DENOISING DIFFUSION IMPLICIT MODELS (DDIM)
从DDPM中我们知道，其扩散过程(前向过程、或加噪过程)被定义为一个马尔可夫过程，其去噪过程(也有叫逆向过程)也是一个马尔可夫过程。对马尔可夫假设的依赖，导致重建每一步都需要依赖上一步的状态，所以推理需要较多的步长。
\[q(x_t|x_{t-1}) := \mathcal{N}(x_t;\sqrt{\alpha_t}x_{t-1},{1-\alpha_t}I) \\ q(x_t|x_{0}) := \mathcal{N}(x_t;\sqrt{\bar{\alpha}t}x{0},{(1-\bar{\alpha}_t})I) \]
\[\begin{align*} q(x_{t-1}|x_t,x_0) &\overset{Bayes}{=} \dfrac{q(x_t|x_{t-1},x_0)q(x_{t-1}|x_0)}{q(x_t|x_0)} \\ &\overset{Markov}{=} \dfrac{q(x_t|x_{t-1})q(x_{t-1}|x_0)}{q(x_t|x_0)} \end{align*} \]
DDPM中对于其逆向分布的建模使用马尔可夫假设，这样做的目的是将式子中的未知项 \(q(x_t|x_{t-1},x_0)\)，转化成了已知项 \(q(x_t|x_{t-1})\)，最后求出 \(q(x_{t-1}|x_t,x_0)\) 的分布也是一个高斯分布 \(\mathcal{N}(x_{t-1};\mu_q(x_t,x_0),\Sigma_q(t))\)。
从DDPM的结论出发，我们不妨直接假设 \(q(x_{t-1}|x_t,x_0)\) 的分布为高斯分布，在不使用马尔可夫假设的情况下，尝试求解 \(q(x_{t-1}|x_t,x_0)\) 。
由 DDPM 中 \(q(x_{t-1}|x_t,x_0)\) 的分布 \(\mathcal{N}(x_{t-1};\mu_q(x_t,x_0),\Sigma_q(t))\) 可知，均值为 一个关于 \(x_t,x_0\) 的函数，方差为一个关于 \(t\) 的函数。
我们可以把 \(q(x_{t-1}|x_t,x_0)\) 设计成如下分布：
\[q(x_{t-1}|x_t,x_0) := \mathcal{N}(x_{t-1}; a x_0 + b x_t,\sigma_t^2 I) \]
这样，只要求解出 \(a,b,\sigma_t\) 这三个待定系数，即可确定 \(q(x_{t-1}|x_t,x_0)\) 的分布。
重参数化 \(q(x_{t-1}|x_t,x_0)\) ：
\[x_{t-1}=a x_0 + b x_t + \sigma_t \varepsilon^{\prime}_{t-1} \]
假设训练模型时输入噪声图片的加噪参数与DDPM完全一致
由 \(q(x_t|x_{0}) := \mathcal{N}(x_t;\sqrt{\bar{\alpha}t}x{0},(1-\bar{\alpha}_t)I)\) ：
\[x_t=\sqrt{\bar{\alpha}t}x{0}+\sqrt{1-\bar{\alpha}t}\varepsilon^{\prime}{t} \]
代入 \(x_t\) 有：
\[\begin{align*} x_{t-1} &=a x_0 + b(\sqrt{\bar{\alpha}t}x{0}+\sqrt{1-\bar{\alpha}t}\varepsilon^{\prime}{t}) + \sigma_t \varepsilon^{\prime}_{t-1} \\ &= (a + b\sqrt{\bar{\alpha}t}) x_0 + (b\sqrt{1-\bar{\alpha}t}\varepsilon^{\prime}{t} + \sigma_t \varepsilon^{\prime}{t-1}) \\ &= (a + b\sqrt{\bar{\alpha}_t}) x_0 + (\sqrt{b^2(1-\bar{\alpha}t)+ \sigma_t^2}) \bar{\varepsilon}{t-1} \end{align*} \]
又：
\[x_{t-1}=\sqrt{\bar{\alpha}{t-1}} x_0 + \sqrt{1-\bar{\alpha}{t-1}} \varepsilon^{\prime}_{t-1} \]
观察系数可以得到方程组：
\[\begin{cases} a + b\sqrt{\bar{\alpha}t} = \sqrt{\bar{\alpha}{t-1}} \\ \sqrt{b^2(1-\bar{\alpha}t)+ \sigma_t^2} = \sqrt{1-\bar{\alpha}{t-1}} \end{cases} \]
三个未知数 两个方程，可以用 \(\sigma_t\) 表示 \(a,b\)：
\[\begin{cases} a = \sqrt{\bar{\alpha}_{t-1}} - \sqrt{\bar{\alpha}t} \sqrt{\dfrac{1-\bar{\alpha}{t-1}-\sigma_t^2}{1-\bar{\alpha}t}} \\ b = \sqrt{\dfrac{1-\bar{\alpha}{t-1}-\sigma_t^2}{1-\bar{\alpha}_t}} \end{cases} \]
