QAbstractItemModel 曲线

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0、前向与逆向过程

  原论文：Denoising Diffusion Probabilistic Models

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1、数据集准备

  选一个数据集，本例采用sklearn自带数据集：

import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import make_s_curve
import torch

s_curve, _ = make_s_curve(10**4, noise=0.1)
s_curve = s_curve[:,[0,2]]/10.0  # 每个点取第0维和第2维, 再除以10
print("shape of s:", np.shape(s_curve))

data = s_curve.T
fig,ax = plt.subplots()
ax.scatter(*data,color='orange',edgecolor='white');
ax.axis('off')

dataset = torch.Tensor(s_curve).float()  # 将S曲线构建成一个张量

  S曲线显示：

---

2、参数计算

  计算前向过程中需要的常数：

# 准备好alpha(αt), beta(βt), αt^, 根号下αt^, 根号下(1-αt^)等值
num_steps = 100  # 设置步长

# 制定每一步的beta
betas = torch.linspace(-6, 6, num_steps)  # 逐渐递增
betas = torch.sigmoid(betas)*(0.5e-2 - 1e-5)+1e-5  # β0,β1,...,βt

# 计算alpha、alpha_prod、alpha_prod_previous、alpha_bar_sqrt等变量的值
alphas = 1 - betas  # αt = 1 - βt
alphas_prod = torch.cumprod(alphas,0)  # αt^ = αt的累乘
# αt^往右平移一位, 原第t步的值维第t-1步的值, 第0步补1
alphas_prod_p = torch.cat([torch.tensor([1]).float(), alphas_prod[:-1]], 0)  # αt-1 ^
alphas_bar_sqrt = torch.sqrt(alphas_prod)  # αt^ 开根号
one_minus_alphas_bar_log = torch.log(1 - alphas_prod)  # log(1-αt^)
one_minus_alphas_bar_sqrt = torch.sqrt(1 - alphas_prod)  # 根号下(1-αt^)

assert alphas.shape==alphas_prod.shape==alphas_prod_p.shape==alphas_bar_sqrt.shape==one_minus_alphas_bar_log.shape==one_minus_alphas_bar_sqrt.shape
print("all the same shape",betas.shape)

---

3、前向过程

  确定扩散过程任意时刻的采样值：

# 计算任意时刻的x采样值，基于x_0和重参数化
def q_x(x_0, t):
    """
    作用：前向过程, 可以基于x[0]得到任意时刻t的x[t]
    输入：x_0:初始干净图像；t：采样步
    输出：x_t:第t步时的x_0已成为的样子
    """
    noise = torch.randn_like(x_0)  # noise为从正态分布中采样的随机噪声
    
    alphas_t = alphas_bar_sqrt[t]  # 根号下αt^
    alphas_1_m_t = one_minus_alphas_bar_sqrt[t]  # 根号下(1-αt^)
    return (alphas_t * x_0 + alphas_1_m_t * noise) # 在x[0]的基础上添加噪声

---

4、前向过程展示

  演示原始数据分布加噪100步后的结果，可观察到从S曲线在100步中逐渐变为高斯分布的过程：

num_shows = 20
fig,axs = plt.subplots(2, 10, figsize=(28,3))
plt.rc('text',color='black')

#共有10000个点，每个点包含两个坐标
#生成100步以内每隔5步加噪声后的图像
for i in range(num_shows):
    j = i//10
    k = i%10
    q_i = q_x(dataset, torch.tensor([i*num_steps//num_shows])) # 生成t时刻的采样数据
    axs[j,k].scatter(q_i[:,0], q_i[:,1], color='green', edgecolor='white')
    axs[j,k].set_axis_off()
    axs[j,k].set_title('$q(\mathbf{x}_{'+str(i*num_steps//num_shows)+'})$')

---

5、模型搭建

  编写拟合逆扩散过程高斯分布的模型，写一个简单的网路，用于预测噪声：

import torch
import torch.nn as nn

class MLPDiffusion(nn.Module):
    
    def __init__(self, n_steps, num_units=128):
        super(MLPDiffusion,self).__init__()
        
        self.linears = nn.ModuleList(
            [
                nn.Linear(2, num_units),
                nn.ReLU(),
                nn.Linear(num_units,num_units),
                nn.ReLU(),
                nn.Linear(num_units,num_units),
                nn.ReLU(),
                nn.Linear(num_units,2),
            ]
        )
        self.step_embeddings = nn.ModuleList(
            [
                nn.Embedding(n_steps,num_units),  # [100,128]
                nn.Embedding(n_steps,num_units),
                nn.Embedding(n_steps,num_units),
            ]
        )
        
