扩散模型代码详细解读

代码地址：denoising-diffusion-pytorch/denoising_diffusion_pytorch.py at main · lucidrains/denoising-diffusion-pytorch (github.com)

前向过程和后向过程的代码都在GaussianDiffusion​这个类中。​

有问题可以一起讨论！

常见问题解决

Why self-conditioning? · Issue #94 · lucidrains/denoising-diffusion-pytorch (github.com)

"pred_x0" preforms better than "pred_noise" · Issue #58 · lucidrains/denoising-diffusion-pytorch (github.com)

What is objective=pred_x0 and how do you use it? · Issue #34 · lucidrains/denoising-diffusion-pytorch (github.com)

Conditional generation · Issue #7 · lucidrains/denoising-diffusion-pytorch (github.com)

Questions About DDPM · Issue #10 · lucidrains/denoising-diffusion-pytorch (github.com)
The difference between pred_x0, pred_v, pred_noise three objectives · Issue #153 · lucidrains/denoising-diffusion-pytorch (github.com)

前向训练过程

p_losses

首先是p_losses函数，这个是训练过程的主体部分。

def p_losses(self, x_start, t, noise = None):
        b, c, h, w = x_start.shape
	# 首先随机生成噪声
        noise = default(noise, lambda: torch.randn_like(x_start))
        # noise sample
	# 噪声采样,注意这个是一次性完成的
        x = self.q_sample(x_start = x_start, t = t, noise = noise)
        # if doing self-conditioning, 50% of the time, predict x_start from current set of times
        # and condition with unet with that
        # this technique will slow down training by 25%, but seems to lower FID significantly
	# 判断是否进行self-condition,就是利用前面步骤预测出的x0来辅助当前的预测
        x_self_cond = None
        if self.self_condition and random() < 0.5:
            with torch.no_grad():
                x_self_cond = self.model_predictions(x, t).pred_x_start
                x_self_cond.detach_()
        # predict and take gradient step
	# 将采样的x和self condition的x一起输入到model当中,这个model是UNet结构
        model_out = self.model(x, t, x_self_cond)
	# 模型预测的目标，分为三种
        if self.objective == 'pred_noise':
            target = noise
        elif self.objective == 'pred_x0':
            target = x_start
        elif self.objective == 'pred_v':
            v = self.predict_v(x_start, t, noise)
            target = v
        else:
            raise ValueError(f'unknown objective {self.objective}')
	# 计算损失
        loss = self.loss_fn(model_out, target, reduction = 'none')
        loss = reduce(loss, 'b ... -> b (...)', 'mean')
        loss = loss * extract(self.p2_loss_weight, t, loss.shape)
        return loss.mean()

对其中的extract函数进行分析，extract函数实现如下：

def extract(a, t, x_shape):
    # Extract some coefficients at specified timesteps,
    # then reshape to [batch_size, 1, 1, 1, 1, ...] for broadcasting purposes.
    b, *_ = t.shape
    # 使用了gather函数
    out = a.gather(-1, t)
    return out.reshape(b, *((1,) * (len(x_shape) - 1)))

q_sample

然后介绍p_losses函数中使用的其他函数，第一个是q_sample函数，它的作用是加上噪声，对应论文的公式：

其中self.sqrt_alphas_cumprod​和self.sqrt_one_minus_alphas_cumprod​分别是alpha的累乘值和1-alpha的累乘值，x_start相当于x0，noise相当于z。

def q_sample(self, x_start, t, noise=None):
        noise = default(noise, lambda: torch.randn_like(x_start))
        return (
            extract(self.sqrt_alphas_cumprod, t, x_start.shape) * x_start +
            extract(self.sqrt_one_minus_alphas_cumprod, t, x_start.shape) * noise
        )

model_predictions

然后是model_predictions函数，它的实现如下：

def model_predictions(self, x, t, x_self_cond = None, clip_x_start = False):
	# 输入到UNet结构中获得输出
        model_output = self.model(x, t, x_self_cond)
        maybe_clip = partial(torch.clamp, min = -1., max = 1.) if clip_x_start else identity
	# 暂不明确它的作用
        if self.objective == 'pred_noise':
            pred_noise = model_output
            x_start = self.predict_start_from_noise(x, t, pred_noise)
            x_start = maybe_clip(x_start)
        elif self.objective == 'pred_x0':
            x_start = model_output
            x_start = maybe_clip(x_start)
            pred_noise = self.predict_noise_from_start(x, t, x_start)
        elif self.objective == 'pred_v':
            v = model_output
            x_start = self.predict_start_from_v(x, t, v)
            x_start = maybe_clip(x_start)
            pred_noise = self.predict_noise_from_start(x, t, x_start)
	# 返回得到的噪声和
        return ModelPrediction(pred_noise, x_start)

