,\dots,\beta_T β 1 , … , β T
q ( x 1 : T ∣ x 0 ) : = ∏ t = 1 T q ( x t ∣ x t − 1 ) , q ( x t ∣ x t − 1 ) : = N ( x t ; 1 − β t x t − 1 , β t I ) q(\mathbf{x}_{1: T} | \mathbf{x}_0):=\prod_{t=1}^T q(\mathbf{x}_t | \mathbf{x}_{t-1}), \quad q(\mathbf{x}_t | \mathbf{x}_{t-1}):=\mathcal{N}(\mathbf{x}_t ; \sqrt{1-\beta_t} \mathbf{x}_{t-1}, \beta_t I)
q ( x 1 : T ∣ x 0 ) : = t = 1 ∏ T q ( x t ∣ x t − 1 ) , q ( x t ∣ x t − 1 ) : = N ( x t ; 1 − β t x t − 1 , β t I )
前向过程的方差时间表可以通过重新参数化来学习，也可以作为超参数保持不变，并且 p θ ( x t − 1 ∣ x t ) p_{\theta}(\mathbf{x}_{t-1} | \mathbf{x}_t) p θ ( x t − 1 ∣ x t ) β t \beta_t β t t t t x t \mathbf{x}_t x t α t : = 1 − β t \alpha_t:=1-\beta_t α t : = 1 − β t α t ˉ : = ∏ s = 1 t α s \bar{\alpha_t}:=\prod_{s=1}^t \alpha_s α t ˉ : = ∏ s = 1 t α s
q ( x t ∣ x 0 ) : = N ( x t ; α t ˉ x 0 , ( 1 − α t ˉ ) I ) q(\mathbf{x}_t | \mathbf{x}_{0}):=\mathcal{N}(\mathbf{x}_t ; \sqrt{\bar{\alpha_t}} \mathbf{x}_{0}, (1-\bar{\alpha_t}) I)
q ( x t ∣ x 0 ) : = N ( x t ; α t ˉ x 0 , ( 1 − α t ˉ ) I )
根据重参数的表示和高斯分布的特性（两个独立高斯随机变量的和仍然是高斯分布，其中均值是两个均值的和，方差是两个方差的和）就能推导出上面这个公式：
x t = α t x t − 1 + 1 − α t ϵ t − 1 = α t ( α t − 1 x t − 2 + 1 − α t − 1 ϵ t − 2 ) + 1 − α t ϵ t − 1 = α t α t − 1 x t − 2 + α t − α t α t − 1 ϵ t − 2 + 1 − α t ϵ t − 1 = α t α t − 1 x t − 2 + ( α t − α t α t − 1 ) 2 + ( 1 − α t ) 2 ϵ t − 2 = α t α t − 1 x t − 2 + α t − α t α t − 1 + 1 − α t ϵ t − 2 = α t α t − 1 x t − 2 + 1 − α t α t − 1 ϵ t − 2 = … = ∏ i = 1 t α i x 0 + 1 − ∏ i = 1 t α i ϵ 0 = α ˉ t x 0 + 1 − α ˉ t ϵ 0 ∼ N ( x t ; α ˉ t x 0 , ( 1 − α ˉ t ) I ) \begin{aligned}
\mathbf{x}_t& =\sqrt{\alpha_t} \mathbf{x}_{t-1}+\sqrt{1-\alpha_t} \epsilon_{t-1} \\
& =\sqrt{\alpha_t}\left(\sqrt{\alpha_{t-1}} \mathbf{x}_{t-2}+\sqrt{1-\alpha_{t-1}} \epsilon_{t-2}\right)+\sqrt{1-\alpha_t} \epsilon_{t-1} \\
& =\sqrt{\alpha_t \alpha_{t-1}} \mathbf{x}_{t-2}+\sqrt{\alpha_t-\alpha_t \alpha_{t-1}} \epsilon_{t-2}+\sqrt{1-\alpha_t} \epsilon_{t-1} \\
& =\sqrt{\alpha_t \alpha_{t-1}} \mathbf{x}_{t-2}+\sqrt{({\sqrt{\alpha_t-\alpha_t \alpha_{t-1}}})^2+{(\sqrt{1-\alpha_t}})^2} \epsilon_{t-2} \\
& =\sqrt{\alpha_t \alpha_{t-1}} \mathbf{x}_{t-2}+\sqrt{\alpha_t-\alpha_t \alpha_{t-1}+1-\alpha_t} \epsilon_{t-2} \\
& =\sqrt{\alpha_t \alpha_{t-1}} \mathbf{x}_{t-2}+\sqrt{1-\alpha_t \alpha_{t-1}} \epsilon_{t-2} \\
& =\dots \\
& =\sqrt{\prod_{i=1}^t \alpha_i} \mathbf{x}_0+\sqrt{1-\prod_{i=1}^t \alpha_i}\epsilon_0 \\
& =\sqrt{\bar{\alpha}_t} \mathbf{x}_0+\sqrt{1-\bar{\alpha}_t} \epsilon_0 \\
& \sim \mathcal{N}\left(\mathbf{x}_t;\sqrt{\bar{\alpha}_t} \mathbf{x}_0,\left(1-\bar{\alpha}_t\right) I\right) \\
\end{aligned}
x t = α t x t − 1 + 1 − α t ϵ t − 1 = α t ( α t − 1 x t − 2 + 1 − α t − 1 ϵ t − 2 ) + 1 − α t ϵ t − 1 = α t α t − 1 x t − 2 + α t − α t α t − 1 ϵ t − 2 + 1 − α t ϵ t − 1 = α t α t − 1 x t − 2 + ( α t − α t α t − 1 ) 2 + ( 1 − α t ) 2 ϵ t − 2 = α t α t − 1 x t − 2 + α t − α t α t − 1 + 1 − α t ϵ t − 2 = α t α t − 1 x t − 2 + 1 − α t α t − 1 ϵ t − 2 = … = i = 1 ∏ t α i x 0 + 1 − i = 1 ∏ t α i ϵ 0 = α ˉ t x 0 + 1 − α ˉ t ϵ 0 ∼ N ( x t ; α ˉ t x 0 , ( 1 − α ˉ t ) I )
其中 α t ˉ \bar{\alpha_t} α t ˉ t t t β t \beta_t β t lim t → ∞ α ˉ t = 0 \lim_{t\rightarrow \infty}\bar\alpha_t =0 lim t → ∞ α ˉ t = 0
lim t → ∞ q ( x t ∣ x 0 ) = N ( 0 , I ) \lim_{t\rightarrow \infty}q(\mathbf{x}_t|\mathbf{x}_0)=\mathcal{N}(0,I)
t → ∞ lim q ( x t ∣ x 0 ) = N ( 0 , I )
也就是说这个条件分布最终会收敛到一个和 x 0 \mathbf{x}_0 x 0
lim t → ∞ q ( x t ) = N ( 0 , I ) \lim_{t\rightarrow\infty}q(\mathbf{x}_t)=\mathcal{N}(0,I)
t → ∞ lim q ( x t ) = N ( 0 , I )
这样一来，只要我们取一个足够大的终止时刻 N N N x 0 \mathbf{x}_0 x 0 x N \mathbf{x}_N x N q ( x t ∣ x t − 1 ) q(\mathbf{x}_t|\mathbf{x}_{t-1}) q ( x t ∣ x t − 1 ) x t − 1 \mathbf{x}_{t-1} x t − 1 x N \mathbf{x}_N x N
总的来说diffusion有三种理解和优化，他们其实是等价的。s θ s_\theta s θ ϵ θ \epsilon_\theta ϵ θ x θ \mathbf{x}_\theta x θ
Diffusion的训练是通过优化负对数似然的ELBO来完成的：
log p ( x ) = log ∫ p ( x 0 : T ) d x 1 : T = log ∫ p ( x 0 : T ) q ( x 1 : T ∣ x 0 ) q ( x 1 : T ∣ x 0 ) d x 1 : T = log E q ( x 1 : T ∣ x 0 ) [ p ( x 0 : T ) q ( x 1 : T ∣ x 0 ) ] ≥ E q ( x 1 : T ∣ x 0 ) [ log p ( x 0 : T ) q ( x 1 : T ∣ x 0 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) ∏ t = 1 T p θ ( x t − 1 ∣ x t ) ∏ t = 1 T q ( x t ∣ x t − 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 2 T p θ ( x t − 1 ∣ x t ) q ( x T ∣ x T − 1 ) ∏ t = 1 T − 1 q ( x t ∣ x t − 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 1 T − 1 p θ ( x t ∣ x t + 1 ) q ( x T ∣ x T − 1 ) ∏ t = 1 T − 1 q ( x t ∣ x t − 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) p θ ( x 0 ∣ x 1 ) q ( x T ∣ x T − 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log ∏ t = 1 T − 1 p θ ( x t ∣ x t + 1 ) q ( x t ∣ x t − 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) q ( x T ∣ x T − 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ ∑ t = 1 T − 1 log p θ ( x t ∣ x t + 1 ) q ( x t ∣ x t − 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) q ( x T ∣ x T − 1 ) ] + ∑ t = 1 T − 1 E q ( x 1 : T ∣ x 0 ) [ log p θ ( x t ∣ x t + 1 ) q ( x t ∣ x t − 1 ) ] = E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x T − 1 , x T ∣ x 0 ) [ log p ( x T ) q ( x T ∣ x T − 1 ) ] + ∑ t = 1 T − 1 E q ( x t − 1 , x t , x t + 1 ∣ x 0 ) [ log p θ ( x t ∣ x t + 1 ) q ( x t ∣ x t − 1 ) ] = E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] ⏟ reconstruction term − E q ( x T − 1 ∣ x 0 ) [ D K L ( q ( x T ∣ x T − 1 ) ∥ p ( x T ) ) ] ⏟ prior matching term − ∑ t = 1 T − 1 E q ( x t − 1 , 1 x t + 1 ∣ x 0 ) [ D K L ( q ( x t ∣ x t − 1 ) ∥ p θ ( x t ∣ x t + 1 ) ) ] ⏟ consistency term \begin{aligned}
\log p(x)&=\log \int p\left(\mathbf{x}_{0: T}\right) d \mathbf{x}_{1: T} \\
& =\log \int \frac{p\left(\mathbf{x}_{0: T}\right) q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)} d \mathbf{x}_{1: T} \\
& =\log \mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\frac{p\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\right] \\
& \geq \mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) \prod_{t=1}^T p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)}{\prod_{t=1}^T q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right) \prod_{t=2}^T p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)}{q\left(\mathbf{x}_T | \mathbf{x}_{T-1}\right) \prod_{t=1}^{T-1} q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right) \prod_{t=1}^{T-1} p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{q\left(\mathbf{x}_T | \mathbf{x}_{T-1}\right) \prod_{t=1}^{T-1} q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)}{q\left(\mathbf{x}_T | \mathbf{x}_{T-1}\right)}\right]+\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \prod_{t=1}^{T-1} \frac{p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)\right]+\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right)}{q\left(\mathbf{x}_T | \mathbf{x}_{T-1}\right)}\right]+\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\sum_{t=1}^{T-1} \log \frac{p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)\right]+\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right)}{q\left(\mathbf{x}_T | \mathbf{x}_{T-1}\right)}\right]+\sum_{t=1}^{T-1} \mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_1 | \mathbf{x}_0\right)}\left[\log p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)\right]+\mathbb{E}_{q\left(\mathbf{x}_{T-1}, \mathbf{x}_T | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right)}{q\left(\mathbf{x}_T | \mathbf{x}_{T-1}\right)}\right]+\sum_{t=1}^{T-1} \mathbb{E}_{q\left(\mathbf{x}_{t-1}, \mathbf{x}_t, \mathbf{x}_{t+1} | \mathbf{x}_0\right)}\left[\log \frac{p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\underbrace{\mathbb{E}_{q\left(\mathbf{x}_1 | \mathbf{x}_0\right)}\left[\log p_\theta\left(\mathbf{x}_0 | \mathbf{x}_1\right)\right]}_{\text {reconstruction term }}-\underbrace{\mathbb{E}_{q\left(\mathbf{x}_{T-1} | \mathbf{x}_0\right)}\left[D_{\mathrm{KL}}\left(q\left(\mathbf{x}_T | \mathbf{x}_{T-1}\right) \| p\left(\mathbf{x}_T\right)\right)\right]}_{\text {prior matching term }} \\