    def forward(self, x, t):
        for idx, embedding_layer in enumerate(self.step_embeddings):
            t_embedding = embedding_layer(t)  # 选第t步的Embedding
            x = self.linears[2*idx](x)  # 先送入Linear层
            x += t_embedding  # 加上Embedding
            x = self.linears[2*idx+1](x)  # 再送入ReLU层
        
        x = self.linears[-1](x)  # 最后一个Linear层, 输出为[10000, 2]
        return x

---

6、损失函数

  编写训练的误差函数，计算网络预测噪声与真实添加噪声的误差：

def diffusion_loss_fn(model, x_0, alphas_bar_sqrt, one_minus_alphas_bar_sqrt, n_steps):
    """
    作用：对任意时刻t进行采样计算loss
    参数：
        model: 模型
        x_0: 干净的图
        alphas_bar_sqrt: 根号下αt^
        one_minus_alphas_bar_sqrt: 根号下(1-αt^)
        n_steps: 采样步
    """
    batch_size = x_0.shape[0]
    
    # 对一个batchsize样本生成随机的时刻t, 覆盖到更多不同的t
    t = torch.randint(0, n_steps, size=(batch_size//2,))  # 在0~99内生成整数采样步
    t = torch.cat([t, n_steps-1-t], dim=0)  # 一个batch的采样步, 尽量让生成的t不重复
    t = t.unsqueeze(-1)  # 增加一个维度(8,1)
    
    # x0的系数
    a = alphas_bar_sqrt[t]  # 根号下αt^
    
    # eps的系数
    aml = one_minus_alphas_bar_sqrt[t]  # 根号下(1-αt^)
    
    # 生成随机噪音eps
    e = torch.randn_like(x_0)  
    
    # 构造模型的输入
    x = x_0*a+e*aml  # 前向过程：根号下αt^ * x0 + 根号下(1-αt^) * eps
    
    # 送入模型，得到t时刻的随机噪声预测值
    output = model(x, t.squeeze(-1))  # 模型预测的是噪声, 噪声维度与x0一样大, [10000,2]
    
    # 与真实噪声一起计算误差，求平均值
    return (e - output).square().mean()

---

7、逆向过程

  编写逆扩散采样函数，从随机噪声生成样本：

def p_sample_loop(model, shape, n_steps, betas, one_minus_alphas_bar_sqrt):
    """
    作用：从x[T]恢复x[T-1]、x[T-2]、...x[0]
    输入：
        model:模型
        shape:数据大小,用于生成随机噪声
        n_steps:逆扩散总步长
        betas: βt
        one_minus_alphas_bar_sqrt: 根号下(1-αt^)
    输出：
        x_seq：一个序列的x, 即 x[T]、x[T-1]、x[T-2]、...x[0]
    """
    cur_x = torch.randn(shape)  # 随机噪声, 对应xt
    x_seq = [cur_x]
    for i in reversed(range(n_steps)):
        cur_x = p_sample(model, cur_x, i, betas, one_minus_alphas_bar_sqrt)
        x_seq.append(cur_x)
    return x_seq


def p_sample(model, x, t, betas, one_minus_alphas_bar_sqrt):
    """
    作用：从x[T]采样t时刻的重构值
    输入：
        model:模型
        x: 采样的随机噪声x[T]
        t: 采样步
        betas: βt
        one_minus_alphas_bar_sqrt: 根号下(1-αt^)
    输出：
        sample: 样本
    """
    t = torch.tensor([t])
    
    coeff = betas[t] / one_minus_alphas_bar_sqrt[t]  # 模型输出的系数：βt/根号下(1-αt^) = 1-αt/根号下(1-αt^)
    
    eps_theta = model(x, t)  # 模型的输出: εθ(xt, t)
        
    # (1/根号下αt) * (xt - (1-αt/根号下(1-αt^))*εθ(xt, t))
    mean = (1/(1-betas[t]).sqrt())*(x-(coeff*eps_theta))  
    
    z = torch.randn_like(x)  # 对应公式中的 z
    sigma_t = betas[t].sqrt()  # 对应公式中的 σt
    
    sample = mean + sigma_t * z 
    
    return (sample)

---

8、模型训练

  开始训练模型，打印loss及中间重构效果：

print('Training model...')
batch_size = 128
dataloader = torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True)
num_epoch = 4000
plt.rc('text',color='blue')

model = MLPDiffusion(num_steps)  # 输出维度是2，输入是x和step
optimizer = torch.optim.Adam(model.parameters(), lr=1e-3) 

for t in range(num_epoch):
    for idx, batch_x in enumerate(dataloader):
        
