Var(N)Var(aN_{1}bN_{2})a^2\sigma_{1}^2b^2\sigma_{2}^2

E(N)E(aN1​bN2​)aμ1​bμ2​Var(N)Var(aN1​bN2​)a2σ12​b2σ22​

相减时

E(N)E(aN_{1}-bN_{2})a\mu_{1}-b\mu_{2}\\

Var(N)Var(aN_{1}-bN_{2})a^2\sigma_{1}^2b^2\sigma_{2}^2

E(N)E(aN1​−bN2​)aμ1​−bμ2​Var(N)Var(aN1​−bN2​)a2σ12​b2σ22​

8、重参数化:从

f(x)\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

f(x)2π

KL(p|q)log\frac{\sigma_2}{\sigma_1}\frac{\sigma^2(\mu_1-\mu_2)^2}{2\sigma_2^2}-\frac{1}{2}

KL(p∣q)logσ1​σ2​​2σ22​σ2(μ1​−μ2​)2​−21​

11、二次函数配方

\displaystyle\int_{-\infty}^{\infty}g(x)p(x)dx

\end{cases}

⎧​k1∑∞​g(xk​)pk​∫−∞∞​g(x)p(x)dx​

13、KL散度公式

KL(p(x)∣q(x))Ex∼p(x)​[q(x)p(x)​]∫p(x)q(x)p(x)​dx

介绍DDPM

Models简称扩散模型是AIGC的核心算法在生成图像的真实性和多样性方面均超越了GAN而且训练过程稳定。

缺点是计算成本较高实时推理比较困难但也有相关技术在时间和空间维度上降低计算量。

扩散模型包括两个过程前向扩散过程(前向加噪过程)和反向去噪过程。

前向过程和反向过程都是马尔可夫链全过程大约需要1000步其中反向过程用来生成数据它的推导过程可以描述成

前向扩散过程是对原始数据逐渐增加高斯噪声直至变成标准高斯分布的过程。

从原始数据集采样

q(x_{1:T}|x_{0})\prod_{t1}^{T}q(x_t|x_{t-1})

\\q(x_{t}|x_{t-1})\mathcal{N}(x_t;\sqrt{\alpha_t}x_{t-1},\beta_{t}I)\\

\end{aligned}

q(x1:T​∣x0​)q(xt​∣xt−1​)​t1∏T​q(xt​∣xt−1​)N(xt​;αt​

​xt−1​,βt​I)​

x_{t}\sqrt{\alpha_{t}}x_{t-1}\sqrt{\beta_{t}}\epsilon_{t}

\space

x_{t-1}\sqrt{\beta_{t}}\epsilon_{t}\\

\sqrt{\alpha_{t}}(\sqrt{\alpha_{t-1}}x_{t-2}\sqrt{\beta_{t-1}}\epsilon_{t-1})\sqrt{\beta_{t}}\epsilon_{t}\\

\sqrt{(\alpha_{t}\dots\alpha_{1})}x_{0}\sqrt{(\alpha_{t}\dots\alpha_{2})\beta_{1}}\epsilon_{1}\sqrt{(\alpha_{t}\dots\alpha_{3})\beta_{2}}\epsilon_{2}\dots\sqrt{\alpha_{t}\beta_{t-1}}\epsilon_{t-1}\sqrt{\beta_{t}}\epsilon_{t}

\end{aligned}

\bar{\alpha_{t}}\alpha_{1}\alpha_{2}\dots\alpha_{t}

αt​ˉ​α1​α2​…αt​

x_{t}\sqrt{\bar{\alpha_{t}}}x_{0}\sqrt{1-\bar{\alpha_{t}}}\epsilon

\space

\textcolor{REd}{q(x_{t}|x_{0})\mathcal{N}(x_{t};\sqrt{\bar{\alpha_{t}}}x_{0},\sqrt{1-\bar{\alpha_{t}}}I)}

xt​αt​ˉ​

反向去噪过程就是数据生成过程它首先是从标准高斯分布中采样得到一个噪声样本再一步步地迭代去噪最后得到数据分布中的一个样本。

q(x_{t-1}|x_t)

q(xt−1​∣xt​)那么从一个随机噪声开始逐步采样就能生成一个真实的样本。

但是真实的条件分布利用贝叶斯公式

q(x_{t-1}|x_{t})

\frac{q(x_{t}|x_{t-1})q(x_{t-1})}{q(x_{t})}

q(xt−1​∣xt​)q(xt​)q(xt​∣xt−1​)q(xt−1​)​

q(x_{t-1})

pθ​(xt−1​∣xt​)来近似。

为了简化起见将反向过程也定义为一个马尔卡夫链且服从高斯分布**建模如下:

p_\***ta(x_{0:T})p(x_T)\prod_{t1}^Tp_\***ta(x_{t-1}|x_t)\\

p_\***ta(x_{t-1}|x_t)N(x_{t-1};\mu_\***ta(x_t,t),\sum_\***ta(x_t,t))

pθ​(x0:T​)p(xT​)t1∏T​pθ​(xt−1​∣xt​)pθ​(xt−1​∣xt​)N(xt−1​;μθ​(xt​,t),θ∑​(xt​,t))