& \quad \quad -\sum_{t=1}^{T-1} \underbrace{\mathbb{E}_{q\left(\mathbf{x}_{t-1,1} \mathbf{x}_{t+1} | \mathbf{x}_0\right)}\left[D_{\mathrm{KL}}\left(q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right) \| p_\theta\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)\right)\right]}_{\text {consistency term }}
\end{aligned}
log p ( x ) = log ∫ p ( x 0 : T ) d x 1 : T = log ∫ q ( x 1 : T ∣ x 0 ) p ( x 0 : T ) q ( x 1 : T ∣ x 0 ) d x 1 : T = log E q ( x 1 : T ∣ x 0 ) [ q ( x 1 : T ∣ x 0 ) p ( x 0 : T ) ] ≥ E q ( x 1 : T ∣ x 0 ) [ log q ( x 1 : T ∣ x 0 ) p ( x 0 : T ) ] = E q ( x 1 : T ∣ x 0 ) [ log ∏ t = 1 T q ( x t ∣ x t − 1 ) p ( x T ) ∏ t = 1 T p θ ( x t − 1 ∣ x t ) ] = E q ( x 1 : T ∣ x 0 ) [ log q ( x T ∣ x T − 1 ) ∏ t = 1 T − 1 q ( x t ∣ x t − 1 ) p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 2 T p θ ( x t − 1 ∣ x t ) ] = E q ( x 1 : T ∣ x 0 ) [ log q ( x T ∣ x T − 1 ) ∏ t = 1 T − 1 q ( x t ∣ x t − 1 ) p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 1 T − 1 p θ ( x t ∣ x t + 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log q ( x T ∣ x T − 1 ) p ( x T ) p θ ( x 0 ∣ x 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log t = 1 ∏ T − 1 q ( x t ∣ x t − 1 ) p θ ( x t ∣ x t + 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log q ( x T ∣ x T − 1 ) p ( x T ) ] + E q ( x 1 : T ∣ x 0 ) [ t = 1 ∑ T − 1 log q ( x t ∣ x t − 1 ) p θ ( x t ∣ x t + 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log q ( x T ∣ x T − 1 ) p ( x T ) ] + t = 1 ∑ T − 1 E q ( x 1 : T ∣ x 0 ) [ log q ( x t ∣ x t − 1 ) p θ ( x t ∣ x t + 1 ) ] = E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x T − 1 , x T ∣ x 0 ) [ log q ( x T ∣ x T − 1 ) p ( x T ) ] + t = 1 ∑ T − 1 E q ( x t − 1 , x t , x t + 1 ∣ x 0 ) [ log q ( x t ∣ x t − 1 ) p θ ( x t ∣ x t + 1 ) ] = reconstruction term E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] − prior matching term E q ( x T − 1 ∣ x 0 ) [ D K L ( q ( x T ∣ x T − 1 ) ∥ p ( x T ) ) ] − t = 1 ∑ T − 1 consistency term E q ( x t − 1 , 1 x t + 1 ∣ x 0 ) [ D K L ( q ( x t ∣ x t − 1 ) ∥ p θ ( x t ∣ x t + 1 ) ) ]
ELBO的推导形式可以通过其各个组成部分进行解释：
第一项可以解释为重建项，预测给定第一步潜变量的原始数据样本的对数概率。这个项在普通的VAE中也出现过，并且可以类似地进行训练。
第二项是一个先验匹配项；当最终潜变量分布与高斯先验相匹配时，它被最小化。这个项不需要优化，因为它没有可训练的参数；此外，我们已经假设T足够大，以使最终分布为高斯分布，因此这个项实际上变为零。
第三项是一个一致性项；它努力使x t \mathbf{x}_t x t
在这个推导下，ELBO的所有项都被计算为期望，因此可以使用蒙特卡罗估计进行近似。然而，实际上使用我们刚刚推导的项来优化ELBO可能不是最优的；因为一致性项是在每个时间步骤上对两个随机变量 x t − 1 , x t + 1 \mathbf{x}_{t-1}, \mathbf{x}_{t+1} x t − 1 , x t + 1 T − 1 T-1 T − 1 T T T q ( x t ∣ x t − 1 ) = q ( x t ∣ x t − 1 , x 0 ) q(\mathbf{x}_t|\mathbf{x}_{t-1}) = q(\mathbf{x}_t|\mathbf{x}_{t-1}, \mathbf{x}_0) q ( x t ∣ x t − 1 ) = q ( x t ∣ x t − 1 , x 0 )
q ( x t ∣ x t − 1 , x 0 ) = q ( x t − 1 ∣ x t , x 0 ) q ( x t ∣ x 0 ) q ( x t − 1 ∣ x 0 ) q\left(\mathbf{x}_t | \mathbf{x}_{t-1}, \mathbf{x}_0\right)=\frac{q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right) q\left(\mathbf{x}_t | \mathbf{x}_0\right)}{q\left(\mathbf{x}_{t-1} | \mathbf{x}_0\right)}
q ( x t ∣ x t − 1 , x 0 ) = q ( x t − 1 ∣ x 0 ) q ( x t − 1 ∣ x t , x 0 ) q ( x t ∣ x 0 )
重新推导ELBO，得到如下公式：
log p ( x ) ≥ E q ( x 1 : T ∣ x 0 ) [ log p ( x 0 : T ) q ( x 1 : T ∣ x 0 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) ∏ t = 1 T p θ ( x t − 1 ∣ x t ) ∏ t = 1 T q ( x t ∣ x t − 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 2 T p θ ( x t − 1 ∣ x t ) q ( x 1 ∣ x 0 ) ∏ t = 2 T q ( x t ∣ x t − 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 1 T − 1 p θ ( x t ∣ x t + 1 ) q ( x 1 ∣ x 0 ) ∏ t = 2 T q ( x t ∣ x t − 1 , x 0 ) ] = 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 ∣ x t + 1 ) q ( x t ∣ x t − 1 , x 0 ) ] = 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 ∣ x t + 1 ) q ( x t − 1 ∣ x t , x 0 ) q ( x t ∣ x 0 ) q ( x t − 1 ∣ x 0 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) p θ ( x 0 ∣ x 1 ) q ( x T ∣ x 0 ) + ∑ t = 2 T log p θ ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] = E q ( x 1 : T ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log p ( x T ) q ( x T ∣ x 0 ) ] + ∑ t = 2 T E q ( x 1 : T ∣ x 0 ) [ log p θ ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] = E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x T ∣ x 0 ) [ log p ( x T ) q ( x T ∣ x 0 ) ] + ∑ t = 2 T E q ( x t , x t − 1 ∣ x 0 ) [ log p θ ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] = E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] ⏟ reconstruction term − D K L ( q ( x T ∣ x 0 ) ∥ p ( x T ) ) ⏟ prior matching term − ∑ t = 2 T E q ( x t ∣ x 0 ) [ D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) ] ⏟ denoising matching term \begin{aligned}
\log p(x)
& \geq \mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) \prod_{t=1}^T p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)}{\prod_{t=1}^T q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right) \prod_{t=2}^T p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)}{q\left(\mathbf{x}_1 | \mathbf{x}_{0}\right) \prod_{t=2}^{T} q\left(\mathbf{x}_t | \mathbf{x}_{t-1}\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right) \prod_{t=1}^{T-1} p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{q\left(\mathbf{x}_1 | \mathbf{x}_{0}\right) \prod_{t=2}^{T} q\left(\mathbf{x}_t | \mathbf{x}_{t-1},\mathbf{x}_0\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 | \mathbf{x}_{0}\right)}+\log \prod_{t=2}^{T} \frac{p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{q\left(\mathbf{x}_t | \mathbf{x}_{t-1},\mathbf{x}_0\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 | \mathbf{x}_{0}\right)}+\log \prod_{t=2}^{T} \frac{p_{\theta}\left(\mathbf{x}_t | \mathbf{x}_{t+1}\right)}{\frac{q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right) q\left(\mathbf{x}_t | \mathbf{x}_0\right)}{q\left(\mathbf{x}_{t-1} | \mathbf{x}_0\right)}}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right) p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)}{q\left(\mathbf{x}_T | \mathbf{x}_0\right)}+\sum_{t=2}^T \log \frac{p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)}{q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)\right]+\mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right)}{q\left(\mathbf{x}_T | \mathbf{x}_0\right)}\right]+\sum_{t=2}^T \mathbb{E}_{q\left(\mathbf{x}_{1: T} | \mathbf{x}_0\right)}\left[\log \frac{p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)}{q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right)}\right] \\