        # 损失计算
        loss = diffusion_loss_fn(model, batch_x, alphas_bar_sqrt, one_minus_alphas_bar_sqrt, num_steps)
        optimizer.zero_grad()  # 梯度清零
        loss.backward()  # 损失回传
        torch.nn.utils.clip_grad_norm_(model.parameters(),1.)  # 梯度裁剪
        optimizer.step()  
        
    if(t % 100 == 0):
        print(loss)
        x_seq = p_sample_loop(model, dataset.shape, num_steps, betas, one_minus_alphas_bar_sqrt)
        
        fig, axs = plt.subplots(1, 10, figsize=(28,3))
        for i in range(1, 11):
            cur_x = x_seq[i*10].detach()
            axs[i-1].scatter(cur_x[:,0],cur_x[:,1],color='red',edgecolor='white');
            axs[i-1].set_axis_off();
            axs[i-1].set_title('$q(\mathbf{x}_{'+str(i*10)+'})$')

  训练过程如下，cpu训练约30min完成：

  重构效果展示（分别为0、1000、2000、3000、4000epoch的结果）：

---

9、动态可视化：

import io
from PIL import Image

# 前向过程
imgs = []
for i in range(100):
    plt.clf()
    q_i = q_x(dataset,torch.tensor([i]))
    plt.scatter(q_i[:,0],q_i[:,1],color='red',edgecolor='white',s=5);
    plt.axis('off');
    
    img_buf = io.BytesIO()
    plt.savefig(img_buf,format='png')
    img = Image.open(img_buf)
    imgs.append(img)

# 逆向过程
reverse = []
for i in range(100):
    plt.clf()
    cur_x = x_seq[i].detach()
    plt.scatter(cur_x[:,0],cur_x[:,1],color='red',edgecolor='white',s=5);
    plt.axis('off')
    
    img_buf = io.BytesIO()
    plt.savefig(img_buf,format='png')
    img = Image.open(img_buf)
    reverse.append(img)

imgs = imgs
imgs[0].save("diffusion_qian.gif", format='GIF', append_images=imgs, save_all=True, duration=100, loop=0)

imgs = reverse
imgs[0].save("diffusion_ni.gif", format='GIF', append_images=imgs, save_all=True, duration=100, loop=0)

前向过程：

逆向过程：

10、代码汇总：

import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import make_s_curve
import torch
import torch.nn as nn
import io
from PIL import Image


### 1、选择一个数据集-----------------------------------------------------------
s_curve, _ = make_s_curve(10**4, noise=0.1)
s_curve = s_curve[:,[0,2]]/10.0  # 每个点取第0维和第2维, 再除以10
print("shape of s:", np.shape(s_curve))

data = s_curve.T
fig,ax = plt.subplots()
ax.scatter(*data,color='orange',edgecolor='white');
ax.axis('off')

dataset = torch.Tensor(s_curve).float()  # 将S曲线构建成一个张量
### -------------------------------------------------------------------------

### 2、确定超参数的值-----------------------------------------------------------
# 准备好alpha(α), beta(β)等值

num_steps = 100  # 设置步长

# 制定每一步的beta
betas = torch.linspace(-6, 6, num_steps)  # 逐渐递增
betas = torch.sigmoid(betas)*(0.5e-2 - 1e-5)+1e-5  # β0,β1,...,βt

# 计算alpha、alpha_prod、alpha_prod_previous、alpha_bar_sqrt等变量的值
alphas = 1 - betas  # αt = 1 - βt
alphas_prod = torch.cumprod(alphas,0)  # αt^ = αt的累乘
# αt^往右平移一位, 原第t步的值维第t-1步的值, 第0步补1
alphas_prod_p = torch.cat([torch.tensor([1]).float(), alphas_prod[:-1]], 0)  # αt-1^
alphas_bar_sqrt = torch.sqrt(alphas_prod)  # αt^ 开根号
one_minus_alphas_bar_log = torch.log(1 - alphas_prod)  # log(1-αt^)
one_minus_alphas_bar_sqrt = torch.sqrt(1 - alphas_prod)  # 根号下(1-αt^)

assert alphas.shape==alphas_prod.shape==alphas_prod_p.shape==\
alphas_bar_sqrt.shape==one_minus_alphas_bar_log.shape\
==one_minus_alphas_bar_sqrt.shape
print("all the same shape",betas.shape)
### ------------------------------------------------------------------------