几种objective

model_predictions函数中有一个难点，就是其中的self.objective，它有三种形式：

• pred_noise：这个相当于是预测噪声，此时UNet模型的输出是噪声
• pred_x0：这个相当于是预测最开始的x，此时UNet模型的输出是去噪的图像
• pred_v：这个相当于是预测速度v，它在这篇文章中提出。然后根据速度求出最开始的x，最后预测出噪声。

如图所示：​

在上面的三种objective中，还涉及到了几种预测方法的实现，具体如下：

（1）predict_start_from_noise：这个函数的作用是根据噪声noise预测最开始的x，也就是去噪的图像。

其中self.sqrt_recip_alphas_cumprod​和self.sqrt_recipm1_alphas_cumprod​来自
公式，它们分别为：
和
。

公式来源文章：DDPM

def predict_start_from_noise(self, x_t, t, noise):
    return (
        extract(self.sqrt_recip_alphas_cumprod, t, x_t.shape) * x_t -
        extract(self.sqrt_recipm1_alphas_cumprod, t, x_t.shape) * noise
    )

它对应论文中的公式如下：

（2）predict_noise_from_start：这个函数的作用是根据图像预测噪声，也就是加噪声。

def predict_noise_from_start(self, x_t, t, x0):
    return (
        (extract(self.sqrt_recip_alphas_cumprod, t, x_t.shape) * x_t - x0) / \
        extract(self.sqrt_recipm1_alphas_cumprod, t, x_t.shape)
    )

它对应论文中的公式如下：

需要注意它是反推过来的，过程如下：

（3）predict_v：预测速度v

 def predict_v(self, x_start, t, noise):
     return (
         extract(self.sqrt_alphas_cumprod, t, x_start.shape) * noise -
         extract(self.sqrt_one_minus_alphas_cumprod, t, x_start.shape) * x_start
     )

它对应论文中的公式：

（4）predict_start_from_v：根据速度v预测最初的x，也就是图像

def predict_start_from_v(self, x_t, t, v):
    return (
        extract(self.sqrt_alphas_cumprod, t, x_t.shape) * x_t -
        extract(self.sqrt_one_minus_alphas_cumprod, t, x_t.shape) * v
    )

它对应论文中的公式如下：其中zt相当于xt。

后向采样过程

sample函数

@torch.no_grad()
def sample(self, batch_size = 16, return_all_timesteps = False):
    image_size, channels = self.image_size, self.channels
    # 采样的函数
    sample_fn = self.p_sample_loop if not self.is_ddim_sampling else self.ddim_sample
    # 调用该函数
    return sample_fn((batch_size, channels, image_size, image_size), return_all_timesteps = return_all_timesteps)

该函数的作用是获取采样的函数然后进行调用，采样函数分成两种：p_sample_loop和ddim_sample。

p_sample_loop函数

 @torch.no_grad()
 def p_sample_loop(self, shape, return_all_timesteps = False):
     batch, device = shape[0], self.betas.device
     # 随机生成噪声图像
     img = torch.randn(shape, device = device)
     imgs = [img]
     x_start = None
     # 遍历所有的t
     for t in tqdm(reversed(range(0, self.num_timesteps)), desc = 'sampling loop time step', total = self.num_timesteps):
         # 判断是否使用self-condition
	 self_cond = x_start if self.self_condition else None
         # 进行采样,得到去噪的图像
         img, x_start = self.p_sample(img, t, self_cond)
         imgs.append(img)
     # 判断是否返回每个步骤的img还是最后一步的img
     ret = img if not return_all_timesteps else torch.stack(imgs, dim = 1)
     # 归一化
     ret = self.unnormalize(ret)
     return ret