& =\mathbb{E}_{q\left(\mathbf{x}_1 | \mathbf{x}_0\right)}\left[\log p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)\right]+\mathbb{E}_{q\left(\mathbf{x}_T | \mathbf{x}_0\right)}\left[\log \frac{p\left(\mathbf{x}_T\right)}{q\left(\mathbf{x}_T | \mathbf{x}_0\right)}\right]+\sum_{t=2}^T \mathbb{E}_{q\left(\mathbf{x}_t, \mathbf{x}_{t-1} | \mathbf{x}_0\right)}\left[\log \frac{p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)}{q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right)}\right] \\
& =\underbrace{\mathbb{E}_{q\left(\mathbf{x}_1 | \mathbf{x}_0\right)}\left[\log p_{\theta}\left(\mathbf{x}_0 | \mathbf{x}_1\right)\right]}_{\text {reconstruction term }}-\underbrace{D_{\mathrm{KL}}\left(q\left(\mathbf{x}_T | \mathbf{x}_0\right) \| p\left(\mathbf{x}_T\right)\right)}_{\text {prior matching term }} \\
& \quad \quad -\sum_{t=2}^T \underbrace{\mathbb{E}_{q\left(\mathbf{x}_t | \mathbf{x}_0\right)}\left[D_{\mathrm{KL}}\left(q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right) \| p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)\right)\right]}_{\text {denoising matching term }} \\
&
\end{aligned}
log p ( x ) ≥ E q ( x 1 : T ∣ x 0 ) [ log q ( x 1 : T ∣ x 0 ) p ( x 0 : T ) ] = E q ( x 1 : T ∣ x 0 ) [ log ∏ t = 1 T q ( x t ∣ x t − 1 ) p ( x T ) ∏ t = 1 T p θ ( x t − 1 ∣ x t ) ] = E q ( x 1 : T ∣ x 0 ) [ log q ( x 1 ∣ x 0 ) ∏ t = 2 T q ( x t ∣ x t − 1 ) p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 2 T p θ ( x t − 1 ∣ x t ) ] = E q ( x 1 : T ∣ x 0 ) [ log q ( x 1 ∣ x 0 ) ∏ t = 2 T q ( x t ∣ x t − 1 , x 0 ) p ( x T ) p θ ( x 0 ∣ x 1 ) ∏ t = 1 T − 1 p θ ( x t ∣ x t + 1 ) ] = E q ( x 1 : T ∣ x 0 ) [ log q ( x 1 ∣ x 0 ) p ( x T ) p θ ( x 0 ∣ x 1 ) + log t = 2 ∏ T q ( x t ∣ x t − 1 , x 0 ) p θ ( x t ∣ x t + 1 ) ] = E q ( x 1 : T ∣ x 0 ) ⎣ ⎡ log q ( x 1 ∣ x 0 ) p ( x T ) p θ ( x 0 ∣ x 1 ) + log t = 2 ∏ T q ( x t − 1 ∣ x 0 ) q ( x t − 1 ∣ x t , x 0 ) q ( x t ∣ x 0 ) p θ ( x t ∣ x t + 1 ) ⎦ ⎤ = E q ( x 1 : T ∣ x 0 ) [ log q ( x T ∣ x 0 ) p ( x T ) p θ ( x 0 ∣ x 1 ) + t = 2 ∑ T log q ( x t − 1 ∣ x t , x 0 ) p θ ( x t − 1 ∣ x t ) ] = E q ( x 1 : T ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x 1 : T ∣ x 0 ) [ log q ( x T ∣ x 0 ) p ( x T ) ] + t = 2 ∑ T E q ( x 1 : T ∣ x 0 ) [ log q ( x t − 1 ∣ x t , x 0 ) p θ ( x t − 1 ∣ x t ) ] = E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] + E q ( x T ∣ x 0 ) [ log q ( x T ∣ x 0 ) p ( x T ) ] + t = 2 ∑ T E q ( x t , x t − 1 ∣ x 0 ) [ log q ( x t − 1 ∣ x t , x 0 ) p θ ( x t − 1 ∣ x t ) ] = reconstruction term E q ( x 1 ∣ x 0 ) [ log p θ ( x 0 ∣ x 1 ) ] − prior matching term D K L ( q ( x T ∣ x 0 ) ∥ p ( x T ) ) − t = 2 ∑ T denoising matching term E q ( x t ∣ x 0 ) [ D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) ]
我们成功地推导出了ELBO的解释，可以用更低的方差进行估计，因为每个项最多只计算一个随机变量的期望。这个公式也有一个优雅的解释，在检查每个单独的项时可以看出：
第一项可以解释为重建项 L 0 L_0 L 0
第二项表示最终噪声输入的分布与标准高斯先验的接近程度 L T L_T L T
第三项是一个去噪匹配项 L t − 1 L_{t-1} L t − 1 p θ ( x t − 1 ∣ x t ) p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_t) p θ ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) q(\mathbf{x}_{t-1}|\mathbf{x}_t, \mathbf{x}_0) q ( x t − 1 ∣ x t , x 0 ) q ( x t − 1 ∣ x t , x 0 ) q(\mathbf{x}_{t-1}|\mathbf{x}_t, \mathbf{x}_0) q ( x t − 1 ∣ x t , x 0 ) x t \mathbf{x}_t x t x 0 \mathbf{x}_0 x 0
在这个ELBO的推导中，大部分的优化成本在求和项中，接下来推导这个求和项里面的每一个项。先计算下面的后验概率，这个概率与神经网络无关，和设定的 β t \beta_t β t
q ( x t − 1 ∣ x t , x 0 ) = N ( x t − 1 ; μ ~ t ( x t , x 0 ) , β ~ t I ) q(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0)=\mathcal{N}(\mathbf{x}_{t-1} ; \tilde{\mu}_t(\mathbf{x}_t, \mathbf{x}_0), \tilde{\beta}_t I)
q ( x t − 1 ∣ x t , x 0 ) = N ( x t − 1 ; μ ~ t ( x t , x 0 ) , β ~ t I )
μ ~ t ( x t , x 0 ) : = α ˉ t − 1 β t 1 − α ˉ t x 0 + α t ( 1 − α ˉ t − 1 ) 1 − α ˉ t x t a n d β ~ t : = 1 − α ˉ t − 1 1 − α ˉ t β t \quad \tilde{\mu}_t(\mathbf{x}_t, \mathbf{x}_0):=\frac{\sqrt{\bar{\alpha}_{t-1}} \beta_t}{1-\bar{\alpha}_t} \mathbf{x}_0+\frac{\sqrt{\alpha_t}(1-\bar{\alpha}_{t-1})}{1-\bar{\alpha}_t} \mathbf{x}_t \quad and \quad \tilde{\beta}_t:=\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_t} \beta_t
μ ~ t ( x t , x 0 ) : = 1 − α ˉ t α ˉ t − 1 β t x 0 + 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t a n d β ~ t : = 1 − α ˉ t 1 − α ˉ t − 1 β t
推导如下：
q ( x t − 1 ∣ x t , x 0 ) = q ( x t ∣ x t − 1 , x 0 ) q ( x t − 1 ∣ x 0 ) q ( x t ∣ x 0 ) = N ( x t ; α t x t − 1 , ( 1 − α t ) I ) N ( x t − 1 ; α ˉ t − 1 x 0 , ( 1 − α ˉ t − 1 ) I ) N ( x t ; α ˉ t x 0 , ( 1 − α ˉ t ) I ) ∝ exp { − [ ( x t − α t x t − 1 ) 2 2 ( 1 − α t ) + ( x t − 1 − α t − 1 x 0 ) 2 2 ( 1 − α ˉ t − 1 ) − ( x t − α ˉ t x 0 ) 2 2 ( 1 − α ˉ t ) ] } = exp { − 1 2 [ ( x t − α t x t − 1 ) 2 1 − α t + ( x t − 1 − α t − 1 x 0 ) 2 1 − α ˉ t − 1 − ( x t − α ˉ t x 0 ) 2 1 − α ˉ t ] } = exp { − 1 2 [ ( − 2 α t x t x t − 1 + α t x t − 1 2 ) 1 − α t + ( x t − 1 2 − 2 α ˉ t − 1 x t − 1 x 0 ) 1 − α ˉ t − 1 + C ( x t , x 0 ) ] } ∝ exp { − 1 2 [ − 2 α t x t x t − 1 1 − α t + α t x t − 1 2 1 − α t + x t − 1 2 1 − α ˉ t − 1 − 2 α ˉ t − 1 x t − 1 x 0 1 − α ˉ t − 1 ] } = exp { − 1 2 [ ( α t 1 − α t + 1 1 − α ˉ t − 1 ) x t − 1 2 − 2 ( α t x t 1 − α t + α t − 1 x 0 1 − α ˉ t − 1 ) x t − 1 ] } = exp { − 1 2 [ α t ( 1 − α ˉ t − 1 ) + 1 − α t ( 1 − α t ) ( 1 − α ˉ t − 1 ) x t − 1 2 − 2 ( α t x t 1 − α t + α t − 1 x 0 1 − α ˉ t − 1 ) x t − 1 ] } = exp { − 1 2 [ α t − α ˉ t + 1 − α t ( 1 − α t ) ( 1 − α ˉ t − 1 ) x t − 1 2 − 2 ( α t x t 1 − α t + α t − 1 x 0 1 − α ˉ t − 1 ) x t − 1 ] } = exp { − 1 2 [ 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) x t − 1 2 − 2 ( α t x t 1 − α t + α t − 1 x 0 1 − α ˉ t − 1 ) x t − 1 ] } = exp { − 1 2 ( 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) ) [ x t − 1 2 − 2 ( α t x t 1 − α t + α t − 1 x 0 1 − α ˉ t − 1 ) 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) x t − 1 ] } = exp { − 1 2 ( 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) ) [ x t − 1 2 − 2 ( α t x t 1 − α t + α t − 1 x 0 1 − α ˉ t − 1 ) ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t x t − 1 ] } = exp { − 1 2 ( 1 ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t ) [ x t − 1 2 − 2 α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 1 − α ˉ t x t − 1 ] } ∝ N ( x t − 1 ; α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 1 − α ˉ t ⏟ μ q ( x t , x 0 ) , ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t I ) ⏟ Σ q ( t ) \begin{aligned}
& q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right)=\frac{q\left(\mathbf{x}_t | \mathbf{x}_{t-1}, \mathbf{x}_0\right) q\left(\mathbf{x}_{t-1} | \mathbf{x}_0\right)}{q\left(\mathbf{x}_t | \mathbf{x}_0\right)} \\
& =\frac{\mathcal{N}\left(\mathbf{x}_t ; \sqrt{\alpha_t} \mathbf{x}_{t-1},\left(1-\alpha_t\right) I\right) \mathcal{N}\left(\mathbf{x}_{t-1} ; \sqrt{\bar{\alpha}_{t-1}} \mathbf{x}_0,\left(1-\bar{\alpha}_{t-1}\right) I\right)}{\mathcal{N}\left(\mathbf{x}_t ; \sqrt{\bar{\alpha}_t} \mathbf{x}_0,\left(1-\bar{\alpha}_t\right) I\right)} \\