\(a, b\) 代入 \(q(x_{t-1}|x_t,x_0) := \mathcal{N}(x_{t-1}; a x_0 + b x_t,\sigma_t^2 I)\)
\[q(x_{t-1}|x_t,x_0) := \mathcal{N}(x_{t-1}; \underbrace{ \left( \sqrt{\bar{\alpha}_{t-1}} - \sqrt{\bar{\alpha}t} \sqrt{\dfrac{1-\bar{\alpha}{t-1}-\sigma_t^2}{1-\bar{\alpha}t}}\right ) x_0 + (\sqrt{\dfrac{1-\bar{\alpha}{t-1}-\sigma_t^2}{1-\bar{\alpha}t}}) x_t}{\mu_q(x_t,x_0,t)},\sigma_t^2 I) \]
又
\[x_t=\sqrt{\bar{\alpha}_t} x_0 + \sqrt{1-\bar{\alpha}_t} \bar{\varepsilon}_0 \\ x_0 = \dfrac{1}{\sqrt{\bar{\alpha}_t}}x_t - \dfrac{\sqrt{1-\bar{\alpha}_t}}{\sqrt{\bar{\alpha}_t}} \bar{\varepsilon}_0 \\ \]
代入 \(x_0\) 有：
\[\mu_q(x_t,x_0,t) = \sqrt{\bar{\alpha}_{t-1}} \dfrac{x_t-\sqrt{1-\bar{\alpha}t} \bar{\varepsilon}0}{\sqrt{\bar{\alpha}{t}}} + \sqrt{1-\bar{\alpha}{t-1}-\sigma_t^2} \bar{\varepsilon}_0 \\ \]
\[\begin{align*} x_{t-1} &= \mu_q(x_t,x_0,t) + \sigma_t \varepsilon_0 \\ &= \sqrt{\bar{\alpha}{t-1}} \underbrace{\dfrac{x_t-\sqrt{1-\bar{\alpha}t} \bar{\varepsilon}0}{\sqrt{\bar{\alpha}{t}}}}{预测的x_0} + \underbrace{\sqrt{1-\bar{\alpha}{t-1}-\sigma_t^2} \bar{\varepsilon}0}{x_t的方向} + \underbrace{\sigma_t \varepsilon_0}_{随机噪声扰动} \end{align*} \]
通过观察 \(x_{t-1}\) 的分布，我们建模采样分布为高斯分布：
\[p_\theta(x_{t-1}|x_t):=\mathcal{N}(x_{t-1};\mu_\theta(x_t,t), \Sigma_\theta(x_t,t)I) \]
并且均值和方差也采用相似的形式：
\[\begin{align*} \mu_\theta(x_t,t) &= \sqrt{\bar{\alpha}{t-1}} \dfrac{x_t-\sqrt{1-\bar{\alpha}t} \epsilon\theta(x_t,t) }{\sqrt{\bar{\alpha}{t}}} + \sqrt{1-\bar{\alpha}{t-1}-\sigma_t^2} \epsilon\theta(x_t,t) \\ \Sigma_\theta(x_t,t) &= \sigma_t^2 \end{align*} \]
其中 \(\epsilon_\theta(x_t,t)\) 为预测的噪声。
此时，确定优化目标只需要 \(q(x_{t-1}|x_t,x_0)\) 和 \(p_\theta(x_{t-1}|x_t)\) 两个分布尽可能相似，使用KL散度来度量，则有：
\[\begin{align*} &\quad \ \underset{\theta}{argmin} D_{KL}(q(x_{t-1}|x_t,x_0)||p_\theta(x_{t-1}|x_t)) \\ &=\underset{\theta}{argmin} D_{KL}(\mathcal{N}(x_{t-1};\mu_q, \Sigma_q(t))||\mathcal{N}(x_{t-1};\mu_\theta, \Sigma_q(t))) \\ &=\underset{\theta}{argmin} \dfrac{1}{2} \left[ log\dfrac{|\Sigma_q(t)|}{|\Sigma_q(t)|} - k + tr(\Sigma_q(t)^{-1}\Sigma_q(t)) + (\mu_q-\mu_\theta)^T \Sigma_q(t)^{-1} (\mu_q-\mu_\theta) \right] \\ &=\underset{\theta}{argmin} \dfrac{1}{2} \left[ 0 - k + k + (\mu_q-\mu_\theta)^T (\sigma_t^2I)^{-1} (\mu_q-\mu_\theta) \right] \\ &\overset{内积公式A^TA}{=} \underset{\theta}{argmin} \dfrac{1}{2\sigma_t^2} \left[ ||\mu_q-\mu_\theta||2^2 \right] \\ &\overset{代入\mu_q,\mu\theta}{=} \underset{\theta}{argmin} \dfrac{1}{2\sigma_t^2} (\sqrt{1-\bar{\alpha}{t-1}-\sigma_t^2} - \dfrac{\sqrt{\bar{\alpha}{t-1}} \sqrt{1-\bar{\alpha}_t}}{\sqrt{\bar{\alpha}_t}}) \left[ ||\bar{\varepsilon}0-\epsilon\theta(x_t,t)||_2^2 \right] \end{align*} \]