其中涉及到归一化函数self.unnormalize​，含有两种

# normalization functions
def normalize_to_neg_one_to_one(img):
    return img * 2 - 1
def unnormalize_to_zero_to_one(t):
    return (t + 1) * 0.5

p_sample函数

@torch.no_grad()
def p_sample(self, x, t: int, x_self_cond = None):
    b, *_, device = *x.shape, x.device
    batched_times = torch.full((b,), t, device = x.device, dtype = torch.long)
    # 获得平均值,方差和x0
    model_mean, _, model_log_variance, x_start = self.p_mean_variance(x = x, t = batched_times, x_self_cond = x_self_cond, clip_denoised = True)
    # 随机生成一个噪声	  
    noise = torch.randn_like(x) if t > 0 else 0. # no noise if t == 0
    # 得到预测的图像,img = 平均值 + exp(0.5 * 方差) * noise
    pred_img = model_mean + (0.5 * model_log_variance).exp() * noise
    return pred_img, x_start

p_mean_variance函数

其中含有p_mean_variance​函数，代码实现如下：

def p_mean_variance(self, x, t, x_self_cond = None, clip_denoised = True):
    # 输入到UNet网络进行预测
    preds = self.model_predictions(x, t, x_self_cond)
    # 得到预测的x0
    x_start = preds.pred_x_start
    # 压缩x0中值的范围至[-1,1]
    if clip_denoised:
        x_start.clamp_(-1., 1.)
    # 得到x0后根据xt和t得到分布的平均值和方差
    model_mean, posterior_variance, posterior_log_variance = self.q_posterior(x_start = x_start, x_t = x, t = t)
    return model_mean, posterior_variance, posterior_log_variance, x_start

q_posterior函数

其中q_posterior​函数的实现如下：

def q_posterior(self, x_start, x_t, t):
    # 计算平均值
    posterior_mean = (
        extract(self.posterior_mean_coef1, t, x_t.shape) * x_start +
        extract(self.posterior_mean_coef2, t, x_t.shape) * x_t
    )
    # 计算方差
    posterior_variance = extract(self.posterior_variance, t, x_t.shape)
    # 获得一个压缩范围的方差,且取对数
    posterior_log_variance_clipped = extract(self.posterior_log_variance_clipped, t, x_t.shape)
    return posterior_mean, posterior_variance, posterior_log_variance_clipped

平均值和方差对应的公式如下：

其中self.posterior_mean_coef1​对应的是x0前面的系数，self.posterior_mean_coef2​对应的是xt前面的系数。

​self.posterior_variance​对应的beta那部分的系数。

ddim_sample函数

@torch.no_grad()
def ddim_sample(self, shape, return_all_timesteps = False):
    batch, device, total_timesteps, sampling_timesteps, eta, objective = shape[0], self.betas.device, self.num_timesteps, self.sampling_timesteps, self.ddim_sampling_eta, self.objective
    times = torch.linspace(-1, total_timesteps - 1, steps = sampling_timesteps + 1)   # [-1, 0, 1, 2, ..., T-1] when sampling_timesteps == total_timesteps
    times = list(reversed(times.int().tolist()))
    time_pairs = list(zip(times[:-1], times[1:])) # [(T-1, T-2), (T-2, T-3), ..., (1, 0), (0, -1)]
    img = torch.randn(shape, device = device)
    imgs = [img]
    x_start = None
    for time, time_next in tqdm(time_pairs, desc = 'sampling loop time step'):
        time_cond = torch.full((batch,), time, device = device, dtype = torch.long)
        self_cond = x_start if self.self_condition else None
        pred_noise, x_start, *_ = self.model_predictions(img, time_cond, self_cond, clip_x_start = True)
        imgs.append(img)
        if time_next < 0:
            img = x_start
            continue
        alpha = self.alphas_cumprod[time]
        alpha_next = self.alphas_cumprod[time_next]
        sigma = eta * ((1 - alpha / alpha_next) * (1 - alpha_next) / (1 - alpha)).sqrt()
        c = (1 - alpha_next - sigma ** 2).sqrt()
        noise = torch.randn_like(img)
        img = x_start * alpha_next.sqrt() + \
              c * pred_noise + \
              sigma * noise
    ret = img if not return_all_timesteps else torch.stack(imgs, dim = 1)
    ret = self.unnormalize(ret)
    return ret

上面部分依据的公式为：（文章）

训练的模型(UNet)

后续会继续更新！
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