& \propto \exp \left\{-\left[\frac{\left(\mathbf{x}_t-\sqrt{\alpha_t} \mathbf{x}_{t-1}\right)^2}{2\left(1-\alpha_t\right)}+\frac{\left(\mathbf{x}_{t-1}-\sqrt{\alpha_{t-1}} \mathbf{x}_0\right)^2}{2\left(1-\bar{\alpha}_{t-1}\right)}-\frac{\left(\mathbf{x}_t-\sqrt{\bar{\alpha}_t} \mathbf{x}_0\right)^2}{2\left(1-\bar{\alpha}_t\right)}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left[\frac{\left(\mathbf{x}_t-\sqrt{\alpha_t} \mathbf{x}_{t-1}\right)^2}{1-\alpha_t}+\frac{\left(\mathbf{x}_{t-1}-\sqrt{\alpha_{t-1}} \mathbf{x}_0\right)^2}{1-\bar{\alpha}_{t-1}}-\frac{\left(\mathbf{x}_t-\sqrt{\bar{\alpha}_t} \mathbf{x}_0\right)^2}{1-\bar{\alpha}_t}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left[\frac{\left(-2 \sqrt{\alpha_t} \mathbf{x}_t \mathbf{x}_{t-1}+\alpha_t \mathbf{x}_{t-1}^2\right)}{1-\alpha_t}+\frac{\left(\mathbf{x}_{t-1}^2-2 \sqrt{\bar{\alpha}_{t-1}} \mathbf{x}_{t-1} \mathbf{x}_0\right)}{1-\bar{\alpha}_{t-1}}+C\left(\mathbf{x}_t, \mathbf{x}_0\right)\right]\right\} \\
& \propto \exp \left\{-\frac{1}{2}\left[-\frac{2 \sqrt{\alpha_t} \mathbf{x}_t \mathbf{x}_{t-1}}{1-\alpha_t}+\frac{\alpha_t \mathbf{x}_{t-1}^2}{1-\alpha_t}+\frac{\mathbf{x}_{t-1}^2}{1-\bar{\alpha}_{t-1}}-\frac{2 \sqrt{\bar{\alpha}_{t-1}} \mathbf{x}_{t-1} \mathbf{x}_0}{1-\bar{\alpha}_{t-1}}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left[\left(\frac{\alpha_t}{1-\alpha_t}+\frac{1}{1-\bar{\alpha}_{t-1}}\right) \mathbf{x}_{t-1}^2-2\left(\frac{\sqrt{\alpha_t} \mathbf{x}_t}{1-\alpha_t}+\frac{\sqrt{\alpha_{t-1}} \mathbf{x}_0}{1-\bar{\alpha}_{t-1}}\right) \mathbf{x}_{t-1}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left[\frac{\alpha_t\left(1-\bar{\alpha}_{t-1}\right)+1-\alpha_t}{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)} \mathbf{x}_{t-1}^2-2\left(\frac{\sqrt{\alpha_t} \mathbf{x}_t}{1-\alpha_t}+\frac{\sqrt{\alpha_{t-1}} \mathbf{x}_0}{1-\bar{\alpha}_{t-1}}\right) \mathbf{x}_{t-1}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left[\frac{\alpha_t-\bar{\alpha}_t+1-\alpha_t}{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)} \mathbf{x}_{t-1}^2-2\left(\frac{\sqrt{\alpha_t} \mathbf{x}_t}{1-\alpha_t}+\frac{\sqrt{\alpha_{t-1}} \mathbf{x}_0}{1-\bar{\alpha}_{t-1}}\right) \mathbf{x}_{t-1}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left[\frac{1-\bar{\alpha}_t}{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)} \mathbf{x}_{t-1}^2-2\left(\frac{\sqrt{\alpha_t} \mathbf{x}_t}{1-\alpha_t}+\frac{\sqrt{\alpha_{t-1}} \mathbf{x}_0}{1-\bar{\alpha}_{t-1}}\right) \mathbf{x}_{t-1}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left(\frac{1-\bar{\alpha}_t}{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)}\right)\left[\mathbf{x}_{t-1}^2-2 \frac{\left(\frac{\sqrt{\alpha_t} \mathbf{x}_t}{1-\alpha_t}+\frac{\sqrt{\alpha_{t-1}} \mathbf{x}_0}{1-\bar{\alpha}_{t-1}}\right)}{\frac{1-\bar{\alpha}_t}{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)}} \mathbf{x}_{t-1}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left(\frac{1-\bar{\alpha}_t}{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)}\right)\left[\mathbf{x}_{t-1}^2-2 \frac{\left(\frac{\sqrt{\alpha_t} \mathbf{x}_t}{1-\alpha_t}+\frac{\sqrt{\alpha_{t-1}} \mathbf{x}_0}{1-\bar{\alpha}_{t-1}}\right)\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_t} \mathbf{x}_{t-1}\right]\right\} \\
& =\exp \left\{-\frac{1}{2}\left(\frac{1}{\frac{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_t}}\right)\left[\mathbf{x}_{t-1}^2-2 \frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \mathbf{x}_0}{1-\bar{\alpha}_t} \mathbf{x}_{t-1}\right]\right\} \\
& \propto \mathcal{N}(\mathbf{x}_{t-1} ; \underbrace{\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \mathbf{x}_0}{1-\bar{\alpha}_t}}_{\mu_q\left(\mathbf{x}_t, \mathbf{x}_0\right)}, \underbrace{\left.\frac{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_t} I\right)}_{\Sigma_q(t)} \\
&
\end{aligned}
q ( x t − 1 ∣ x t , x 0 ) = q ( x t ∣ x 0 ) q ( x t ∣ x t − 1 , x 0 ) q ( x t − 1 ∣ x 0 ) = N ( x t ; α ˉ t x 0 , ( 1 − α ˉ t ) I ) N ( x t ; α t x t − 1 , ( 1 − α t ) I ) N ( x t − 1 ; α ˉ t − 1 x 0 , ( 1 − α ˉ t − 1 ) I ) ∝ exp { − [ 2 ( 1 − α t ) ( x t − α t x t − 1 ) 2 + 2 ( 1 − α ˉ t − 1 ) ( x t − 1 − α t − 1 x 0 ) 2 − 2 ( 1 − α ˉ t ) ( x t − α ˉ t x 0 ) 2 ] } = exp { − 2 1 [ 1 − α t ( x t − α t x t − 1 ) 2 + 1 − α ˉ t − 1 ( x t − 1 − α t − 1 x 0 ) 2 − 1 − α ˉ t ( x t − α ˉ t x 0 ) 2 ] } = exp { − 2 1 [ 1 − α t ( − 2 α t x t x t − 1 + α t x t − 1 2 ) + 1 − α ˉ t − 1 ( x t − 1 2 − 2 α ˉ t − 1 x t − 1 x 0 ) + C ( x t , x 0 ) ] } ∝ exp { − 2 1 [ − 1 − α t 2 α t x t x t − 1 + 1 − α t α t x t − 1 2 + 1 − α ˉ t − 1 x t − 1 2 − 1 − α ˉ t − 1 2 α ˉ t − 1 x t − 1 x 0 ] } = exp { − 2 1 [ ( 1 − α t α t + 1 − α ˉ t − 1 1 ) x t − 1 2 − 2 ( 1 − α t α t x t + 1 − α ˉ t − 1 α t − 1 x 0 ) x t − 1 ] } = exp { − 2 1 [ ( 1 − α t ) ( 1 − α ˉ t − 1 ) α t ( 1 − α ˉ t − 1 ) + 1 − α t x t − 1 2 − 2 ( 1 − α t α t x t + 1 − α ˉ t − 1 α t − 1 x 0 ) x t − 1 ] } = exp { − 2 1 [ ( 1 − α t ) ( 1 − α ˉ t − 1 ) α t − α ˉ t + 1 − α t x t − 1 2 − 2 ( 1 − α t α t x t + 1 − α ˉ t − 1 α t − 1 x 0 ) x t − 1 ] } = exp { − 2 1 [ ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t x t − 1 2 − 2 ( 1 − α t α t x t + 1 − α ˉ t − 1 α t − 1 x 0 ) x t − 1 ] } = exp ⎩ ⎨ ⎧ − 2 1 ( ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t ) ⎣ ⎡ x t − 1 2 − 2 ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t ( 1 − α t α t x t + 1 − α ˉ t − 1 α t − 1 x 0 ) x t − 1 ⎦ ⎤ ⎭ ⎬ ⎫ = exp ⎩ ⎨ ⎧ − 2 1 ( ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t ) ⎣ ⎡ x t − 1 2 − 2 1 − α ˉ t ( 1 − α t α t x t + 1 − α ˉ t − 1 α t − 1 x 0 ) ( 1 − α t ) ( 1 − α ˉ t − 1 ) x t − 1 ⎦ ⎤ ⎭ ⎬ ⎫ = exp { − 2 1 ( 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 ) [ x t − 1 2 − 2 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 x t − 1 ] } ∝ N ( x t − 1 ; μ q ( x t , x 0 ) 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 , Σ q ( t ) 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) I )
接下来就是计算几个散度，根据假设ELBO里面的散度都是两个高斯分布间的散度，所以我们需要推导高斯分布的散度：
D K L ( N ( x ; μ x , Σ x ) ∥ N ( y ; μ y , Σ y ) ) = 1 2 [ log ∣ Σ y ∣ ∣ Σ x ∣ − d + tr ( Σ y − 1 Σ x ) + ( μ y − μ x ) T Σ y − 1 ( μ y − μ x ) ] D_{\mathrm{KL}}\left(\mathcal{N}\left(x ; \mu_x, \Sigma_x\right) \| \mathcal{N}\left(y ; \mu_y, \Sigma_y\right)\right)=\frac{1}{2}\left[\log \frac{\left|\Sigma_y\right|}{\left|\Sigma_x\right|}-d+\operatorname{tr}\left(\Sigma_y^{-1} \Sigma_x\right)+\left(\mu_y-\mu_x\right)^T \Sigma_y^{-1}\left(\mu_y-\mu_x\right)\right]
D K L ( N ( x ; μ x , Σ x ) ∥ N ( y ; μ y , Σ y ) ) = 2 1 [ log ∣ Σ x ∣ ∣ Σ y ∣ − d + t r ( Σ y − 1 Σ x ) + ( μ y − μ x ) T Σ y − 1 ( μ y − μ x ) ]
arg min θ D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) = arg min θ D K L ( N ( x t − 1 ; μ q , Σ q ( t ) ) ∥ N ( x t − 1 ; μ θ , Σ q ( t ) ) ) = arg min θ 1 2 [ log ∣ Σ q ( t ) ∣ ∣ Σ q ( t ) ∣ − d + tr ( Σ q ( t ) − 1 Σ q ( t ) ) + ( μ θ − μ q ) T Σ q ( t ) − 1 ( μ θ − μ q ) ] = arg min θ 1 2 [ log 1 − d + d + ( μ θ − μ q ) T Σ q ( t ) − 1 ( μ θ − μ q ) ] = arg min θ 1 2 [ ( μ θ − μ q ) T Σ q ( t ) − 1 ( μ θ − μ q ) ] = arg min θ 1 2 [ ( μ θ − μ q ) T ( σ q 2 ( t ) I ) − 1 ( μ θ − μ q ) ] = arg min θ 1 2 σ q 2 ( t ) [ ∥ μ θ − μ q ∥ 2 2 ] \begin{aligned}