### 3、确定扩散过程任意时刻的采样值----------------------------------------------
# 计算任意时刻的x采样值，基于x_0和重参数化
def q_x(x_0, t):
    
    """
    作用：可以基于x[0]得到任意时刻t的x[t]
    输入：x_0:初始干净图像；t：采样步
    输出：x_t:第t步时的x_0的样子
    """
    noise = torch.randn_like(x_0)  # noise为从正态分布中采样的随机噪声
    
    alphas_t = alphas_bar_sqrt[t]  
    alphas_1_m_t = one_minus_alphas_bar_sqrt[t]
    return (alphas_t * x_0 + alphas_1_m_t * noise) # 在x[0]的基础上添加噪声
### ------------------------------------------------------------------------

### 4、演示原始数据分布加噪100步后的结果-----------------------------------------
num_shows = 20
fig,axs = plt.subplots(2, 10, figsize=(28,3))
plt.rc('text',color='black')

#共有10000个点，每个点包含两个坐标
#生成100步以内每隔5步加噪声后的图像
for i in range(num_shows):
    j = i//10
    k = i%10
    q_i = q_x(dataset, torch.tensor([i*num_steps//num_shows])) # 生成t时刻的采样数据
    axs[j,k].scatter(q_i[:,0], q_i[:,1], color='green', edgecolor='white')
    axs[j,k].set_axis_off()
    axs[j,k].set_title('$q(\mathbf{x}_{'+str(i*num_steps//num_shows)+'})$')
### ------------------------------------------------------------------------

### 5、编写拟合逆扩散过程高斯分布的模型-----------------------------------------

class MLPDiffusion(nn.Module):
    
    def __init__(self, n_steps, num_units=128):
        super(MLPDiffusion,self).__init__()
        
        self.linears = nn.ModuleList(
            [
                nn.Linear(2, num_units),
                nn.ReLU(),
                nn.Linear(num_units,num_units),
                nn.ReLU(),
                nn.Linear(num_units,num_units),
                nn.ReLU(),
                nn.Linear(num_units,2),
            ]
        )
        self.step_embeddings = nn.ModuleList(
            [
                nn.Embedding(n_steps,num_units),  # [100,128]
                nn.Embedding(n_steps,num_units),
                nn.Embedding(n_steps,num_units),
            ]
        )
        
    def forward(self, x, t):
        for idx, embedding_layer in enumerate(self.step_embeddings):
            t_embedding = embedding_layer(t)  # 选第t步的Embedding
            x = self.linears[2*idx](x)  # 先送入Linear层
            x += t_embedding  # 加上Embedding
            x = self.linears[2*idx+1](x)  # 再送入ReLU层
        
        x = self.linears[-1](x)  # 最后一个Linear层, 输出为[10000, 2]
        return x
### ------------------------------------------------------------------------

### 6、编写训练的误差函数------------------------------------------------------
def diffusion_loss_fn(model, x_0, alphas_bar_sqrt, one_minus_alphas_bar_sqrt, n_steps):
    """
    作用：对任意时刻t进行采样计算loss
    参数：
        model: 模型
        x_0: 干净的图
        alphas_bar_sqrt: αt^开根号
        one_minus_alphas_bar_sqrt: 根号下(1-αt^)
        n_steps: 采样步
    """
    batch_size = x_0.shape[0]
    
    # 对一个batchsize样本生成随机的时刻t, 覆盖到更多不同的t
    t = torch.randint(0, n_steps, size=(batch_size//2,))  # 在0~99内生成整数采样步
    t = torch.cat([t, n_steps-1-t], dim=0)  # 一个batch的采样步, 尽量让生成的t不重复
    t = t.unsqueeze(-1)  # 增加一个维度(8,1)
    