恰好与DDPM的优化目标一致，所以我们可以直接复用DDPM训练好的模型。
\(p_{\theta}\) 的采样步骤则为：
\[x_{t-1} = \sqrt{\bar{\alpha}{t-1}} \underbrace{\dfrac{x_t-\sqrt{1-\bar{\alpha}t} \epsilon\theta(x_t,t)}{\sqrt{\bar{\alpha}{t}}}}{预测的x_0} + \underbrace{\sqrt{1-\bar{\alpha}{t-1}-\sigma_t^2} \epsilon_\theta(x_t,t)}{x_t的方向} + \underbrace{\sigma_t \varepsilon}{随机噪声扰动} \]
令 \(\sigma_t=\eta \sqrt{\dfrac{(1-{\alpha}{t})(1-\bar{\alpha}{t-1})}{1-\bar{\alpha}_{t}}}\)
当 \(\eta =1\) 时，前向过程为 Markovian ，采样过程变为 DDPM 。
当 \(\eta =0\) 时，采样过程为确定过程，此时的模型 称为 隐概率模型(implicit probabilstic model)。
DDIM如何加速采样：
在 DDPM 中，基于马尔可夫链 \(t\) 与 \(t-1\) 是相邻关系，例如 \(t=100\) 则 \(t-1=99\)；
在 DDIM 中，\(t\) 与 \(t-1\) 只表示前后关系，例如 \(t=100\) 时，\(t-1\) 可以是 90 也可以是 80、70，只需保证 \(t-1 < t\) 即可。
此时构建的采样子序列 \(\tau=[\tau_i,\tau_{i-1},\cdots,\tau_{1}] \ll [t,t-1,\cdots,1]\) 。
例如，原序列 \(\Tau=[100,99,98,\cdots,1]\)，采样子序列为 \(\tau=[100,90,80,\cdots,1]\) 。
DDIM 采样公式为：
\[x_{\tau_{i-1}} = \sqrt{\bar{\alpha}{\tau{i-1}}} {\dfrac{x_{\tau_{i}}-\sqrt{1-\bar{\alpha}{\tau{i}}} \epsilon_\theta(x_{\tau_{i}},{\tau_{i}})}{\sqrt{\bar{\alpha}{\tau{i}}}}} + {\sqrt{1-\bar{\alpha}{\tau{i-1}}-\sigma_{\tau_{i}}^2} \epsilon_\theta(x_{\tau_{i}},{\tau_{i}})} + {\sigma_{\tau_{i}} \varepsilon} \]
当 \(\eta= 0\) 时，DDIM 采样公式为：
\[ x_{\tau_{i-1}} = \dfrac{\sqrt{\bar{\alpha}{\tau{i-1}}}}{\sqrt{\bar{\alpha}{\tau{i}}}} x_{\tau_{i}} + \left( \sqrt{1-\bar{\alpha}{\tau{i-1}}} - \dfrac{\sqrt{\bar{\alpha}{\tau{i-1}}}}{\sqrt{\bar{\alpha}{\tau{i}}}} \sqrt{1-\bar{\alpha}{\tau{i}}} \right) \epsilon_\theta(x_{\tau_i},\tau_i) \]
代码实现
训练过程与 DDPM 一致，代码参考上一篇文章。采样代码如下：
py
device = 'cuda'
torch.cuda.empty_cache()
model = Unet().to(device)
model.load_state_dict(torch.load('ddpm_T1000_l2_epochs_300.pth'))
model.eval()
image_size=96
epochs = 500
batch_size = 128
T=1000
betas = torch.linspace(0.0001, 0.02, T).to('cuda') # torch.Size([1000])
# 每隔20采样一次
tau_index = list(reversed(range(0, T, 20))) #[980, 960, ..., 20, 0]
eta = 0.003
# train
alphas = 1 - betas # 0.9999 -> 0.98