& \underset{\theta}{\arg \min } D_{\mathrm{KL}}\left(q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right) \| p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)\right) \\
& =\underset{\theta}{\arg \min } D_{\mathrm{KL}}\left(\mathcal{N}\left(\mathbf{x}_{t-1} ; \mu_q, \Sigma_q(t)\right) \| \mathcal{N}\left(\mathbf{x}_{t-1} ; \mu_{\theta}, \Sigma_q(t)\right)\right) \\
& =\underset{\theta}{\arg \min } \frac{1}{2}\left[\log \frac{\left|\Sigma_q(t)\right|}{\left|\Sigma_q(t)\right|}-d+\operatorname{tr}\left(\Sigma_q(t)^{-1} \Sigma_q(t)\right)+\left(\mu_{\theta}-\mu_q\right)^T \Sigma_q(t)^{-1}\left(\mu_{\theta}-\mu_q\right)\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2}\left[\log 1-d+d+\left(\mu_{\theta}-\mu_q\right)^T \Sigma_q(t)^{-1}\left(\mu_{\theta}-\mu_q\right)\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2}\left[\left(\mu_{\theta}-\mu_q\right)^T \Sigma_q(t)^{-1}\left(\mu_{\theta}-\mu_q\right)\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2}\left[\left(\mu_{\theta}-\mu_q\right)^T\left(\sigma_q^2(t) I\right)^{-1}\left(\mu_{\theta}-\mu_q\right)\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2 \sigma_q^2(t)}\left[\left\|\mu_{\theta}-\mu_q\right\|_2^2\right]
\end{aligned}
θ arg min D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) = θ arg min D K L ( N ( x t − 1 ; μ q , Σ q ( t ) ) ∥ N ( x t − 1 ; μ θ , Σ q ( t ) ) ) = θ arg min 2 1 [ log ∣ Σ q ( t ) ∣ ∣ Σ q ( t ) ∣ − d + t r ( Σ q ( t ) − 1 Σ q ( t ) ) + ( μ θ − μ q ) T Σ q ( t ) − 1 ( μ θ − μ q ) ] = θ arg min 2 1 [ log 1 − d + d + ( μ θ − μ q ) T Σ q ( t ) − 1 ( μ θ − μ q ) ] = θ arg min 2 1 [ ( μ θ − μ q ) T Σ q ( t ) − 1 ( μ θ − μ q ) ] = θ arg min 2 1 [ ( μ θ − μ q ) T ( σ q 2 ( t ) I ) − 1 ( μ θ − μ q ) ] = θ arg min 2 σ q 2 ( t ) 1 [ ∥ μ θ − μ q ∥ 2 2 ]
这里我们用 μ q \mu_q μ q μ q ( x t , x 0 ) \mu_q(\mathbf{x}_t, \mathbf{x}_0) μ q ( x t , x 0 ) μ θ \mu_\theta μ θ μ θ ( x t , t ) \mu_\theta(\mathbf{x}_t, t) μ θ ( x t , t ) μ θ ( x t , t ) \mu_\theta(\mathbf{x}_t, t) μ θ ( x t , t ) μ q ( x t , x 0 ) \mu_q(\mathbf{x}_t, \mathbf{x}_0) μ q ( x t , x 0 )
第一种方法是让神经网络直接输出后验分布的均值 μ θ ( x t , t ) \mu_\theta(\mathbf{x}_t, t) μ θ ( x t , t )
第二种方法就是直接预测μ ~ t ( x t , x 0 ) \tilde{\mu}_t(\mathbf{x}_t, \mathbf{x}_0) μ ~ t ( x t , x 0 ) x 0 \mathbf{x}_0 x 0
第三种方法是把x 0 \mathbf{x}_0 x 0 x t \mathbf{x}_t x t ϵ \epsilon ϵ ϵ \epsilon ϵ
去噪模型的形式为:
arg min θ D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) = arg min θ D K L ( N ( x t − 1 ; μ q , Σ q ( t ) ) ∥ N ( x t − 1 ; μ θ , Σ q ( t ) ) ) = arg min θ 1 2 σ q 2 ( t ) [ ∥ α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x ^ θ ( x t , t ) 1 − α ˉ t − α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 1 − α ˉ t ∥ 2 ] = arg min θ 1 2 σ q 2 ( t ) [ ∥ α ˉ t − 1 ( 1 − α t ) x ^ θ ( x t , t ) 1 − α ˉ t − α ˉ t − 1 ( 1 − α t ) x 0 1 − α ˉ t ∥ 2 2 ] = arg min θ 1 2 σ q 2 ( t ) [ ∥ α ˉ t − 1 ( 1 − α t ) 1 − α ˉ t ( x ^ θ ( x t , t ) − x 0 ) ∥ 2 2 ] = arg min θ 1 2 σ q 2 ( t ) α ˉ t − 1 ( 1 − α t ) 2 ( 1 − α ˉ t ) 2 [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t α ˉ t − 1 ( 1 − α t ) 2 ( 1 − α ˉ t ) 2 [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) α ˉ t − 1 ( 1 − α t ) 2 ( 1 − α ˉ t ) 2 [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 α ˉ t − 1 ( 1 − α t ) ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 α ˉ t − 1 − α ˉ t ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 α ˉ t − 1 − α ˉ t − 1 α ˉ t + α ˉ t − 1 α ˉ t − α ˉ t ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 α ˉ t − 1 ( 1 − α ˉ t ) − α ˉ t ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 ( α ˉ t − 1 ( 1 − α ˉ t ) ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) − α ˉ t ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 ( α ˉ t − 1 1 − α ˉ t − 1 − α ˉ t 1 − α ˉ t ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = arg min θ 1 2 ( S N R ( t − 1 ) − S N R ( t ) ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] \begin{aligned}
& \underset{\theta}{\arg \min } D_{\mathrm{KL}}\left(q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right) \| p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)\right) \\
& =\underset{\theta}{\arg \min } D_{\mathrm{KL}}\left(\mathcal{N}\left(\mathbf{x}_{t-1} ; \mu_q, \Sigma_q(t)\right) \| \mathcal{N}\left(\mathbf{x}_{t-1} ; \mu_{\theta}, \Sigma_q(t)\right)\right) \\
& =\underset{\theta}{\arg \min } \frac{1}{2 \sigma_q^2(t)}\left[\left\|\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \hat{x}_{\theta}\left(\mathbf{x}_t, t\right)}{1-\bar{\alpha}_t}-\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \mathbf{x}_0}{1-\bar{\alpha}_t}\right\|_2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2 \sigma_q^2(t)}\left[\left\|\frac{\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \hat{x}_{\theta}\left(\mathbf{x}_t, t\right)}{1-\bar{\alpha}_t}-\frac{\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \mathbf{x}_0}{1-\bar{\alpha}_t}\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2 \sigma_q^2(t)}\left[\left\|\frac{\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right)}{1-\bar{\alpha}_t}\left(\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right)\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2 \sigma_q^2(t)} \frac{\bar{\alpha}_{t-1}\left(1-\alpha_t\right)^2}{\left(1-\bar{\alpha}_t\right)^2}\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
&=\underset{\theta}{\arg \min } \frac{1}{2 \frac{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_t}} \frac{\bar{\alpha}_{t-1}\left(1-\alpha_t\right)^2}{\left(1-\bar{\alpha}_t\right)^2}\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2} \frac{1-\bar{\alpha}_t}{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_{t-1}\right)} \frac{\bar{\alpha}_{t-1}\left(1-\alpha_t\right)^2}{\left(1-\bar{\alpha}_t\right)^2}\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min }\frac{1}{2} \frac{\bar{\alpha}_{t-1}\left(1-\alpha_t\right)}{\left(1-\bar{\alpha}_{t-1}\right)\left(1-\bar{\alpha}_t\right)}\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min }\frac{1}{2} \frac{\bar{\alpha}_{t-1}-\bar{\alpha}_t}{\left(1-\bar{\alpha}_{t-1}\right)\left(1-\bar{\alpha}_t\right)}\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2} \frac{\bar{\alpha}_{t-1}-\bar{\alpha}_{t-1} \bar{\alpha}_t+\bar{\alpha}_{t-1} \bar{\alpha}_t-\bar{\alpha}_t}{\left(1-\bar{\alpha}_{t-1}\right)\left(1-\bar{\alpha}_t\right)}\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2} \frac{\bar{\alpha}_{t-1}\left(1-\bar{\alpha}_t\right)-\bar{\alpha}_t\left(1-\bar{\alpha}_{t-1}\right)}{\left(1-\bar{\alpha}_{t-1}\right)\left(1-\bar{\alpha}_t\right)}\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2}\left(\frac{\bar{\alpha}_{t-1}\left(1-\bar{\alpha}_t\right)}{\left(1-\bar{\alpha}_{t-1}\right)\left(1-\bar{\alpha}_t\right)}-\frac{\bar{\alpha}_t\left(1-\bar{\alpha}_{t-1}\right)}{\left(1-\bar{\alpha}_{t-1}\right)\left(1-\bar{\alpha}_t\right)}\right)\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& =\underset{\theta}{\arg \min } \frac{1}{2}\left(\frac{\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t-1}}-\frac{\bar{\alpha}_t}{1-\bar{\alpha}_t}\right)\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right] \\