--------------------下面这段讲解与上面有些跳脱是为损失函数做铺垫------------------------------

虽然真实条件分布

x0​的后验分布$q(x_{t-1}|x_{t},x_{0})

\frac{q(x_{t}|x_{t-1},x_{0})q(x_{t-1}|x_{0})}{q(x_{t}|x_{0})}\frac{q(x_{t}|x_{t-1})q(x_{t-1}|x_{0})}{q(x_{t}|x_{0})}

q(xt−1​∣xt​,x0​)q(xt​∣x0​)q(xt​∣xt−1​,x0​)q(xt−1​∣x0​)​q(xt​∣x0​)q(xt​∣xt−1​)q(xt−1​∣x0​)​

\begin{aligned}

q(x_{t-1}|x_{0})\sqrt{\bar{\alpha}_{t-1}}x_{0}\sqrt{1-\bar{\alpha}_{t-1}}\epsilon\sim

\mathcal{N}(\sqrt{\bar{\alpha}_{t-1}}x_{0},(1-\bar{\alpha}_{t-1})I)\\

q(x_{t}|x_{0})\sqrt{\bar{\alpha}_{t}}x_{0}\sqrt{1-\bar{\alpha}_{t}}\epsilon\sim

\mathcal{N}(\sqrt{\bar{\alpha}_{t}}x_{0},(1-\bar{\alpha}_{t})I)\\

q(x_{t}|x_{t-1})\sqrt{\alpha}_{t}x_{t-1}\beta_{t}\epsilon\sim

\mathcal{N}(\sqrt{\alpha}_{t}x_{t-1},\beta_{t}I)

\end{aligned}

q(xt−1​∣x0​)q(xt​∣x0​)q(xt​∣xt−1​)​αˉt−1​

​x0​1−αˉt−1​

exp(-\frac{1}{2}(\frac{(x_{t}-\sqrt{\alpha_{t}}x_{t-1})^2}{\beta_{t}})\frac{(x_{t-1}-\sqrt{\bar{\alpha}}_{t-1}x_{0})^2}{1-\bar{\alpha}_{t-1}}-\frac{(x_{t}-\sqrt{\bar{\alpha}_{t}}x_{0})^2}{1-\bar{\alpha}_{t}})\\

exp(-\frac{1}{2}(\frac{\alpha_{t}}{\beta_{t}}\frac{1}{1-\bar{\alpha}_{t-1}})x_{t-1}^2-(\frac{2\sqrt{\alpha_{t}}}{\beta_{t}}x_{t}\frac{2\sqrt{\bar{\alpha_{t}}}}{1-\bar{\alpha_{t}}}x_{0})x_{t-1}C(x_{t},x_{0}))

\end{aligned}

q(xt−1​∣xt​,x0​)​∝exp(−21​(βt​(xt​−αt​

​x0​)2​)exp(−21​(βt​αt​​1−αˉt−1​1​)xt−12​−(βt​2αt​

​​xt​1−αt​ˉ​2αt​ˉ​

\widetilde{\beta}_t1/(\frac{\alpha_{t}}{\beta_{t}}\frac{1}{1-\bar{\alpha}_{t-1}})\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t}}\beta_{t}\\

\widetilde{\mu}_t(\frac{\sqrt\alpha_{t}}{\beta_{t}}x_{t}\frac{\sqrt{\bar{\alpha}_{t}}}{1-\bar{\alpha_{t}}}x_{0})/(\frac{\alpha_{t}}{\beta_{t}}\frac{1}{1-\bar{\alpha}_{t-1}})\frac{\sqrt{\alpha_{t}}(1-\bar{\alpha}_{t-1})}{1-\bar{\alpha_{t}}}x_{t}\frac{\sqrt{\bar{\alpha}_{t-1}}\beta_{t}}{1-\bar{\alpha}_{t}}x_{0}

​t​1/(βt​αt​​1−αˉt−1​1​)1−αˉt​1−αˉt−1​​βt​μ

​t​(βt​α

​​x0​)/(βt​αt​​1−αˉt−1​1​)1−αt​ˉ​αt​

​βt​​x0​

\frac{1}{\sqrt{\bar\alpha_t}}(x_t-

\alpha_t}

\widetilde{\mu}_t\frac{1}{\sqrt{\alpha_t}

}(x_t-\frac{\beta_t}{\sqrt{(1-\alpha_t)}}\epsilon)

​t​αt​

----------------------------------------------------------------------------------------------