alphas_cumprod = torch.cumprod(alphas, axis=0) # 0.9999 -> 0.0000
sqrt_alphas_cumprod = torch.sqrt(alphas_cumprod)
sqrt_one_minus_alphas_cumprod = torch.sqrt(1-alphas_cumprod)
def get_val_by_index(val, t, x_shape):
batch_t = t.shape[0]
out = val.gather(-1, t)
return out.reshape(batch_t, *((1,) * (len(x_shape) - 1))) # torch.Size([batch_t, 1, 1, 1])
def p_sample_ddim(model):
def step_denoise(model, x_tau_i, tau_i, tau_i_1):
sqrt_alphas_bar_tau_i = get_val_by_index(sqrt_alphas_cumprod, tau_i, x_tau_i.shape)
sqrt_alphas_bar_tau_i_1 = get_val_by_index(sqrt_alphas_cumprod, tau_i_1, x_tau_i.shape)
denoise = model(x_tau_i, tau_i)
if eta == 0:
sqrt_1_minus_alphas_bar_tau_i = get_val_by_index(sqrt_one_minus_alphas_cumprod, tau_i, x_tau_i.shape)
sqrt_1_minus_alphas_bar_tau_i_1 = get_val_by_index(sqrt_one_minus_alphas_cumprod, tau_i_1, x_tau_i.shape)
x_tau_i_1 = sqrt_alphas_bar_tau_i_1 / sqrt_alphas_bar_tau_i * x_tau_i \
+ (sqrt_1_minus_alphas_bar_tau_i_1 - sqrt_alphas_bar_tau_i_1 / sqrt_alphas_bar_tau_i * sqrt_1_minus_alphas_bar_tau_i) \
* denoise
return x_tau_i_1
sigma = eta * torch.sqrt((1-get_val_by_index(alphas, tau_i, x_tau_i.shape)) * \
(1-get_val_by_index(sqrt_alphas_cumprod, tau_i_1, x_tau_i.shape)) / get_val_by_index(sqrt_one_minus_alphas_cumprod, tau_i, x_tau_i.shape))
noise_z = torch.randn_like(x_tau_i, device=x_tau_i.device)
# 整个式子由三部分组成
c1 = sqrt_alphas_bar_tau_i_1 / sqrt_alphas_bar_tau_i * (x_tau_i - get_val_by_index(sqrt_one_minus_alphas_cumprod, tau_i, x_tau_i.shape) * denoise)
c2 = torch.sqrt(1 - get_val_by_index(alphas_cumprod, tau_i_1, x_tau_i.shape) - sigma) * denoise
c3 = sigma * noise_z
x_tau_i_1 = c1 + c2 + c3
return x_tau_i_1
img_pred = torch.randn((4, 3, image_size, image_size), device=device)
for k in range(0, len(tau_index)):
# print(tau_index)
# 因为 tau_index 是倒序的，tau_i = k, tau_i_1 = k+1，这里不能弄反
tau_i_1 = torch.tensor([tau_index[k+1]], device=device, dtype=torch.long)
tau_i = torch.tensor([tau_index[k]], device=device, dtype=torch.long)
img_pred = step_denoise(model, img_pred, tau_i, tau_i_1)
torch.cuda.empty_cache()
if tau_index[k+1] == 0: return img_pred
return img_pred
with torch.no_grad():
img = p_sample_ddim(model)
img = torch.clamp(img, -1.0, 1.0)
show_img_batch(img.detach().cpu())
DDIM
https://arxiv.org/pdf/2010.02502
https://github.com/ermongroup/ddim