& = \underset{\theta}{\arg \min } \frac{1}{2} \left(SNR(t-1)-SNR(t)\right)\left[\left\|\hat{x}_{\theta}\left(\mathbf{x}_t, t\right)-\mathbf{x}_0\right\|_2^2\right]
\end{aligned}
θ arg min D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) = θ arg min D K L ( N ( x t − 1 ; μ q , Σ q ( t ) ) ∥ N ( x t − 1 ; μ θ , Σ q ( t ) ) ) = θ arg min 2 σ q 2 ( t ) 1 [ ∥ ∥ ∥ ∥ 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x ^ θ ( x t , t ) − 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 ∥ ∥ ∥ ∥ 2 ] = θ arg min 2 σ q 2 ( t ) 1 [ ∥ ∥ ∥ ∥ 1 − α ˉ t α ˉ t − 1 ( 1 − α t ) x ^ θ ( x t , t ) − 1 − α ˉ t α ˉ t − 1 ( 1 − α t ) x 0 ∥ ∥ ∥ ∥ 2 2 ] = θ arg min 2 σ q 2 ( t ) 1 [ ∥ ∥ ∥ ∥ 1 − α ˉ t α ˉ t − 1 ( 1 − α t ) ( x ^ θ ( x t , t ) − x 0 ) ∥ ∥ ∥ ∥ 2 2 ] = θ arg min 2 σ q 2 ( t ) 1 ( 1 − α ˉ t ) 2 α ˉ t − 1 ( 1 − α t ) 2 [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 − α ˉ t ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 ( 1 − α ˉ t ) 2 α ˉ t − 1 ( 1 − α t ) 2 [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( 1 − α t ) ( 1 − α ˉ t − 1 ) 1 − α ˉ t ( 1 − α ˉ t ) 2 α ˉ t − 1 ( 1 − α t ) 2 [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α ˉ t − 1 ( 1 − α t ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α ˉ t − 1 − α ˉ t [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α ˉ t − 1 − α ˉ t − 1 α ˉ t + α ˉ t − 1 α ˉ t − α ˉ t [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α ˉ t − 1 ( 1 − α ˉ t ) − α ˉ t ( 1 − α ˉ t − 1 ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α ˉ t − 1 ( 1 − α ˉ t ) − ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α ˉ t ( 1 − α ˉ t − 1 ) ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( 1 − α ˉ t − 1 α ˉ t − 1 − 1 − α ˉ t α ˉ t ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ] = θ arg min 2 1 ( S N R ( t − 1 ) − S N R ( t ) ) [ ∥ x ^ θ ( x t , t ) − x 0 ∥ 2 2 ]
其中信噪比(SNR)定义如下，因为q ( x t ∣ x 0 ) : = N ( x t ; α t ˉ x 0 , ( 1 − α t ˉ ) I ) q(\mathbf{x}_t | \mathbf{x}_{0}):=\mathcal{N}(\mathbf{x}_t ; \sqrt{\bar{\alpha_t}} \mathbf{x}_{0}, (1-\bar{\alpha_t}) I) q ( x t ∣ x 0 ) : = N ( x t ; α t ˉ x 0 , ( 1 − α t ˉ ) I )
S N R ( t ) = μ 2 σ 2 = α ˉ t 1 − α ˉ t SNR(t)=\frac{\mu^2}{\sigma^2}=\frac{\bar{\alpha}_t}{1-\bar{\alpha}_{t}}
S N R ( t ) = σ 2 μ 2 = 1 − α ˉ t α ˉ t
信噪比代表了原始信号与存在的噪声量之间的比率。较高的信噪比代表较多的信号，较低的信噪比代表较多的噪声。在扩散模型中，我们要求信噪比随着时间步长t t t
α ˉ t 1 − α ˉ t = exp ( − ω η ( t ) ) α ˉ t = sigmoid ( − ω η ( t ) ) 1 − α ˉ t = sigmoid ( ω η ( t ) ) \begin{aligned}
\frac{\bar{\alpha}_t}{1-\bar{\alpha}_t}=\exp \left(-\omega_{\boldsymbol{\eta}}(t)\right) \\
\bar{\alpha}_t=\operatorname{sigmoid}\left(-\omega_{\boldsymbol{\eta}}(t)\right) \\
1-\bar{\alpha}_t=\operatorname{sigmoid}\left(\omega_{\boldsymbol{\eta}}(t)\right)
\end{aligned}
1 − α ˉ t α ˉ t = exp ( − ω η ( t ) ) α ˉ t = s i g m o i d ( − ω η ( t ) ) 1 − α ˉ t = s i g m o i d ( ω η ( t ) )
此外，最小化我们导出的ELBO目标在所有噪声水平上的求和项可以通过最小化所有时间步上的期望来逼近，然后可以在时间步长上使用随机样本进行优化:
arg min θ E t ∼ U { 2 , T } [ E q ( x t ∣ x 0 ) [ D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) ] ] \underset{\theta}{\arg \min } \mathbb{E}_{t \sim U\{2, T\}}\left[\mathbb{E}_{q\left(\mathbf{x}_t | \mathbf{x}_0\right)}\left[D_{\mathrm{KL}}\left(q\left(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0\right) \| p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t\right)\right)\right]\right]
θ arg min E t ∼ U { 2 , T } [ E q ( x t ∣ x 0 ) [ D K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) ] ]
另外，将x t = α ˉ t x 0 + 1 − α ˉ t ϵ 0 \mathbf{x}_t=\sqrt{\bar{\alpha}_t} \mathbf{x}_0+\sqrt{1-\bar{\alpha}_t} \epsilon_0 x t = α ˉ t x 0 + 1 − α ˉ t ϵ 0
μ q ( x t , x 0 ) = α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 1 − α ˉ t = α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x t − 1 − α ˉ t ϵ 0 α ˉ t 1 − α ˉ t = α t ( 1 − α ˉ t − 1 ) x t + ( 1 − α t ) x t − 1 − α ˉ t ϵ 0 α t 1 − α ˉ t = α t ( 1 − α ˉ t − 1 ) x t 1 − α ˉ t + ( 1 − α t ) x t ( 1 − α ˉ t ) α t − ( 1 − α t ) 1 − α ˉ t ϵ 0 ( 1 − α ˉ t ) α t = ( α t ( 1 − α ˉ t − 1 ) 1 − α ˉ t + 1 − α t ( 1 − α ˉ t ) α t ) x t − ( 1 − α t ) 1 − α ˉ t ( 1 − α ˉ t ) α t ϵ 0 = ( α t ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α t + 1 − α t ( 1 − α ˉ t ) α t ) x t − 1 − α t 1 − α ˉ t α t ϵ 0 = α t − α ˉ t + 1 − α t ( 1 − α ˉ t ) α t x t − 1 − α t 1 − α ˉ t α t ϵ 0 = 1 − α ˉ t ( 1 − α ˉ t ) α t x t − 1 − α t 1 − α ˉ t α t ϵ 0 = 1 α t x t − 1 − α t 1 − α ˉ t α t ϵ 0 = 1 α t ( x t − 1 − α t 1 − α ˉ t ϵ 0 ) \begin{aligned}
\mu_q\left(\mathbf{x}_t, \mathbf{x}_0\right) & =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \mathbf{x}_0}{1-\bar{\alpha}_t} \\
& =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \frac{\mathbf{x}_t-\sqrt{1-\bar{\alpha}_t} \epsilon_0}{\sqrt{\bar{\alpha}_t}}}{1-\bar{\alpha}_t} \\
& =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\left(1-\alpha_t\right) \frac{\mathbf{x}_t-\sqrt{1-\bar{\alpha}_t} \epsilon_0}{\sqrt{\alpha_t}}}{1-\bar{\alpha}_t} \\
& =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t}{1-\bar{\alpha}_t}+\frac{\left(1-\alpha_t\right) \mathbf{x}_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}-\frac{\left(1-\alpha_t\right) \sqrt{1-\bar{\alpha}_t} \epsilon_0}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}} \\
& =\left(\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_t}+\frac{1-\alpha_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}\right) \mathbf{x}_t-\frac{\left(1-\alpha_t\right) \sqrt{1-\bar{\alpha}_t}}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}} \epsilon_0 \\
& =\left(\frac{\alpha_t\left(1-\bar{\alpha}_{t-1}\right)}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}+\frac{1-\alpha_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}\right) \mathbf{x}_t-\frac{1-\alpha_t}{\sqrt{1-\bar{\alpha}_t} \sqrt{\alpha_t}} \epsilon_0 \\
& =\frac{\alpha_t-\bar{\alpha}_t+1-\alpha_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}} \mathbf{x}_t-\frac{1-\alpha_t}{\sqrt{1-\bar{\alpha}_t} \sqrt{\alpha_t}} \epsilon_0 \\
& =\frac{1-\bar{\alpha}_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}} \mathbf{x}_t-\frac{1-\alpha_t}{\sqrt{1-\bar{\alpha}_t} \sqrt{\alpha_t}} \epsilon_0 \\
& =\frac{1}{\sqrt{\alpha_t}} \mathbf{x}_t-\frac{1-\alpha_t}{\sqrt{1-\bar{\alpha}_t} \sqrt{\alpha_t}} \epsilon_0 \\
& =\frac{1}{\sqrt{\alpha_t}}\left(\mathrm{x}_t-\frac{1-\alpha_t}{\sqrt{1-\bar{\alpha}_t}} \epsilon_0\right)
\end{aligned}