    # x0的系数
    a = alphas_bar_sqrt[t]  # 根号下αt^
    
    # eps的系数
    aml = one_minus_alphas_bar_sqrt[t]  # 根号下(1-αt^)
    
    # 生成随机噪音eps
    e = torch.randn_like(x_0)  
    
    # 构造模型的输入
    x = x_0*a+e*aml  # 前向过程：根号下αt^ * x0 + 根号下(1-αt^) * eps
    
    # 送入模型，得到t时刻的随机噪声预测值
    output = model(x, t.squeeze(-1))  # 模型预测的是噪声, 噪声维度与x0一样大, [10000,2]
    
    # 与真实噪声一起计算误差，求平均值
    return (e - output).square().mean()

### ------------------------------------------------------------------------

### 7、编写逆扩散采样函数（inference）------------------------------------------
def p_sample_loop(model, shape, n_steps, betas, one_minus_alphas_bar_sqrt):
    """
    作用：从x[T]恢复x[T-1]、x[T-2]、...x[0]
    输入：
        model:模型
        shape:数据大小,用于生成随机噪声
        n_steps:逆扩散总步长
        betas: βt
        one_minus_alphas_bar_sqrt: 根号下(1-αt^)
    输出：
        x_seq：一个序列的x, 即 x[T]、x[T-1]、x[T-2]、...x[0]
    """
    cur_x = torch.randn(shape)  # 随机噪声, 对应xt
    x_seq = [cur_x]
    for i in reversed(range(n_steps)):
        cur_x = p_sample(model, cur_x, i, betas, one_minus_alphas_bar_sqrt)
        x_seq.append(cur_x)
    return x_seq


def p_sample(model, x, t, betas, one_minus_alphas_bar_sqrt):
    """
    作用：从x[T]采样t时刻的重构值
    输入：
        model:模型
        x: 采样的随机噪声x[T]
        t: 采样步
        betas: βt
        one_minus_alphas_bar_sqrt: 根号下(1-αt^)
    输出：
        sample: 样本
    """
    t = torch.tensor([t])
    
    coeff = betas[t] / one_minus_alphas_bar_sqrt[t]  # 模型输出的系数：βt/根号下(1-αt^) = 1-αt/根号下(1-αt^)
    
    eps_theta = model(x, t)  # 模型的输出: εθ(xt, t)
        
    # 均值: (1/根号下αt) * (xt - (1-αt/根号下(1-αt^))*εθ(xt, t))
    mean = (1/(1-betas[t]).sqrt())*(x-(coeff*eps_theta))  
    
    z = torch.randn_like(x)  # 对应公式中的 z
    sigma_t = betas[t].sqrt()  # 对应公式中的 σt
    
    sample = mean + sigma_t * z 
    
    return (sample)

### ------------------------------------------------------------------------

### 8、开始训练模型，打印loss及中间重构效果---------------------------------------    
print('Training model...')
batch_size = 128
dataloader = torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True)
num_epoch = 4000
plt.rc('text',color='blue')

model = MLPDiffusion(num_steps)  # 输出维度是2，输入是x和step
optimizer = torch.optim.Adam(model.parameters(), lr=1e-3) 

for t in range(num_epoch):
    for idx, batch_x in enumerate(dataloader):
        
        # 损失计算
        loss = diffusion_loss_fn(model, batch_x, alphas_bar_sqrt, one_minus_alphas_bar_sqrt, num_steps)
        optimizer.zero_grad()  # 梯度清零
        loss.backward()  # 损失回传
        torch.nn.utils.clip_grad_norm_(model.parameters(),1.)  # 梯度裁剪
        optimizer.step()  
        
    if(t % 100 == 0):
        print(loss)
        x_seq = p_sample_loop(model, dataset.shape, num_steps, betas, one_minus_alphas_bar_sqrt)
        
        fig, axs = plt.subplots(1, 10, figsize=(28,3))
        for i in range(1, 11):
            cur_x = x_seq[i*10].detach()
            axs[i-1].scatter(cur_x[:,0],cur_x[:,1],color='red',edgecolor='white');
            axs[i-1].set_axis_off();
            axs[i-1].set_title('$q(\mathbf{x}_{'+str(i*10)+'})$')

### ------------------------------------------------------------------------

### 9、动画演示扩散过程和逆扩散过程----------------------------------------------
# 前向过程
imgs = []
for i in range(100):
    plt.clf()
    q_i = q_x(dataset,torch.tensor([i]))
    plt.scatter(q_i[:,0],q_i[:,1],color='red',edgecolor='white',s=5);
    plt.axis('off');
    
    img_buf = io.BytesIO()
    plt.savefig(img_buf,format='png')
    img = Image.open(img_buf)
    imgs.append(img)

# 逆向过程
reverse = []
for i in range(100):
    plt.clf()
    cur_x = x_seq[i].detach()
    plt.scatter(cur_x[:,0],cur_x[:,1],color='red',edgecolor='white',s=5);
    plt.axis('off')
    
    img_buf = io.BytesIO()
    plt.savefig(img_buf,format='png')
    img = Image.open(img_buf)
    reverse.append(img)

imgs = imgs
imgs[0].save("diffusion_qian.gif", format='GIF', append_images=imgs, save_all=True, duration=100, loop=0)

imgs = reverse
imgs[0].save("diffusion_ni.gif", format='GIF', append_images=imgs, save_all=True, duration=100, loop=0)

### ------------------------------------------------------------------------

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