#

Denoising Diffusion Probabilistic Models (DDPM)

This is a PyTorch implementation/tutorial of the paper Denoising Diffusion Probabilistic Models.

In simple terms, we get an image from data and add noise step by step. Then We train a model to predict that noise at each step and use the model to generate images.

The following definitions and derivations show how this works. For details please refer to the paper.

Forward Process

The forward process adds noise to the data $x_{0} \sim q (x_{0})$ , for $T$ timesteps.

q (x_{t} ∣ x_{t - 1}) = N (x_{t}; 1 - β_{t} x_{t - 1}, β_{t} I) q (x_{1 : T} ∣ x_{0}) = t = 1 \prod T q (x_{t} ∣ x_{t - 1})

where $β_{1}, \dots, β_{T}$ is the variance schedule.

We can sample $x_{t}$ at any timestep $t$ with,

q (x_{t} ∣ x_{0}) = N (x_{t}; \overset{α_{t}}{ˉ} x_{0}, (1 - \overset{α_{t}}{ˉ}) I)

where $α_{t} = 1 - β_{t}$ and $\overset{α_{t}}{ˉ} = \prod_{s = 1}^{t} α_{s}$

Reverse Process

The reverse process removes noise starting at $p (x_{T}) = N (x_{T}; 0, I)$ for $T$ time steps.

p_{θ} (x_{t - 1} ∣ x_{t}) p_{θ} (x_{0 : T}) p_{θ} (x_{0}) = N (x_{t - 1}; μ_{θ} (x_{t}, t), Σ_{θ} (x_{t}, t)) = p_{θ} (x_{T}) t = 1 \prod T p_{θ} (x_{t - 1} ∣ x_{t}) = \int p_{θ} (x_{0 : T}) d x_{1 : T}

$θ$ are the parameters we train.

Loss

We optimize the ELBO (from Jenson's inequality) on the negative log likelihood.

E [- lo g p_{θ} (x_{0})] \leq E_{q} [- lo g \frac{p _{θ} ( x _{0 : T} )}{q ( x _{1 : T} ∣ x _{0} )}] = L

The loss can be rewritten as follows.

L = E_{q} [- lo g \frac{p _{θ} ( x _{0 : T} )}{q ( x _{1 : T} ∣ x _{0} )}] = E_{q} [- lo g p (x_{T}) - t = 1 \sum T lo g \frac{p _{θ} ( x _{t - 1} ∣ x _{t} )}{q ( x _{t} ∣ x _{t - 1} )}] = E_{q} [- lo g \frac{p ( x _{T} )}{q ( x _{T} ∣ x _{0} )} - t = 2 \sum T lo g \frac{p _{θ} ( x _{t - 1} ∣ x _{t} )}{q ( x _{t - 1} ∣ x _{t} , x _{0} )} - l o g p_{θ} (x_{0} ∣ x_{1})] = E_{q} [D_{K L} (q (x_{T} ∣ x_{0}) ∥ p (x_{T})) + t = 2 \sum T D_{K L} (q (x_{t - 1} ∣ x_{t}, x_{0}) ∥ p_{θ} (x_{t - 1} ∣ x_{t})) - l o g p_{θ} (x_{0} ∣ x_{1})]

$D_{K L} (q (x_{T} ∣ x_{0}) ∥ p (x_{T}))$ is constant since we keep $β_{1}, \dots, β_{T}$ constant.

Computing $L_{t - 1} = D_{K L} (q (x_{t - 1} ∣ x_{t}, x_{0}) ∥ p_{θ} (x_{t - 1} ∣ x_{t}))$

The forward process posterior conditioned by $x_{0}$ is,

q (x_{t - 1} ∣ x_{t}, x_{0}) \tilde{μ}_{t} (x_{t}, x_{0}) \tilde{β_{t}} = N (x_{t - 1}; \tilde{μ}_{t} (x_{t}, x_{0}), \tilde{β_{t}} I) = \frac{α ˉ _{t - 1} β _{t}}{1 - α _{t} ˉ} x_{0} + \frac{α _{t} ( 1 - α ˉ _{t - 1} )}{1 - α _{t} ˉ} x_{t} = \frac{1 - α ˉ _{t - 1}}{1 - α _{t} ˉ} β_{t}

The paper sets $Σ_{θ} (x_{t}, t) = σ_{t}^{2} I$ where $σ_{t}^{2}$ is set to constants $β_{t}$ or $\tilde{β_{t}}$ .