μ q ( x t , x 0 ) = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) α ˉ t x t − 1 − α ˉ t ϵ 0 = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + ( 1 − α t ) α t x t − 1 − α ˉ t ϵ 0 = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + ( 1 − α ˉ t ) α t ( 1 − α t ) x t − ( 1 − α ˉ t ) α t ( 1 − α t ) 1 − α ˉ t ϵ 0 = ( 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) + ( 1 − α ˉ t ) α t 1 − α t ) x t − ( 1 − α ˉ t ) α t ( 1 − α t ) 1 − α ˉ t ϵ 0 = ( ( 1 − α ˉ t ) α t α t ( 1 − α ˉ t − 1 ) + ( 1 − α ˉ t ) α t 1 − α t ) x t − 1 − α ˉ t α t 1 − α t ϵ 0 = ( 1 − α ˉ t ) α t α t − α ˉ t + 1 − α t x t − 1 − α ˉ t α t 1 − α t ϵ 0 = ( 1 − α ˉ t ) α t 1 − α ˉ t x t − 1 − α ˉ t α t 1 − α t ϵ 0 = α t 1 x t − 1 − α ˉ t α t 1 − α t ϵ 0 = α t 1 ( x t − 1 − α ˉ t 1 − α t ϵ 0 )
我们尝试使用 μ θ ( x t , t ) \mu_\theta(\mathbf{x}_t, t) μ θ ( x t , t ) μ q ( x t , x 0 ) \mu_q(\mathbf{x}_t, \mathbf{x}_0) μ q ( x t , x 0 ) x t \mathbf{x}_t x t ϵ 0 \epsilon_0 ϵ 0
μ θ ( x t , t ) = 1 α t ( x t − 1 − α t 1 − α ˉ t ϵ θ ( x t , t ) ) \mu_\theta(\mathbf{x}_t, t) = \frac{1}{\sqrt{\alpha_t}} \left( \mathbf{x}_t - \frac{1 - \alpha_t}{\sqrt{1 - \bar{\alpha}_t}} \epsilon_\theta(\mathbf{x}_t, t) \right)
μ θ ( x t , t ) = α t 1 ( x t − 1 − α ˉ t 1 − α t ϵ θ ( x t , t ) )
x t − 1 = N ( x t − 1 ; 1 α t ( x t − 1 − α t 1 − α ˉ t ϵ θ ( x t , t ) ) , Σ θ ( x t , t ) ) \mathbf{x}_{t-1} = \mathcal{N}\left(\mathbf{x}_{t-1}; \frac{1}{\sqrt{\alpha_t}} \left( \mathbf{x}_t - \frac{1 - \alpha_t}{\sqrt{1 - \bar{\alpha}_t}} \epsilon_\theta(\mathbf{x}_t, t) \right), \Sigma_\theta(\mathbf{x}_t, t)\right)
x t − 1 = N ( x t − 1 ; α t 1 ( x t − 1 − α ˉ t 1 − α t ϵ θ ( x t , t ) ) , Σ θ ( x t , t ) )
这里，ϵ θ ( x t , t ) \epsilon_{\theta}(\mathbf{x}_t, t) ϵ θ ( x t , t ) ϵ 0 \epsilon_0 ϵ 0 x 0 \mathbf{x}_0 x 0
L simple ( θ ) : = E t , x 0 , ϵ [ ∥ ϵ − ϵ θ ( α ˉ t x 0 + 1 − α ˉ t ϵ , t ) ∥ 2 ] L_{\text {simple }}(\theta):=\mathbb{E}_{t, \mathbf{x}_0, \epsilon}\left[\left\|\epsilon-\epsilon_\theta\left(\sqrt{\bar{\alpha}_t} \mathbf{x}_0+\sqrt{1-\bar{\alpha}_t} \epsilon, t\right)\right\|^2\right]
L simple ( θ ) : = E t , x 0 , ϵ [ ∥ ∥ ϵ − ϵ θ ( α ˉ t x 0 + 1 − α ˉ t ϵ , t ) ∥ ∥ 2 ]
我们先推导Tweedie公式。在参数估计里面，经典贝叶斯派的公式很多，Tweedie公式就是其中一个。假设p ( x ∣ θ ) = N ( θ , σ 2 ) p(x|\theta)=\mathcal{N}(\theta,\sigma^2) p ( x ∣ θ ) = N ( θ , σ 2 )
E [ θ ∣ x ] = ∫ θ p ( θ ∣ x ) d θ = ∫ θ p ( x ∣ θ ) p ( θ ) p ( x ) d θ = ∫ θ p ( x ∣ θ ) p ( θ ) d θ p ( x ) = ∫ θ 1 2 π σ 2 e − ( x − θ ) 2 2 σ 2 p ( θ ) d θ p ( x ) = ∫ [ σ 2 θ − x σ 2 1 2 π σ 2 e − ( x − θ ) 2 2 σ 2 p ( θ ) + x 1 2 π σ 2 e − ( x − θ ) 2 2 σ 2 p ( θ ) ] d θ p ( x ) = ∫ σ 2 θ − x σ 2 1 2 π σ 2 e − ( x − θ ) 2 2 σ 2 p ( θ ) d θ + ∫ x 1 2 π σ 2 e − ( x − θ ) 2 2 σ 2 p ( θ ) d θ p ( x ) = σ 2 ∫ d [ 1 2 π σ 2 e − ( x − θ ) 2 2 σ 2 ] d x p ( θ ) d θ + ∫ x 1 2 π σ 2 e − ( x − θ ) 2 2 σ 2 p ( θ ) d θ p ( x ) = σ 2 ∫ d p ( x ∣ θ ) d x p ( θ ) d θ + ∫ x p ( x ∣ θ ) p ( θ ) d θ p ( x ) = σ 2 d d x ∫ p ( x ∣ θ ) p ( θ ) d θ + x ∫ p ( x ∣ θ ) p ( θ ) d θ p ( x ) = σ 2 d p ( x ) d x + x p ( x ) p ( x ) = x + σ 2 d d x log p ( x ) \begin{aligned}
\mathbb{E}[\theta | x]& =\int \theta p(\theta | x) \mathrm{d} \theta \\
& =\int \theta \frac{p(x | \theta) p(\theta)}{p(x)} \mathrm{d} \theta \\
& =\frac{\int \theta p(x | \theta) p(\theta) \mathrm{d} \theta}{p(x)} \\
& =\frac{\int \theta \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\theta)^2}{2 \sigma^2}} p(\theta) \mathrm{d} \theta}{p(x)} \\
& =\frac{\int\left[\sigma^2 \frac{\theta-x}{\sigma^2} \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\theta)^2}{2 \sigma^2}} p(\theta)+x \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\theta)^2}{2 \sigma^2}} p(\theta)\right] \mathrm{d} \theta}{p(x)} \\
& =\frac{\int \sigma^2 \frac{\theta-x}{\sigma^2} \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\theta)^2}{2 \sigma^2}} p(\theta) \mathrm{d} \theta+\int x \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\theta)^2}{2 \sigma^2}} p(\theta) \mathrm{d} \theta}{p(x)} \\
& =\frac{\sigma^2 \int \frac{\mathrm{d}\left[\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\theta)^2}{2 \sigma^2}}\right]}{\mathrm{d} x} p(\theta) \mathrm{d} \theta+\int x \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\theta)^2}{2 \sigma^2}} p(\theta) \mathrm{d} \theta}{p(x)} \\
& =\frac{\sigma^2 \int \frac{\mathrm{d} p(x | \theta)}{\mathrm{d} x} p(\theta) \mathrm{d} \theta+\int x p(x | \theta) p(\theta) \mathrm{d} \theta}{p(x)} \\
& =\frac{\sigma^2 \frac{\mathrm{d}}{\mathrm{d} x} \int p(x | \theta) p(\theta) \mathrm{d} \theta+x \int p(x | \theta) p(\theta) \mathrm{d} \theta}{p(x)} \\
& =\frac{\sigma^2 \frac{\mathrm{d} p(x)}{\mathrm{d} x}+x p(x)}{p(x)} \\
& =x+\sigma^2 \frac{\mathrm{d}}{\mathrm{d} x} \log p(x) \\
&
\end{aligned}
E [ θ ∣ x ] = ∫ θ p ( θ ∣ x ) d θ = ∫ θ p ( x ) p ( x ∣ θ ) p ( θ ) d θ = p ( x ) ∫ θ p ( x ∣ θ ) p ( θ ) d θ = p ( x ) ∫ θ 2 π σ 2 1 e − 2 σ 2 ( x − θ ) 2 p ( θ ) d θ = p ( x ) ∫ [ σ 2 σ 2 θ − x 2 π σ 2 1 e − 2 σ 2 ( x − θ ) 2 p ( θ ) + x 2 π σ 2 1 e − 2 σ 2 ( x − θ ) 2 p ( θ ) ] d θ = p ( x ) ∫ σ 2 σ 2 θ − x 2 π σ 2 1 e − 2 σ 2 ( x − θ ) 2 p ( θ ) d θ + ∫ x 2 π σ 2 1 e − 2 σ 2 ( x − θ ) 2 p ( θ ) d θ = p ( x ) σ 2 ∫ d x d [ 2 π σ 2 1 e − 2 σ 2 ( x − θ ) 2 ] p ( θ ) d θ + ∫ x 2 π σ 2 1 e − 2 σ 2 ( x − θ ) 2 p ( θ ) d θ = p ( x ) σ 2 ∫ d x d p ( x ∣ θ ) p ( θ ) d θ + ∫ x p ( x ∣ θ ) p ( θ ) d θ = p ( x ) σ 2 d x d ∫ p ( x ∣ θ ) p ( θ ) d θ + x ∫ p ( x ∣ θ ) p ( θ ) d θ = p ( x ) σ 2 d x d p ( x ) + x p ( x ) = x + σ 2 d x d log p ( x )
在diffusion的公式里面，我们应用它来预测给定样本的 x t \mathbf{x}_t x t
E [ μ x t ∣ x t ] = x t + ( 1 − α ˉ t ) ∇ x t log p ( x t ) \mathbb{E}\left[\mu_{\mathbf{x}_t} | \mathbf{x}_t\right]=\mathbf{x}_t+\left(1-\bar{\alpha}_t\right) \nabla_{\mathbf{x}_t} \log p\left(\mathbf{x}_t\right)
E [ μ x t ∣ x t ] = x t + ( 1 − α ˉ t ) ∇ x t log p ( x t )
根据Tweedie’s Formula，对 x t \mathbf{x}_t x t μ x t = α ˉ t x 0 \mu_{\mathbf{x}_t} =\sqrt{\bar\alpha_t}\mathbf{x}_0 μ x t = α ˉ t x 0
x 0 = x t + ( 1 − α ˉ t ) ∇ log p ( x t ) α ˉ t \mathbf{x}_0=\frac{\mathbf{x}_t+\left(1-\bar{\alpha}_t\right) \nabla \log p\left(\mathbf{x}_t\right)}{\sqrt{\bar{\alpha}_t}}
x 0 = α ˉ t x t + ( 1 − α ˉ t ) ∇ log p ( x t )
那么就能推导出新的公式：
μ q ( x t , x 0 ) = α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 1 − α ˉ t = α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x t + ( 1 − α ˉ t ) ∇ log p ( x t ) α ˉ t 1 − α ˉ t = α t ( 1 − α ˉ t − 1 ) x t + ( 1 − α t ) x t + ( 1 − α ˉ t ) ∇ log p ( x t ) α t 1 − α ˉ t = α t ( 1 − α ˉ t − 1 ) x t 1 − α ˉ t + ( 1 − α t ) x t ( 1 − α ˉ t ) α t + ( 1 − α t ) ( 1 − α ˉ t ) ∇ log p ( x t ) ( 1 − α ˉ t ) α t = ( α t ( 1 − α ˉ t − 1 ) 1 − α ˉ t + 1 − α t ( 1 − α ˉ t ) α t ) x t + 1 − α t α t ∇ log p ( x t ) = ( α t ( 1 − α ˉ t − 1 ) ( 1 − α ˉ t ) α t + 1 − α t ( 1 − α ˉ t ) α t ) x t + 1 − α t α t ∇ log p ( x t ) = α t − α ˉ t + 1 − α t ( 1 − α ˉ t ) α t x t + 1 − α t α t ∇ log p ( x t ) = 1 − α ˉ t ( 1 − α ˉ t ) α t x t + 1 − α t α t ∇ log p ( x t ) = 1 α t x t + 1 − α t α t ∇ log p ( x t ) \begin{aligned}
\mu_q\left(\mathbf{x}_t, \mathbf{x}_0\right) & =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \mathbf{x}_0}{1-\bar{\alpha}_t} \\
& =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\sqrt{\bar{\alpha}_{t-1}}\left(1-\alpha_t\right) \frac{\mathbf{x}_t+\left(1-\bar{\alpha}_t\right) \nabla \log p\left(\mathbf{x}_t\right)}{\sqrt{\bar{\alpha}_t}}}{1-\bar{\alpha}_t} \\