\mu_\***ta(x_t,t)\frac{1}{\sqrt{\alpha_t}

}(x_t-\frac{\beta_t}{\sqrt{(1-\alpha_t)}}\epsilon_\***ta(x_t,t))\\

x_{t-1}\sim

x_{t-1}\frac{1}{\sqrt{\alpha_t}

}(x_t-\frac{\beta_t}{\sqrt{(1-\alpha_t)}}\epsilon_\***ta(x_t,t))\sqrt{\widetilde{\beta}_t}z

\space

​βt​​ϵθ​(xt​,t))xt−1​∼pθ​(xt−1​∣xt​)xt−1​αt​

​1​(xt​−(1−αt​)

https://blog.csdn.net/weixin_45453121/article/details/131223653

Code

num_samples:breakplt.subplot(int(num_samples/cols)

cols,

linear_beta_schedule(timesteps,

start0.0001,

t.cpu())#print(out:,out)#print(out.shape:,out.shape)return

*((1,)

torch.randn_like(x_0)sqrt_alphas_cumprod_t

get_index_from_list(sqrt_alphas_cumprod,

x_0.shape)sqrt_one_minus_alphas_cumprod_t

get_index_from_list(sqrt_one_minus_alphas_cumprod,

x_0.shape)#

sqrt_alphas_cumprod_t.to(device)

x_0.to(device)

sqrt_one_minus_alphas_cumprod_t.to(device)

noise.to(device),

load_transformed_dataset(IMG_SIZE):data_transforms

IMG_SIZE)),transforms.ToTensor(),

Scales

transforms.Compose(data_transforms)train

torchvision.datasets.MNIST(root./Data,transformdata_transform,trainTrue)test

torchvision.datasets.MNIST(root./Data,

trainFalse)return

torch.utils.data.ConcatDataset([train,

test])def

show_tensor_image(image):reverse_transforms

transforms.Compose([transforms.Lambda(lambda

t.permute(1,

t.numpy().astype(np.uint8)),transforms.ToPILImage(),])#Take

first

:]plt.imshow(reverse_transforms(image))class

Block(nn.Module):def

upFalse):super().__init__()self.time_mlp

out_ch)if

nn.BatchNorm2d(out_ch)self.bnorm2

nn.BatchNorm2d(out_ch)self.relu

nn.ReLU()def

self.bnorm1(self.relu(self.conv1(x)))#

Time

self.bnorm2(self.relu(self.conv2(h)))#

Down

SinusoidalPositionEmbeddings(nn.Module):def

__init__(self,

dim):super().__init__()self.dim

dimdef

torch.exp(torch.arange(half_dim,

devicedevice)

__init__(self):super().__init__()image_channels

(64,

nn.Sequential(SinusoidalPositionEmbeddings(time_emb_dim),nn.Linear(time_emb_dim,

Initial

nn.ModuleList([Block(down_channels[i],

down_channels[i1],

nn.ModuleList([Block(up_channels[i],

up_channels[i1],

x.shape)sqrt_one_minus_alphas_cumprod_t

get_index_from_list(sqrt_one_minus_alphas_cumprod,

get_index_from_list(sqrt_recip_alphas,

x.shape)#

sqrt_one_minus_alphas_cumprod_t)posterior_variance_t

get_index_from_list(posterior_variance,

x.shape)if

torch.sqrt(posterior_variance_t)

noisetorch.no_grad()

#生成第T步的图片plt.figure(figsize(15,15))plt.axis(off)num_images

10stepsize

dtypetorch.long)#print(t:,t)img

sample_timestep(img,

int(i/stepsize)1)plt.title(str(i))show_tensor_image(img.detach().cpu())plt.show()if

__name__

linear_beta_schedule(timestepsT)#

Pre-calculate

print(alphas_cumprod.shape)alphas_cumprod_prev

(1,

print(alphas_cumprod_prev.shape)sqrt_recip_alphas

torch.sqrt(1.0

torch.sqrt(alphas_cumprod)sqrt_one_minus_alphas_cumprod

torch.sqrt(1.

alphas_cumprod)posterior_variance

betas

print(posterior_variance.shape)IMG_SIZE

32BATCH_SIZE

load_transformed_dataset(IMG_SIZE)dataloader

DataLoader(data,

是包含标签的所以取batch[0]#print(batch[0].shape)optimizer.zero_grad()t

torch.randint(0,

t)loss.backward()optimizer.step()if

epoch

)sample_plot_image(IMG_SIZE)参考文献

https://zhuanlan.zhihu.com/p/630354327](https://zhuanlan.zhihu.com/p/630354327)

https://blog.csdn.net/weixin_45453121/article/details/131223653

https://www.cnblogs.com/risejl/p/17448442.html

https://zhuanlan.zhihu.com/p/569994589?utm_id0