Then, $p_{θ} (x_{t - 1} ∣ x_{t}) = N (x_{t - 1}; μ_{θ} (x_{t}, t), σ_{t}^{2} I)$

For given noise $ϵ \sim N (0, I)$ using $q (x_{t} ∣ x_{0})$

x_{t} (x_{0}, ϵ) x_{0} = \overset{α_{t}}{ˉ} x_{0} + 1 - \overset{α_{t}}{ˉ} ϵ = \frac{1}{α _{t} ˉ} (x_{t} (x_{0}, ϵ) - 1 - \overset{α_{t}}{ˉ} ϵ)

This gives,

L_{t - 1} = D_{K L} (q (x_{t - 1} ∣ x_{t}, x_{0}) ∥ p_{θ} (x_{t - 1} ∣ x_{t})) = E_{q} [\frac{1}{2 σ _{t}^{2}} ∥ ∥ \tilde{μ} (x_{t}, x_{0}) - μ_{θ} (x_{t}, t) ∥ ∥^{2}] = E_{x_{0}, ϵ} [\frac{1}{2 σ _{t}^{2}} ∥ ∥ \frac{1}{α _{t}} (x_{t} (x_{0}, ϵ) - \frac{β _{t}}{1 - α _{t} ˉ} ϵ) - μ_{θ} (x_{t} (x_{0}, ϵ), t) ∥ ∥^{2}]

Re-parameterizing with a model to predict noise

μ_{θ} (x_{t}, t) = \tilde{μ} (x_{t}, \frac{1}{α _{t} ˉ} (x_{t} - 1 - \overset{α_{t}}{ˉ} ϵ_{θ} (x_{t}, t))) = \frac{1}{α _{t}} (x_{t} - \frac{β _{t}}{1 - α _{t} ˉ} ϵ_{θ} (x_{t}, t))

where $ϵ_{θ}$ is a learned function that predicts $ϵ$ given $(x_{t}, t)$ .

This gives,

L_{t - 1} = E_{x_{0}, ϵ} [\frac{β _{t} ^{2}}{2 σ _{t}^{2} α _{t} ( 1 - α _{t} ˉ )} ∥ ∥ ϵ - ϵ_{θ} (\overset{α_{t}}{ˉ} x_{0} + 1 - \overset{α_{t}}{ˉ} ϵ, t) ∥ ∥^{2}]

That is, we are training to predict the noise.

Simplified loss

$L_{simple} (θ) = E_{t, x_{0}, ϵ} [∥ ∥ ϵ - ϵ_{θ} (\overset{α_{t}}{ˉ} x_{0} + 1 - \overset{α_{t}}{ˉ} ϵ, t) ∥ ∥^{2}]$

This minimizes $- l o g p_{θ} (x_{0} ∣ x_{1})$ when $t = 1$ and $L_{t - 1}$ for $t > 1$ discarding the weighting in $L_{t - 1}$ . Discarding the weights $\frac{β _{t} ^{2}}{2 σ _{t}^{2} α _{t} ( 1 - α _{t} ˉ )}$ increase the weight given to higher $t$ (which have higher noise levels), therefore increasing the sample quality.

This file implements the loss calculation and a basic sampling method that we use to generate images during training.

Here is the UNet model that gives $ϵ_{θ} (x_{t}, t)$ and training code. This file can generate samples and interpolations from a trained model.

162from typing import Tuple, Optional
163
164import torch
165import torch.nn.functional as F
166import torch.utils.data
167from torch import nn
168
169from labml_nn.diffusion.ddpm.utils import gather

#

Denoise Diffusion

172class DenoiseDiffusion:

#

eps_model is $ϵ_{θ} (x_{t}, t)$ model
n_steps is $t$
device is the device to place constants on

177    def __init__(self, eps_model: nn.Module, n_steps: int, device: torch.device):

#

183        super().__init__()
184        self.eps_model = eps_model

#

Create $β_{1}, \dots, β_{T}$ linearly increasing variance schedule

187        self.beta = torch.linspace(0.0001, 0.02, n_steps).to(device)

#

$α_{t} = 1 - β_{t}$

190        self.alpha = 1. - self.beta

#

$\overset{α_{t}}{ˉ} = \prod_{s = 1}^{t} α_{s}$

192        self.alpha_bar = torch.cumprod(self.alpha, dim=0)

#

$T$

194        self.n_steps = n_steps

#

$σ^{2} = β$

196        self.sigma2 = self.beta

#

Get $q (x_{t} ∣ x_{0})$ distribution

q (x_{t} ∣ x_{0}) = N (x_{t}; \overset{α_{t}}{ˉ} x_{0}, (1 - \overset{α_{t}}{ˉ}) I)