& =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t+\left(1-\alpha_t\right) \frac{\mathbf{x}_t+\left(1-\bar{\alpha}_t\right) \nabla \log p\left(\mathbf{x}_t\right)}{\sqrt{\alpha_t}}}{1-\bar{\alpha}_t} \\
& =\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right) \mathbf{x}_t}{1-\bar{\alpha}_t}+\frac{\left(1-\alpha_t\right) \mathbf{x}_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}+\frac{\left(1-\alpha_t\right)\left(1-\bar{\alpha}_t\right) \nabla \log p\left(\mathbf{x}_t\right)}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}} \\
& =\left(\frac{\sqrt{\alpha_t}\left(1-\bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_t}+\frac{1-\alpha_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}\right) \mathbf{x}_t+\frac{1-\alpha_t}{\sqrt{\alpha_t}} \nabla \log p\left(\mathbf{x}_t\right) \\
& =\left(\frac{\alpha_t\left(1-\bar{\alpha}_{t-1}\right)}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}+\frac{1-\alpha_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}}\right) \mathbf{x}_t+\frac{1-\alpha_t}{\sqrt{\alpha_t}} \nabla \log p\left(\mathbf{x}_t\right) \\
& =\frac{\alpha_t-\bar{\alpha}_t+1-\alpha_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}} \mathbf{x}_t+\frac{1-\alpha_t}{\sqrt{\alpha_t}} \nabla \log p\left(\mathbf{x}_t\right) \\
& =\frac{1-\bar{\alpha}_t}{\left(1-\bar{\alpha}_t\right) \sqrt{\alpha_t}} \mathbf{x}_t+\frac{1-\alpha_t}{\sqrt{\alpha_t}} \nabla \log p\left(\mathbf{x}_t\right) \\
& =\frac{1}{\sqrt{\alpha_t}} \mathbf{x}_t+\frac{1-\alpha_t}{\sqrt{\alpha_t}} \nabla \log p\left(\mathbf{x}_t\right)
\end{aligned}
μ q ( x t , x 0 ) = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) x 0 = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + α ˉ t − 1 ( 1 − α t ) α ˉ t x t + ( 1 − α ˉ t ) ∇ log p ( x t ) = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + ( 1 − α t ) α t x t + ( 1 − α ˉ t ) ∇ log p ( x t ) = 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) x t + ( 1 − α ˉ t ) α t ( 1 − α t ) x t + ( 1 − α ˉ t ) α t ( 1 − α t ) ( 1 − α ˉ t ) ∇ log p ( x t ) = ( 1 − α ˉ t α t ( 1 − α ˉ t − 1 ) + ( 1 − α ˉ t ) α t 1 − α t ) x t + α t 1 − α t ∇ log p ( x t ) = ( ( 1 − α ˉ t ) α t α t ( 1 − α ˉ t − 1 ) + ( 1 − α ˉ t ) α t 1 − α t ) x t + α t 1 − α t ∇ log p ( x t ) = ( 1 − α ˉ t ) α t α t − α ˉ t + 1 − α t x t + α t 1 − α t ∇ log p ( x t ) = ( 1 − α ˉ t ) α t 1 − α ˉ t x t + α t 1 − α t ∇ log p ( x t ) = α t 1 x t + α t 1 − α t ∇ log p ( x t )
在score-based里面我们使用如下式子去近似上面的式子：
μ θ ( x t , t ) = 1 α t x t + 1 − α t α t s θ ( x t , t ) \mu_{\theta}\left(\mathbf{x}_t, t\right)=\frac{1}{\sqrt{\alpha_t}} \mathbf{x}_t+\frac{1-\alpha_t}{\sqrt{\alpha_t}} s_{\theta}\left(\mathbf{x}_t, t\right)
μ θ ( x t , t ) = α t 1 x t + α t 1 − α t s θ ( x t , t )
这里s θ ( x t , t ) s_{\theta}\left(\mathbf{x}_t, t\right) s θ ( x t , t ) x t \mathbf{x}_t x t ϵ 0 \epsilon_0 ϵ 0
除了 L 0 L_0 L 0 L 0 L_0 L 0 p θ ( x 0 ∣ x 1 ) p_{\theta}(\mathbf{x}_0|\mathbf{x}_1) p θ ( x 0 ∣ x 1 ) L T L_T L T θ \theta θ L T L_T L T
改变了q ( x t − 1 ∣ x t , x 0 ) q(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0) q ( x t − 1 ∣ x t , x 0 )
q σ ( x t − 1 ∣ x t , x 0 ) = N ( x t − 1 ; α t − 1 x 0 + 1 − α t − 1 − σ t 2 x t − α t x 0 1 − α t , σ t 2 I ) q_\sigma(\mathbf{x}_{t-1} \vert \mathbf{x}_t, \mathbf{x}_0) = \mathcal{N}(\mathbf{x}_{t-1}; \sqrt{\alpha_{t-1}}\mathbf{x}_0 + \sqrt{1 - \alpha_{t-1} - \sigma_t^2} \frac{\mathbf{x}_t - \sqrt{\alpha_t}\mathbf{x}_0}{\sqrt{1 - \alpha_t}}, \sigma_t^2 \mathbf{I})
q σ ( x t − 1 ∣ x t , x 0 ) = N ( x t − 1 ; α t − 1 x 0 + 1 − α t − 1 − σ t 2 1 − α t x t − α t x 0 , σ t 2 I )
x t − 1 = α t − 1 ( x t − 1 − α t ϵ θ ( x t , t ) α t ⏟ predicted x 0 ) + 1 − α t − 1 − σ t 2 ⋅ ϵ θ ( x t , t ) ⏟ direction pointing to x t + σ t ϵ t ⏟ random noise \mathbf{x}_{t-1} = \sqrt{\alpha_{t-1}}\Big(\underbrace{\frac{\mathbf{x}_t-\sqrt{1-\alpha_{t}}\mathbf{\epsilon}_\theta(\mathbf{x}_t, t)}{\sqrt{\alpha_{t}}}}_{\text{predicted}\ \mathbf{x}_0}\Big) + \underbrace{\sqrt{1 - \alpha_{t-1} - \sigma_t^2} \cdot \mathbf{\epsilon}_\theta(\mathbf{x}_t, t)}_{\text{direction pointing to }\ \mathbf{x}_t} + \underbrace{\sigma_t\epsilon_t}_{\text {random noise}}
x t − 1 = α t − 1 ( predicted x 0 α t x t − 1 − α t ϵ θ ( x t , t ) ) + direction pointing to x t 1 − α t − 1 − σ t 2 ⋅ ϵ θ ( x t , t ) + random noise σ t ϵ t
注意：其中的α \alpha α α ˉ \bar{\alpha} α ˉ x 0 \mathbf{x}_0 x 0 x t \mathbf{x}_t x t ϵ t \epsilon_t ϵ t x t \mathbf{x}_t x t
上面的推导只关注于数据分布 p ( x ) p(\mathbf{x}) p ( x ) p ( x ∣ y ) p(\mathbf{x}|y) p ( x ∣ y ) y y y
p ( x 0 : T ∣ y ) = p ( x T ) ∏ t = 1 T p θ ( x t − 1 ∣ x t , y ) p\left(\mathbf{x}_{0: T} | y\right)=p\left(\mathbf{x}_T\right) \prod_{t=1}^T p_{\theta}\left(\mathbf{x}_{t-1} | \mathbf{x}_t, y\right)
p ( x 0 : T ∣ y ) = p ( x T ) t = 1 ∏ T p θ ( x t − 1 ∣ x t , y )
其中 y y y y y y
分类器指导
∇ log p ( x t ∣ y ) = ∇ log ( p ( x t ) p ( y ∣ x t ) p ( y ) ) = ∇ log p ( x t ) + ∇ log p ( y ∣ x t ) − ∇ log p ( y ) = ∇ log p ( x t ) ⏟ unconditional score + ∇ log p ( y ∣ x t ) ⏟ adversarial gradient \begin{aligned}
\nabla \log p\left(\mathbf{x}_t | y\right) & =\nabla \log \left(\frac{p\left(\mathbf{x}_t\right) p\left(y | \mathbf{x}_t\right)}{p(y)}\right) \\
& =\nabla \log p\left(\mathbf{x}_t\right)+\nabla \log p\left(y | \mathbf{x}_t\right)-\nabla \log p(y) \\
& =\underbrace{\nabla \log p\left(\mathbf{x}_t\right)}_{\text {unconditional score }}+\underbrace{\nabla \log p\left(y | \mathbf{x}_t\right)}_{\text {adversarial gradient }}
\end{aligned}
∇ log p ( x t ∣ y ) = ∇ log ( p ( y ) p ( x t ) p ( y ∣ x t ) ) = ∇ log p ( x t ) + ∇ log p ( y ∣ x t ) − ∇ log p ( y ) = unconditional score ∇ log p ( x t ) + adversarial gradient ∇ log p ( y ∣ x t )
我们也可以选择控制后面那个训练的分类器牺牲的多样性代价：
∇ log p ( x t ∣ y ) = ∇ log p ( x t ) + γ ∇ log p ( y ∣ x t ) \nabla \log p\left(\mathbf{x}_t | y\right) = \nabla \log p\left(\mathbf{x}_t\right) + \gamma \nabla \log p\left(y | \mathbf{x}_t\right)
∇ log p ( x t ∣ y ) = ∇ log p ( x t ) + γ ∇ log p ( y ∣ x t )
分类器指南的一个显著缺点是它依赖于一个单独的学习分类器。因为分类器必须处理任意噪声的输入，这是大多数现有的预先训练的分类模型都没有优化过的，所以它必须与扩散模型一起学习。
无分类器指导
∇ log p ( x t ∣ y ) = ∇ log p ( x t ) + γ ( ∇ log p ( x t ∣ y ) − ∇ log p ( x t ) ) = ∇ log p ( x t ) + γ ∇ log p ( x t ∣ y ) − γ ∇ log p ( x t ) = γ ∇ log p ( x t ∣ y ) ⏟ conditional score + ( 1 − γ ) ∇ log p ( x t ) ⏟ unconditional score \begin{aligned}
\nabla \log p\left(\mathbf{x}_t | y\right) & =\nabla \log p\left(\mathbf{x}_t\right)+\gamma\left(\nabla \log p\left(\mathbf{x}_t | y\right)-\nabla \log p\left(\mathbf{x}_t\right)\right) \\
& =\nabla \log p\left(\mathbf{x}_t\right)+\gamma \nabla \log p\left(\mathbf{x}_t | y\right)-\gamma \nabla \log p\left(\mathbf{x}_t\right) \\
& =\underbrace{\gamma \nabla \log p\left(\mathbf{x}_t | y\right)}_{\text {conditional score }}+\underbrace{(1-\gamma) \nabla \log p\left(\mathbf{x}_t\right)}_{\text {unconditional score }}
\end{aligned}
∇ log p ( x t ∣ y ) = ∇ log p ( x t ) + γ ( ∇ log p ( x t ∣ y ) − ∇ log p ( x t ) ) = ∇ log p ( x t ) + γ ∇ log p ( x t ∣ y ) − γ ∇ log p ( x t ) = conditional score γ ∇ log p ( x t ∣ y ) + unconditional score ( 1 − γ ) ∇ log p ( x t )