198    def q_xt_x0(self, x0: torch.Tensor, t: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor]:

#

gather $α_{t}$ and compute $\overset{α_{t}}{ˉ} x_{0}$

208        mean = gather(self.alpha_bar, t) ** 0.5 * x0

#

$(1 - \overset{α_{t}}{ˉ}) I$

210        var = 1 - gather(self.alpha_bar, t)

#

212        return mean, var

#

Sample from $q (x_{t} ∣ x_{0})$

q (x_{t} ∣ x_{0}) = N (x_{t}; \overset{α_{t}}{ˉ} x_{0}, (1 - \overset{α_{t}}{ˉ}) I)

214    def q_sample(self, x0: torch.Tensor, t: torch.Tensor, eps: Optional[torch.Tensor] = None):

#

$ϵ \sim N (0, I)$

224        if eps is None:
225            eps = torch.randn_like(x0)

#

get $q (x_{t} ∣ x_{0})$

228        mean, var = self.q_xt_x0(x0, t)

#

Sample from $q (x_{t} ∣ x_{0})$

230        return mean + (var ** 0.5) * eps

#

Sample from $p_{θ} (x_{t - 1} ∣ x_{t})$

p_{θ} (x_{t - 1} ∣ x_{t}) μ_{θ} (x_{t}, t) = N (x_{t - 1}; μ_{θ} (x_{t}, t), σ_{t}^{2} I) = \frac{1}{α _{t}} (x_{t} - \frac{β _{t}}{1 - α _{t} ˉ} ϵ_{θ} (x_{t}, t))

232    def p_sample(self, xt: torch.Tensor, t: torch.Tensor):

#

$ϵ_{θ} (x_{t}, t)$

246        eps_theta = self.eps_model(xt, t)

#

gather $\overset{α_{t}}{ˉ}$

248        alpha_bar = gather(self.alpha_bar, t)

#

$α_{t}$

250        alpha = gather(self.alpha, t)

#

$\frac{β}{1 - α _{t} ˉ}$

252        eps_coef = (1 - alpha) / (1 - alpha_bar) ** .5

#

$\frac{1}{α _{t}} (x_{t} - \frac{β _{t}}{1 - α _{t} ˉ} ϵ_{θ} (x_{t}, t))$

255        mean = 1 / (alpha ** 0.5) * (xt - eps_coef * eps_theta)

#

$σ^{2}$

257        var = gather(self.sigma2, t)

#

$ϵ \sim N (0, I)$

260        eps = torch.randn(xt.shape, device=xt.device)

#

Sample

262        return mean + (var ** .5) * eps

#

Simplified Loss

$L_{simple} (θ) = E_{t, x_{0}, ϵ} [∥ ∥ ϵ - ϵ_{θ} (\overset{α_{t}}{ˉ} x_{0} + 1 - \overset{α_{t}}{ˉ} ϵ, t) ∥ ∥^{2}]$

264    def loss(self, x0: torch.Tensor, noise: Optional[torch.Tensor] = None):

#

Get batch size

273        batch_size = x0.shape[0]

#

Get random $t$ for each sample in the batch

275        t = torch.randint(0, self.n_steps, (batch_size,), device=x0.device, dtype=torch.long)

#

$ϵ \sim N (0, I)$

278        if noise is None:
279            noise = torch.randn_like(x0)

#

Sample $x_{t}$ for $q (x_{t} ∣ x_{0})$

282        xt = self.q_sample(x0, t, eps=noise)

#

Get $ϵ_{θ} (\overset{α_{t}}{ˉ} x_{0} + 1 - \overset{α_{t}}{ˉ} ϵ, t)$

284        eps_theta = self.eps_model(xt, t)

#

MSE loss

287        return F.mse_loss(noise, eps_theta)

Denoising Diffusion Probabilistic Models (DDPM)

Forward Process

Reverse Process

Loss

Computing Lt−1​=DKL​(q(xt−1​∣xt​,x0​)∥pθ​(xt−1​∣xt​))

Simplified loss

Denoise Diffusion

Get q(xt​∣x0​) distribution

Sample from q(xt​∣x0​)

Sample from pθ​(xt−1​∣xt​)

Simplified Loss

Computing $L_{t - 1} = D_{K L} (q (x_{t - 1} ∣ x_{t}, x_{0}) ∥ p_{θ} (x_{t - 1} ∣ x_{t}))$

Get $q (x_{t} ∣ x_{0})$ distribution

Sample from $q (x_{t} ∣ x_{0})$

Sample from $p_{θ} (x_{t - 1} ∣ x_{t})$